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The paper presents a new algorithm of elastic stress predictor in non linear stochastic finite element method using the Generalized Polynomial Chaos. The statistical moments of strains calculated based on the displacement Polynomial Chaos expansion. To descretise the stochastic process of material the Karhunen-Loeve Expansion was used and it is presented. Using the strains and the material Karhunen-Loeve Expansion the stress components are calculated. A numerical example of shallow foundation was carried out and the results of stress and strain of the new algorithm were compared with those raised from Monte Carlo method which is treated as the exact solution. A great accuracy was presented.

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’ 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 25 years or so. These include the Stochastic Finite Element Method (SFEM) [

In this paper we present SFEM [

A numerical example of shallow foundation is given in the last part of the paper. The results of the two methods of stress and strains calculation are compared and presented.

Considering an arbitrary body and a the sample space

number M of random variables

statistical moments of the results we perform a change of variable

The expected value of a quantity of the problem is given by the following norm:

The author has presented a stochastic finite element procedure to solve boundary problems using polynomial chaos [

where the order Q and the formula

In order to propagate the uncertainties from input parameters to the results for an elastoplastic problem throw the constitutive equation the strains must be calculated first.

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:

Using the displacement polynomial chaos expansion the Equation (4) leads to:

Solving for each increment the boundary problem the strain Polynomial Chaos Expansion can be calculated as before. At each increment n + 1 they are also known the stress from the previous state

The mean value of elastic predictor and of the trial stress are given:

The 4^{th}-order stochastic elasticity tensor of elastic module is given by the equation:

Based on that the stochastic process of Young modulus over the spatial domain with a known mean value

where:

Assuming one dimension and 3^{rd} order polynomial chaos and plain strain conditions:

Considering as

of Elasticity modulus , the mean values and the variance of the lognormal distribution are equal to:

Using the Chaos polynomial expansion the stochastic equation of each component of stress is given:

The expected value after some algebra gives:

The variance of stress:

where:

Similarly for the other components.

Expected value of

Based on the stochastic Equation (13) of

Variance value of

Using again the stochastic Equation (13) of

where:

As an example the variance of

The expected value of

The variance value of

The analysis are carried out similar as the invariant of

A shallow foundation problem for various values of variation’s coefficient

The geometry of the finite elements used for the simulation of the problem presented in

Poisson ratio equal to 0.25. Calculations have been made for ten different coefficients

modulus with a minimum value of 0.1 and then with step 0.1 to a maximum value equal to 1. For SFEM one dimensional Hermite GPC with order 5 [

To propagate the uncertainties of input parameters to constitutive relations of strain and stress where arises due

to spatial variability of mechanical parameters in engineering problems, a new algorithm of Stochastic Finite Element Method has been presented.

An algorithm of Stochastic Finite Element using Polynomial Chaos has been developed and the elastic predictor of stress in a non linear problem is calculated.

A numerical example of shallow foundation was carried out and the results of stress and strain of the new algorithm were compared with those raised from Monte Carlo method which is treated as the exact solution. A great accuracy was presented.

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.

Drakos Stefanos, (2015) Elastic Stress Predictor for Stochastic Finite Element Problems. World Journal of Mechanics,05,222-233. doi: 10.4236/wjm.2015.511021

In order to solve the problem we have to create the new space

Assuming that the

The tensor product of the M

And using (A2)

where

And

Xiu & Karniadakis [

where:

For a 3^{rd} order of one dimension of uncertainty the Hermite Polynomial Chaos is given by: