A Galerkin finite element technique for stochastic field problems

A new finite element method is developed to analyse the structures with more than one parameter behaving in a stochastic manner. The Galerkin weighted residual method is used. As a generalization, this paper treats the random eigenvalue problems arising when the material property values of the structural systems are having stochastic fluctuations i.e. the measurement errors can be accounted for when probabilistic modelling is used. The free vibration problem of a stochastic beam whose Young's modulus and mass density are distributed stochastically is considered. Using Galerkin's weighted residual procedure, a stochastic finite element method is developed and implemented to arrive at a random algebraic eigenvalue problem. The stochastic characteristics of eigensolutions are derived in terms of the stochastic material property variations. Numerical examples are given. It is demonstrated that through this formulation the finite element discretization need not be dependent on the characteristics of stochastic fields describing the fluctuations in the material property values. Further, calculation of covariances between stiffness elements, mass elements, etc., is carried out in a relatively simple manner and is more general in nature.

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