High order beam elements for the stability and non-linear analysis of frame structures

Abstract

The Euler-Bernoulli beam elements presented in this paper are based on the improvement of the accuracy of the finite element solution by increasing the polynomial degree of the shape functions. This approach, called the p-version of the finite element method has become more and more attractive in the last two decades. It was shown in the literature, that the p-version for linear elliptic problems with smooth solutions converges exponentially in the energy norm. In the practical analyses, the p-version yields an accuracy which can hardly be obtained by the classical h-version, based on the refinement of the finite element mesh. The elements studied in this paper show very high convergence in buckling and nonlinear analyses. The final result of the studies is a beam element with six degree polynomial shape functions, which yields engineering accuracy in the practical buckling and non-linear frame analyses with only one element per column. Although this type of element introduces three supplementary non-nodal degrees of freedom, the computational costs of the analyses are much lover than in the case of the classical beam elements based on cubic shape functions