Stochastic Finite Element Method Based On Physical Response Bases

When structural parameters are subjected to random and space variations, a stochasticfield function should be used to describe them. Correspondingly, the structural analysis shouldbe carried out by Stochastic Finite Element Methods (SFEM). The general procedures ofSFEM are as follows: firstly, stochastic field functions which are employed to describe thestructural properties have to be discretized; secondly, the structural response bases areconstructed in tems of which the structural responses are expressed; finally, based on thestandard format of the finite element method and the approximation principles, the system oflinear equations is set up.In the traditional SFEM, the random field function is discretized based on K-Lexpansion of its covariance function and the polynomial chaos is employed to constructorthogonal response bases. This kind of SFEM has a sound mathematic background, it hasuntil now been the well-established method. However, a lot of random variables areintroduced when discretizing random field functions and constructing orthogonal responsebases, which makes the dimensions of equations increase enormously. Hence, solving theequations becomes very difficult, which limits the application of the kind of SFEM.In view of the drawbacks of the traditional SFEM, a new method is developed in thepaper. In the method developed, the optimal linear estimation method is adopted for thediscretization of the random field. The physical response bases are constructed based on thephysical relation of responses and random functions and the chosen point values of randomfunctions which are employed to express the structural response. The approximationprinciples are used to set up the system of linear equations. Finally, the appropriate method isconstructed to resolve the equations.By use of the optimal linear estimation method, the number of random variables can bereduced which are used to discretize the random function. Moreover, the expression ofstructural responses based on physical bases can decrease the strong coupling effects ofrandom and numerical analyses. Hence, theoretically, the method developed has a highercomputational efficiency than the traditional one.The structural analyses of the clamped beam at two ends with the bending rigidity beingmodeled as a random function are carried out by means of the developed method. Thenumerical results show that, when the coefficient of variation of the random field function isnot larger than0.3, the good convergence of responses can be achieved. In addition, the number of the physical bases can be reduced by the rational choice of discretized point valuesof random function for a given calculation accuracy.In conclusion, numerical analyses show that, the stochastic finite element method basedon physical bases have a high calculation efficiency. However, the further investigationsshould be carried out when applying to a general structure.