Constitutive Relations of Stress and Strain in Stochastic Finite Element Method

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1 American Journal of Computational and Applied Mathematics 15, 5(6): DOI: 1.593/j.ajcam Constitutive Relations of Stress and Strain in Stochastic Finite Element Method Drakos Stefanos International Centre for Computational Engineering, Rhodes, Greece Abstract The analysis and design in structural and geotechnical engineering problems requires the calculation of stress and strain which is generally a difficult task because of the uncertainty and spatial variability of the properties of soil materials. This paper presents a procedure of conducting Stochastic Finite Element Analysis using Polynomial Chaos in order to propagate the uncertainties of input to constitutive relation of stress and strain. The problem is dominated by highly non linearity. Among other methods the procedure leads to an efficient computational cost for real practical problems. This is achieved by polynomial chaos expansion displacement field, stress and strain also. An example of a plane-strain strip load on a semi-infinite elastic foundation is presented and the results of settlement are compared to those obtained from the closed form solution method. A close matching of the two is observed. The constitutive relation of stress and strain is presented as result of the Polynomial Chaos expansion and Monte Carlo method. A close matching of the two method is observed also. Keywords Stochastic Finite Element Method, Constitutive Relations, Polynomial Chaos, Quantification of Uncertainty 1. Introduction The analysis and design in structural and geotechnical engineering problems requires the calculation of stress and strain which is generally a difficult task because of the uncertainty and spatial variability of the materials s properties. Various forms of uncertainties arise which depend on the nature of geological formation or construction method, the site investigation, the type and the accuracy of design calculations etc. In recent years there has been considerable interest amongst engineers and researchers in the issues related to quantification of uncertainty as it affects safety, design as well as the cost of projects. A number of approaches using statistical concepts have been proposed in engineering in the past 5 years or so. These include the Stochastic Finite Element Method (SFEM) [1-3], and the Random Finite Element Method (RFEM) [4-8]. The RFEM involves generating a random field of soil or structure properties with controlled mean, standard deviation and spatial correlation length, which is then mapped onto a finite element mesh. However the number of works on the stochastic stress and strain calculation and their statistical moments are limited. An essential paper on the field is presented by Ghosh & Farhat [9] where the constitutive relation of stress and strain calculated by different approaches. * Corresponding author: stefanos.drakos@gmail.com (Drakos Stefanos) Published online at Copyright 15 Scientific & Academic Publishing. All Rights Reserved In the past SFEM has been developed using different expansions of stochastic variables. In this paper we present SFEM [11-13] using the method of Generalized Polynomial Chaos (GPC) [14]. To descretise the stochastic process of material the Karhunen-Loeve Expansion was used and it is presented. The constitutive relation of stress and strain calculated using the Generalized Polynomial Chaos and verified against Monte Carlo simulation which is treated as the exact solution based on a series of computational experiment. A numerical example of