Numerical computation of the flexibility matrix of variable cross section beams, with finite elements

Abstract

In this work, the flexibility properties of variable cross section beams are derived, through the application of the second theorem of Castigliano; considering the complementary energy by bending and share forces. To perform the integration of the flexibility coefficients, a numerical method, which considers the discretization of the beam domain with first order rectangular finite elements, in conjunction with the Gauss rule, is proposed. At the end of the work, the proposed method is applied to a tapered beam that has been discretized with a maximum of five finite elements. It is shown that the method is general, and that it can be applied to beams of variable section in which the cross section can be complex. The results shown that no more than 3 finite elements are needed to discretize the domain of beams in which, the ratio height-length is of the order of ten.