New formulation of FEM for deterministic and stochastic beams through generalization of Fuchsʹ approach Original Research Article

This paper proposes an alternative way of constructing the global stiffness matrix of the finite element method for bending beams, it also applies the new formulation to first and second moment analysis of stochastic beams, which involve spatially uncertain bending stiffness. Originating from Fuchsʹ 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. The mean vector and covariance matrix of the displacement of the beam are then obtained in terms of probabilistic characteristics of the uncertain bending stiffness. This is in contrast to the conventional finite element method in stochastic setting, which is based on the perturbation technique. The example is given to illustrate the efficacy of the new formulation and its application to bending of stochastic beams.