foundation settlement given in the last part of the paper and the results of settlement compared with those arises from closed form solution. The two methods of stress and strains constitutive relation compared also and the results are presented.. Problem Description and Model Formulation Let us consider a general boundary value problem of computation of probable deformation of a body of arbitrary shape having randomly varying material properties caused by the application of a randomly varying load as shown in Fig. 1. According to the elasticity theory a boundary value problem can be described as follow: σσ iiii,jj (xx, ωω) = ff(xx, ωω) iiii DD ΩΩ σσ iiii (xx, ωω) = CC iiiiiiii (xx, ωω)εε kkkk (xx, ωω) iiii DD ΩΩ (1) uu(xx, ωω) = gg DD iiii BB DD σσ iiii (xx, ωω)nn jj = gg NN iiii BB DD

2 American Journal of Computational and Applied Mathematics 15, 5(6): And in the weak form as: aa(uu, vv) = ll(vv) () aa(uu, vv) = εε ΤΤ (vv)cc(xx, ωω)εε(uu) dddd (3) DD ll(vv) = ff(xx, ωω) vvvvvv + gg NN vvvvvv DD BB NN εε ΤΤ (vv)cc(xx, ωω)εε(uu) dddd gg BB DD DD (4) Figure 1. Body of arbitrary shape In order to model the problem assuming the sample space (Ω, F, P) where F is the σ-algebra and is considered to contain all the information that is available, P is the probability measure and the spatial domain of the soil or the structure is DD R. The Elasticity modulus {EE( xx, ωω) : DD ΩΩ} and the external load {ff( xx, ωω) : DD ΩΩ} considered as second order random fields and their functions are determined EE, ff: DD ΩΩ R VV = LL ΩΩ, LL (DD) and characterized by specific distribution where in our case as lognormal. Considering as μμ kk, σσ kk and vv kk = σσ kk the μμ kk mean value the standard deviation and the coefficient of variation of Elasticity modulus, the lognormal distribution is given [8]: EE = exp(μμ llllll + σσ llllll ZZ(ωω)) (5) Where the mean values and the variance of the distribution are equal to: σσ llllll = ln(1 + vv kk ) μμ llllll = ln(μμ kk ) 1 σσ (6) llllll And ωω ΩΩ, ΖΖ ΝΝ(,1) To separate the deterministic part from the stochastic part of the formulation the Karhunen-Loeve expansion has been used. It is considered as the most efficient method for the discretization of a random field, requiring the smallest number of random variables to represent the field within a given level of accuracy. Based on that the stochastic process of Young modulus over the spatial domain with a known mean value E (x) and covariance matrix CCCCCC(x 1, x ) assuming lognormal distribution is given by: EE xx, ξξ(ωω) = exp(εε (xx) + κκ λλ κκ ww κκ (xx)ξξ κκ (ωω)) (7) In practice, calculations were carried out over a finite number of summations (for example 1-5) so the approximate stochastic representation is given by the trancuated part of expansion: EE xx, ξξ(ωω) = exp(εε (xx) + ΚΚ κκ=1 λλ κκ ww κκ ξξ κκ (ωω)) (8) λλ κκ : are the eingenvalues of the covariance function ww κκ (xx): are the eingenfunctions of the covariance function CCCCCC(x 1, x ) x D and ω Ω and ξξ = [ξξ 1, ξξ 1,, ξξ ΜΜ ]: ΩΩ R ΜΜ = 1 1 ΜΜ The pairs of eingenvalues and eingenfunctions arised by the equation: CC(xx DD 1, xx )φφ κκ (xx ) = λλ κκ ww κκ (xx 1 ) (9) Using the Karhunen-Loeve expansion the stochastic elasticity tensor is given by: CC iiiiiiii (xx, yy) = EE(xx)CC iiiiiiii (xx), ii, jj, kk, ll = 1,,3 (1) CC iiiiiiii (xx): is expressed in terms of (deterministic) Poisson's ratio as CC iiiiiiii (xx) = vv δδ (1+vv) iiii δδ kkkk + 1 (δδ (1+vv) iiiiδδ jjjj + δδ llll δδ jjjj ) (11) To compute the statistical moments of the calculations output we perform a change of variable yy kk : = ξξ kk (ωω) and yy = [yy 1, yy, yy MM ] [1]. If the random variables are