# The Weighted Integral Method for Calculating the Stochastic Stiffness Matrix of Stochastic Systems

*by*George Deodatis, Princeton Univ, Princeton, United States,

Masanobu Shinozuka, Princeton Univ, Princeton, United States,

**Abstract:**This paper introduces the 'weighted integral method' for calculating an exact expression of the stochastic stiffness matrix of a stochastic structure. Using this method, the stochastic stiffness matrix is calculated in terms of integrals of the stochastic field describing the random material property multiplied by a deterministic function. These integrals are random variables called weighted integrals. As a consequence, the finite element mesh that would be used in a deterministic analysis can be used for any value of the correlation distance of the stochastic field involved in the problem. Two approaches are used to derive the stochastic element stiffness matrix. The first approach is based on the principle of stationary potential energy and the second on the principle of virtual work. The potential energy approach produces a stochastic stiffness matrix which is an approximation of the corresponding exact one obtained using the virtual work approach. Finally, stochastic shape functions are introduced describing the stochastic displacement field of the beam element with random material properties.

**Subject Headings:**Stochastic processes | Integrals | Stiffening | Matrix (mathematics) | Material properties | Finite element method | Mesh generation | Correlation

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