This paper, introducing local integration, formulates a new stochastic finite element method for estimating the response variability of multi-dimensional stochastic systems. Young's modulus is assumed to have a spatial variation and is idealized as a multi-dimensional continuous Gaussian stochastic field. An essential feature of the proposed method is that the continuous stochastic field is rigorously taken care of by means of local integrations to construct element stiffness matrices, as the results, the issue involving the stochastic field is transformed into a problem involving only a few random variables, and the perturbation technique is then utilized with considerable ease. This may lead to substantial improvement in computational efficiency. And the accuracy of the solution from the proposed method appears to be independent of the way in which discretization is performed, whereas the problem associated with the convergence of the solution from conventional methods, which are based on a discretized stochastic field, always remains in any case. In this paper, it is shown that stochastic stiffness matrices can be easily derived from the principle of complementary virtual work, when the spatially varying Young's modulus is idealized as mentioned above. In order to examine the validity of the proposed method, two kinds of stochastic structures subjected to deterministic loads; a portal frame and a plane stress plate, are analyzed. As the results, the proposed method has a great advantage not only in the computational cost but also in the solution accuracy. Finally, it should be emphasized that the formulation of this method is so systematic that it can be easily extended to various engineering problems.