independent and ρρ ii denote the density of ξξ ii then the joint density is given by: ρρ(yy) = ρρ 1 (yy 1 )ρρ (yy ) ρρ MM (yy MM ) (1) Drakos & Pande [1, 13] developed a new algorithm of Stochastic Finite method using the Generalized Polynomial Chaos (appendix A) and proved that the problem formulation has the final form: mm KK mm = qq ff + tt gggg mm KK BBBB gg dd (13) mm = ρρ(yy)ψψ κκ (yy)ψψ pp (yy)exp( ΚΚ λλ κκ φφ κκ yy κκ KK mm = BB TT exp(εε (xx))cc (xx)bbbbbbbbbb DD tt gggg = κκ=1 ) qq = ρρ(yy)ψψ pp (yy)ψψ 1 (yy) dddd KK BBBB = BBBB ff = φφ ΤΤ ff(xx)dddd DD φφ ΤΤ gg NN dddd BBBB B is strain displacement matrix. φφ is the hat function. ψψ is the Polynomial Chaos dddd BB TT exp(εε (xx))cc (xx)bbbbbbbbbb (14) (15)

3 166 Drakos Stefanos: Constitutive Relations of Stress and Strain in Stochastic Finite Element Method 3. Constitutive Relations of Stress and Strain The calculation of the constitutive relation of stress and strain in the case of stochastic problems is quite complicated especially when the invariant of them are needed where the equations become highly nonlinear. In [9] several numerical integration schemes to evaluate the statistical moments of strains and stresses in a random system is presented. In the current work the propagation of the input uncertainty to the stress and strain relation is modelled by the polynomial Chaos expansion and verified against Monte Carlo simulation which is treated as the exact solution of the problems. The computational implementation of the Monte Carlo Method leads to the random field generation and the requested function u k (x) gets a new value for each realization. At the end of all simulations the statistical εε iiii (xx, yy) = 1 uu (kk) ii (xx)ψψ κκ (yy) moment are calculated. The expected value and the variance are given by: EE uu(xx) = 1 KK uu KK kk(xx) VVVVVV(uu(xx)) = 1 (uu (16) KK KK 1 kk(xx) EE(uu(xx)) In an elastostatic problem of homogeneous isotropic body one of the field equations that must be satisfied at all interior points of the body is the Strain-Displacement relations:,jj εε iiii (xx, yy) = 1 uu(xx, yy) ii,jj + uu(xx, yy) jj,ii ii, jj = 1,,3 (17) Using the displacement polynomial chaos expansion uu(xx, yy) = uu kk (xx)ψψ κκ (yy) (18) Q and ψ are given in appendix A The equation 17 leads to: + uu (kk) jj (xx)ψψ κκ (yy) εε iiii (xx, yy) = 1 uu (kk) ii,jj (xx)ψψ κκ (yy) + uu (kk) jj,ii (xx)ψψ κκ (yy) = εε (kk) iiii ψψ κκ (yy) 3.1. Expected Value of Strains According to the polynomial chaos expansion of strains the expected value can be evaluated by the following: EE[εε iiii (xx, yy)] = EE 1 uu (kk) ii,jj (xx)ψψ κκ (yy) + uu (kk) jj,ii (xx)ψψ κκ (yy) = 1 () uuii,jj (xx) EE[ψψ (yy)] + uu (kk) ii,jj (xx)ee[ψψ κκ (yy)] + uu () jj,ii (xx) EE[ψψ (yy)] + uu (kk) jj,ii (xx)ee[ψψ κκ (yy)] 1 1 EE εε iiii (xx, yy) = 1 uu () (xx) ii,jj + uu () () jj,ii (xx) = εε iiii () 3.. Variance of Strains Respected to the expected value evaluation and the Chaos Polynomial characteristics the variance of the strain tensor can be calculated as: σσ = EE εε iiii (xx, yy) (EE εε iiii (xx, yy ) PP = [εε (κκ) iiii (xx)] ρρ(yy)ψψ kk (yy)ddyy [εε () iiii (xx)] = [εε () iiii (xx)] ρρ(yy)ψψ (yy)ddyy + [εε (κκ) iiii (xx)] ρρ(yy)ψψ kk (yy)ddyy 1,ii [εε () iiii (xx)] σσ = (κκ) (xx)] < ψψ kk (yy) > (1) [εε iiii (19)

