GBT-Based Finite Element Formulation to Analyse the Buckling Behaviour of Thin-Walled Members Subjected to Non-Uniform Bending

Abstract

In this paper, one investigates the local-plate, distortional and global buckling behaviour (critical bifurcation loads and buckling mode shapes) of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e., under the presence of longitudinal stress gradients. In order to achieve this goal, one begins by developing and numerically implementing a beam finite element formulation based on Generalised Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. After presenting the main concepts, procedures and assumptions involved in the above formulation, one addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Particular attention is devoted to the main steps involved in the determination of the elementary linear and geometric stiffness matrices, as they must incorporate the stiffness reduction stemming from the presence of the non-uniform bending moments (longitudinal stress gradients) and also of the pre-buckling shear stresses caused by them - the inclusion of this last effect constitutes an original contribution within the context of GBT buckling analyses. Then, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, one presents and discusses numerical results concerning thin-walled steel Ibeams acted by various (uniform and non-uniform) bending moment diagrams. In particular, one analyses (i) cantilevers subjected to uniform major axis bending (Fig. 1(a)), tip point loads (Fig. 1(b)) and uniformly distributed loads (Fig. 1(c)), as well as (ii) simply supported lipped beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads by taking full advantage of the GBT modal features, one is able to acquire a much deeper understanding about the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling mode shapes. For validation purposes, some GBT-based critical loads/moments and buckling mode shapes are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, one assesses the computational efficiency of the buckling analyses carried out using the proposed GBT-based beam finite element, by comparing the number of degrees of freedom involved with those required to obtain equally accurate results with discretisations in shell finite elements (note that “uniform stress” GBT-based beam finite elements are no longer applicable). Open image in new window Figure 1 Local-plate buckling mode shapes of I-section cantilevers subjected to (a) uniform major axis bending, (b) a tip point load and (c) a uniformly distributed load.