Non-Perturbative Fem for Deterministic and Stochastic Beams Through Inverse of Stiffness Matrix

Abstract

This paper proposes an alternative way of constructing the global stiffness matrix in the finite element analysis of bending beams, which involve spatially deterministic or stochastical bending stiffness. Originating from Fuchs’ idea of decoupling the shear and bending components in the bending beam, the element level stiffness matrix is diagonalized. The generalized stress-strain, strain-displacement and equilibrium relationships are assembled, respectively, and then are combined to form the global stiffness matrix. The advantage of the new formulation is that the bending stiffness explicitly appears in the global stiffness matrix, which can be inverted exactly without application of perturbation based expansion. The mean vector and correlation matrix of the displacement of the beam are then obtained in terms of probabilistic characteristics of the uncertain bending stiffness. The example is given to illustrate the efficacy of the new formulation and its application to bending of stochastic beams.