4 American Journal of Computational and Applied Mathematics 15, 5(6): Expected Value of Stress Tensor The constitutive relation of stress and strain given by the well known equation of Hooke s law equation: Using the polynomial chaos expansion of strains we get: σσ iiii (xx, yy) = EE(xx, yy)cc iiiiiiii εε mmmm () σσ iiii (xx, yy) = EE(xx, yy)cc iiiiiiii εε (kk) mmmm ψψ κκ (yy) (3) According to the elasticity modulus distribution and the strain tensor Chaos Polynomial expansion the expected value of stress tensor takes the following form: < σσ iiii (xx, yy) >=< EE(xx, yy)cc iiiiiiii εε mmmm > =< eeee pp(mmmm llllll + σσ llllll yy) CC iiiiiiii = ee mmmm llllll CC iiiiiiii εε (kk) mmmm ψψ kk (yy) > εε (kk) mmmm ρρ(yy)ee σσ llllll yy ψψ kk (yy)ddyy ) < σσ iiii (xx, yy) >= ee mmmm llllll CC iiiiiiii εε (kk) mmmm < ee σσ llllll yy ψψ kk (yy) > (4) 3.4. Variance of Stress Tensor Having calculated the expected value of stress tensor and knowing its stochastic equation of by the Hook s law constitutive relation, the variance of stress tensor can be calculated as: σσ = EE σσ iiii (xx, yy) (EE σσ iiii (xx, yy ) =< EE(xx, yy)cc iiiiiiii PP εε (kk) mmmm ψψ κκ (yy) > [ee mmmm llllll CC iiiiiiii Given the elasticity modulus distribution the variance of stress tensor is equal to: PP εε (kk) mmmm < ee σσ llllll yy ψψ kk (yy) > σσ =< exp(mmmm llllll + σσ llllll yy) CC iiiiiiii εε (kk) mmmm ψψ κκ (yy) > ee mmmm llllll (kk) CC iiiiiiii εε mmmm ρρ(yy)ee σσ llllll yy ψψ kk (yy)ddyy (5) ] ) 3.5. Pore Pressure Calculation A major issue in a wide range of geotechnical and geomechanics problem is the estimation of the excess pore pressure in the ground. Using the Chaos Polynomial expansion the statistical moments of pore pressure can be evaluated as following. The pore pressure is given by the following equation: pp = KK aa εε vv = KK aa (εε 11 + εε + εε 33 ) (5) εε vv : is the volumetric strain KK aa : is shown [15] as Assuming a minimum value of KK aa = EE (1 vv) KK aa EE (1 vv) Expected Value of Pore Pressure The expected value of pore pressure is the result of the summation of the expected value of the strain s Chaos expansion multiplied by the stochastic fluid modulus: EE[pp] = EE[KK aa (εε 11 + εε + εε 33 )] (7) Replacing the longnormal distribution o elasticity modulus we get: EE[pp] = EE eeeeee(mmmm llllll + σσ llllll yy) (1 vv) εε (kk) 11 ψψ κκ (yy) + εε (kk) ψψ κκ (yy) + εε (kk) 33 ψψ κκ (yy) PP (6)

5 168 Drakos Stefanos: Constitutive Relations of Stress and Strain in Stochastic Finite Element Method And finally = eeeeee(mmmm llllll ) (1 vv) EE ee σσ llllll yy εε (kk) 11 ψψ κκ (yy) + εε (kk) ψψ κκ (yy) + εε (kk) 33 ψψ κκ (yy) EE[pp] = eeeeee (mmmm llllll ) (1 vv) [εε (kk) ] vv ee σσ llllll yy ψψ κκ (yy)dddd (8) Variance of Pore Pressure Using the outcome of the pore pressure expected value its variance is given by: VVVVVV[pp] = EE[pp ] (EE[pp]) EE[pp ] = EE eeeeee (mmmm llllll ) ee σσyy (1 vv) (EE[pp]) = eeeeee (mmmm llllll ) (1 vv) [εε (kk) vv ] εε (kk) vv ψψ κκ (yy) ee σσ llllll yy ψψ κκ (yy)dddd 3.6. Invariants of Stress Tensor It is well known that to solve engineering problem the model itself should be defined independent of the coordinate system attached to the material. Thus, it is necessary to define the model in terms of stress invariants which are, by definition, independent of the coordinate system selected. The physical content of a stress tensor is reflected exclusively in the stress invariants (II 1, II, II 3 ). (9) II 1 = 1 3 σσ kkkk (3) II = 1 (σσ kkkk ) σσ iiii σσ iiii (31) ΙΙ 3 = εε iiiiii σσ ii1 σσ jj σσ kk3 (3) Solving a stochastic problem the statistical moments of stress invariants are needed. In the following paragraphs the calculation of the expected and variance value of each of invariants are presented Expected value of II 11 According to the linearity of the excepted value of II 1 can be calculated: ΕΕ[II 1 ] = ΕΕ[ 1 3 σσ qqqq ] and EE σσ qqqq = ΕΕ[II 1 ] = 1 3 EE σσ qqqq (kk) CC qqqqqqqq εε mmmm ρρ(yy)ee σσ llllll yy ψψ κκ (yy)dddd (33) CC = exp(mmmm llllll ) CC (34) Variance of II 11 Knowing the mean value of II 1 the variance is: VVVVVV(II 1 ) = EE[II 1 ] (EE[II 1 ]) But due to linearity: VVVVVV(II 1 ) = 1 9 (VVVVVV σσ qqqq ) = EE ee σσ llllll yy CC qqqqqqqq εε (kk) mmmm ψψ κκ (yy) EE[II 1 ] = 1 EE σσ 9 qqqq (EE[II 1 ]) = 1 EE[σσ 9 qqqq ] = (kk) CC qqqqqqqq εε mmmm (35) ρρ(yy)ee σσσσ ψψ κκ (yy)dddd

