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Structural Engineering and Mechanics
  Volume 28, Number 2, January30 2008, pages 129-152

Solution of randomly excited stochastic differential equations with stochastic operator using spectral stochastic finite element method (SSFEM)
Hussein A., El-Tawil M., El-Tahan W. and Mahmoud, A. A.

    This paper considers the solution of the stochastic differential equations (SDEs) with random operator and/or random excitation using the spectral SFEM. The random system parameters (involved in the operator) and the random excitations are modeled as second order stochastic processes defined only by their means and covariance functions. All random fields dealt with in this paper are continuous and do
not have known explicit forms dependent on the spatial dimension. This fact makes the usage of the finite element (FE) analysis be difficult. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is used to represent these processes to overcome this difficulty. Then, a spectral approximation for the stochastic response (solution) of the SDE is obtained based on the
implementation of the concept of generalized inverse defined by the Neumann expansion. This leads to an explicit expression for the solution process as a multivariate polynomial functional of a set of uncorrelated random variables that enables us to compute the statistical moments of the solution vector. To check the validity of this method, two applications are introduced which are, randomly loaded simply supported reinforced concrete beam and reinforced concrete cantilever beam with random bending rigidity. Finally, a more general application, randomly loaded simply supported reinforced concrete beam with random bending rigidity, is presented to illustrate the method.
Key Words
    stochastic differential equation; stochastic finite element method(SFEM); spectral SFEM; Karhunen-Loeve expansion; Neumann expansion; Bernoulli-beam equation; stiffness matrix; bending rigidity; Exponential covariance model; Triangular covariance model; Wiener model.
Hussein A.: Higher Technical Institute, 10th of Ramadan City, Basic Sciences Department, Egypt
El-Tawil M., El-Tahan W. and Mahmoud, A. A.: Engineering Mathematics Department, Faculty of Engineering, Cairo University, Giza, Egypt

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