An engineering application of the h- p Version of the finite element method to the static analysis of a Euler-Bernoulli beam

Abstract

A hybrid h- p finite element technique for the static analysis of Euler-Bernoulli beams is described in this paper. The standard h-version method of discretizing the problem is retained, but modified to allow the use of polynomially-enriched beam elements. Specially chosen K-orthogonal polynomials facilitate this enrichment; they also fully decouple the h- and p-contributions to the global stiffness matrix [ K]. This design feature permits the resulting matrix equation of static equilibrium, [ K]{ q} = { F}, to be solved with optimum efficiency, and moreover, enables a hierarchical re-analysis to be carried out at the post-processing stage. A novel method of calculating the loading vector F, indirectly based on a generalization of the “parallel axes theorem”, is described for arbitrarily distributed loading actions. It is shown that such loads may adequately be represented by a relatively sparse set of discrete data points, such as typically might arise from experimental test results. In contrast to the conventional h-version, in which such loading information would dictate the mesh design, no such restrictions apply to the current formulation. Such versatility permits the full interplay between h-refinement and p-enrichment to be studied. The [hitherto unknown] physical interpretation of the hierarchical generalized coordinates follows from the methodology evolved to determine the loading vector. A simple error estimator, based on discontinuities in the bending moment and shear force distributions at element interfaces, is used to assess the accuracy of the solution, and the relative merits of mesh refinement vs hierarchical enrichment. The main conclusion which can be drawn from this work is that it is computationally most economical to use the coarsest h-mesh commensurate with the boundary conditions and loading actions, and then hierarchically enrich each element.