6 American Journal of Computational and Applied Mathematics 15, 5(6): Expected Value of II As shown before: EE[II ] = EE[ 1 σσ qqqq σσ iiii σσ iiii ] This gives EE[II ] = 1 EE eeσσ llllll yy CC qqqqqqqq εε (kk) mmmm ψψ κκ (yy) EE ee σσ llllll yy CC iiiiiiii εε (kk) mmmm ψψ κκ (yy) ee σσ llllll yy CC iiiiiiii εε (kk) mmmm ψψ κκ (yy) (36) Variance of II The variance of II due to the stress product on its equation become highly nonlinear. Thus VVVVVV(II ) = EE[II ] (EE[II ]) EE[II ] = 1 EE ee σσ llllll yy CC qqqqqqqq εε (kk) mmmm ψψ κκ (yy) 4 EE ee σσ llllll yy CC iiiiiiii εε (kk) mmmm ψψ κκ (yy) ee σσ llllll yy CC iiiiiiii εε (kk) mmmm ψψ κκ (yy) (EE[II ]) = 1 EE ee σσ llllll yy CC qqqqqqqq εε (kk) mmmm ψψ κκ (yy) EE ee σσ llllll yy CC iiiiiiii εε (kk) mmmm ψψ κκ (yy) ee σσ llllll yy CC iijjjjjj εε (kk) mmmm ψψ κκ (yy) (37) Expected Value of ΙΙ 33 The high non linearity presented also in the statistical moments of II 3. EE[ΙΙ 3 ] = EE[ee iiiiii σσ ii1 σσ jj σσ kk3 ] This leads to: EE[ΙΙ 3 ] = ee iiiiii EE ee 3σσ llllll yy CC ii1mmmm εε (kk) mmmm ψψ κκ (yy) CC jj mmmm εε (kk) mmmm ψψ κκ (yy) CC kk3mmmm εε (kk) mmmm ψψ κκ (yy) (38) Variance of ΙΙ 33 Similarly as documented above the variance of II 3 is equal to: VVVVVV(II 3 ) = EE[II 3 ] (EE[II 3 ]) EE ee iiiiii ee 3σσ llllll yy CC ii1mmmm εε (kk) mmmm ψψ κκ (yy) CC jj mmmm εε (kk) mmmm ψψ κκ (yy) CC kk3mmmm εε (kk) mmmm ψψ κκ (yy) EE ee iiiiii ee 3σσ llllll yy CC ii1mmmm εε (kk) mmmm ψψ κκ (yy) CC jj mmmm εε (kk) mmmm ψψ κκ (yy) CC kk3mmmm εε (kk) mmmm ψψ κκ (yy) (39) 4. Numerical Example A shallow foundation problem for various values of variation s coefficient vv ee is solved taken to account the randomness of the ground. To estimate the statistical moments of the soil deformation the numerical algorithm of SFEM using the Generalized Polynomial Chaos as described in the previous paragraphs is applied and the results are compared to those obtained by the closed form solution. To avoid the negative values of the elastic modulus assumed to have lognormal. It is known that the settlement beneath a foundation with uniform but random elastic modulus is given by the equation [8]: ss = ss dddddd μμ EE (4) EE ss dddddd is the deterministic value of settlement with EE = μμ ΕΕ everywhere. Assuming lognormal distribution for the settlements the mean values is equal to μμ llll (ss) = llll(ss dddddd ) + 1 σσ llll (EE) (41) The geometry of the finite elements used for the simulation of the problem presented in Fig.. The input data of the problem is the random field modulus with a constant

7 17 Drakos Stefanos: Constitutive Relations of Stress and Strain in Stochastic Finite Element Method average value equal to 1 Mpa and a fixed Poisson ratio equal to.5. Calculations have been made for ten different coefficients vv ee = σσ ΕΕ of the elastic modulus with a minimum μμ ΕΕ value of.1 and then with step.1 to a maximum value equal to 1. The randomness of Elasticity modulus in Fig. 3 is shown. For SFEM one dimensional Hermite GPC with order 5 [14] were used. In the Fig. B1 the results of SFEM method comparatively with the closed form solution are shown and they present great accuracy. It is observed that for of v e =.5 the error is equal to.8%. In the figures B-B13 the strains and stress components, the pore pressure and the stress tensor invariants are presented as resulted by the Chaos Polynomial expansion and compared with those raised by the Monte Carlo Method. Simulations of 1-5 samples were carried and the convergence of the outcomes decreases as the number of Monte Carlo simulations increases. presented. An algorithm of Stochastic Finite Element using Polynomial Chaos has been developed. An analysis of settlement of a plane strain strip load on an elastic foundation has been given as an example of the proposed approach. It is shown that the results of SFEM using polynomial chaos compare well with those obtained from closed form solution. The stress and strain constitutive relation the pore pressure and the stress invariants are modeled by the polynomial Chaos expansion and verified against Monte Carlo simulation which is treated as the exact solution of the problems. The main advantage in using the proposed methodology is that a large number of realizations which have to be made for (Random Finite Element Method) avoided, thus making the procedure viable for realistic practical problems. Appendix A Figure. Finite element mesh Figure 3. Modulus of Random Elasticity of two different realizations 5. Conclusions To propagate the uncertainties of input parameters to constitutive relations of strain and stress where arises due to spatial variability of mechanical parameters of soil/rock in geotechnical and geomechanics problems, a procedure of conducting a Stochastic Finite Element Analysis has been Galerkin approximation and Generalized Polynomial of chaos In order to solve the problem 1 we have to create the new space LL pp (, ΗΗ 1 (DD)). For that reason the subspace SS kk LL pp () is considered as [1]. SS kk = ssssssss{ψψ 1, ψψ,, ψψ κ } (A.1) Using the dyadic product of the space VV h, SS kk the space LL pp (, ΗΗ 1 (DD)) created. Thus VV hkk = VV h VV kk = ssssssss{φφ ii ψψ jj, i = 1 NN, j = 1, } (A.) The space VV hkk has dimension QN and regards the test function v. In the case where exists NN BB finite element supported by boundaries condition then the subspace of solution belongs is: WW hkk = VV hkk ssssssss{φφ NN+1, φφ NN+,, φφ ΝΝ+NNNN } (A.3) kk Assuming that the SS ii represents a space of univariate orthonormal polynomial of variable yy ii ιι R with order k or lower and: SS kk ii = ssssssss PP ii aaii (yy ii ), aa ii =,1,, kk, ii = 1, MM (A.4) kk The tensor product of the M SS ii subspace results the space of the Generalized Polynomial Chaos: SS kk = SS 1 SS SS MM (A.5) And using (A4) SS kk MM = ssssssss PP aa ii ii (yy ii ): aa ii =,1, kk, ii = 1 MM, ii=1 (A.6) aa kk Where aa = And MM ii=1 aa ii = dim(ss kk ) = (MM+kk)! (A.7) MM!kk! Xiu & Karniadakis [14] show the application of the method for different kind of orthonormal polynomials and in the current paper the Hermite polynomial was used with the following characteristics:

8 American Journal of Computational and Applied Mathematics 15, 5(6): PP = 1, < PP ii >=, ii > < PP mm PP nn >= PP mm (yy)pp nn (yy)ρρ(yy)ddyy = γγ nn δδ mmmm (A.8) γγ nn =< PP nn > : are the normalization factors, δδ mmmm is the Kronecker delta ρρ(yy) = 1 yy : is the density function and ππ ee PP nn = ( 1) nn ee yy ddnn ddyy nn ee yy (A.9) For a 3 rd order of one dimension of uncertainty the Hermite Polynomial Chaos is given by: ψψ οο (yy) = PP (yy) = 1, ψψ 1 (yy) = PP 1 (yy) = yy, ψψ (yy) = PP (yy) = yy 1 Appendix B Results of Numerical Example 1 1 Std[I1] Figure B3. Standard deviation of stress tensor complements. (MC 1 E[strain] E[ε],Chaos E[ε],Random 1 E[ε11],Chaos E[ε11],Random std[σ], Chaos std[σ],random std[σ11], Chaos std[σ11],random E[Stress] Figure B1. Closed form solution and SFEM results E[σ],GPC E[σ],Random E[σ11],GPC E[σ11],Random Figure B4. Expected value of strain tensor complements. (MC 1 Std[Srtain] Std[ε],Chaos Std[ε],Random 1 Std[ε11], Chaos Std[ε11],Random Figure B. Expected value of stress tensor complements (MC 1 Figure B5. Standard deviation of strain tensor complements. (MC 1

9 17 Drakos Stefanos: Constitutive Relations of Stress and Strain in Stochastic Finite Element Method E[p] Figure B6. Expected value of pore pressure. (MC 1 std[p] Figure B7. Standard deviation of pore pressure. (MC 1 E[I1] Figure B8. Expected value of stress tensor invariant II 1. (MC 1, 3 E[p],chaos E[p],random-1 samples std[p],chaos std[p],random 1 samples E[I1],chaos E[I1],random 1 samples E[I1],random 3 samples std[i1] Figure B9. Standard deviation of stress tensor invariant II 1. (MC 3 E[I] std[i1], Chaos std[i1], Random 3 samples Figure B1. Expected value of stress tensor invariant II. (MC 1, 5 Std[I] Figure B11. Standard deviation of stress tensor invariant II. (MC 1, 5 E[I],chaos E[I],random, 1 samples E[I],random, 5 samples std[i],chaos std[i],random, 1 samples std[i],random,5 samples

10 American Journal of Computational and Applied Mathematics 15, 5(6): E[I3] Figure B1. Expected value of stress tensor invariant II 3. (MC 1, 5 Std[I3] Figure B13. Standard deviation of stress tensor invariant II 3. (MC 1, 5 REFERENCES E[I3],Chaos E[I3],Random 1 E[I3],Random [1] Phoon, K., Quek, S., Chow, Y. and Lee, S (199), Reliability analysis of pile settlements. Journal of Geotechnical Engineering, ASCE, 116(11), std[i3],chaos std[i3],random 1 std[i3],random 5 [] Mellah, R., Auvinet, G. and Masrouri, F. (), Stochastic finite element method applied to non-linear analysis of embankments. Probabilistic Engineering Mechanics, 15 (3), [3] Eloseily, K., Ayyub, B. and Patev, R. (), Reliability assessment of pile groups in sands. Journal of Structural Engineering, ASCE, 18(1), [4] Fenton, G. A., and Vanmarcke, E. H. (199), Simulation of random fields via local average subdivision. J. Eng. Mech., 116(8), [5] Paice, G. M., Griffiths, D. V., and Fenton, G. A. (1996), Finite element modeling of settlements on spatially random soil. J. Geotech. Eng., 1(9), [6] Fenton, G. A., and Griffiths, D. V. (), Probabilistic foundation settlement on spatially random soil. J. Geotech. Geoenviron. Eng.,18(5), [7] Fenton, G. A., and Griffiths, D. V. (5), Three-dimensional probabilistic foundation settlement. J. Geotech. Geoenviron. Eng., 131(), [8] Fenton, G. A., and Griffiths, D. V. (8), Risk assessment in geotechnical engineering, Wiley, Hoboken, NJ. [9] Ghosh Debraj and Farhat Charbel (8), Strain and stress computations in stochastic finite element methods, International Journal for Numerical Method In Engineering, Vol. 74, Iss. 8, [1] Lord G., Powel C., Shardlow T. (14), An Introduction to Computational Stochastic PDEs, Cambridge Texts in Applied Mathematics. [11] Ghanem, R. G., and Spanos, P. D. (1991), Stochastic finite elements: A spectral approach, Springer-rlag, New York. [1] Drakos I. S. & Pande G. N. Quantitative of uncertainties in Earth Structures, World Congress on Advances in Structural Engineering and Mechanics, Incheon Korea 5-9 August, 15. [13] Drakos I. S. & Pande G. N. Stochastic Finite Element Analysis using Polynomial Chaos Studia Geotechnica et Mechanica Journal.(Under Review). [14] D Xiu, G Em Karniadakis. (3), Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos Computer Methods in Applied Mechanics and Engineering 191 (43), [15] Griffiths, D.V (1985). The effect of pore-fluid compressibility on failure loads in elasto-plastic soil. Int J Numer Anal Methods Geomech, vol.9, no.3, pp