Determination of buckling load by adaptive shape functions

Abstract

This study proposes the development of an accurate and powerful numerical technique capable of determining the flexural buckling load of two-dimensional skeleton structures. The key formulation is based upon the principle of stationary total potential energy whereas the discretization is adopted using standard Rayleigh-Ritz approximation. The crucial feature of the current technique is that a space of trial functions used in the approximation of the buckling shape is constructed from shape functions containing an adaptive parameter directly related to the member axial force and, more importantly, this space will contain the exact buckling shape if the member axial force is identical to its buckling load. Such special basis functions can be constructed, in an element-wise fashion, from a solution of the ordinary differential equation governing the buckling shape of a single element. In the present study, we propose a set of basis functions for several types of elements such as elements with and without lateral constraints, elements with consideration of shear deformation, and elements made of nonlinear materials. This integrated component enhances the capability of the buckling analysis of various types of structures (e.g. plane frames with and without lateral bracings, columns rested on an elastic foundation, inelastic buckling of columns, structures with significant shear deformation, etc.). With use of adaptive interpolation functions along with a proper iterative procedure, the approximate buckling shape and approximate buckling load automatically converge to the exact solutions without the need for mesh refinement. For any iteration, the minimum eigenvalue is efficiently and accurately computed by a selected numerical technique. The proposed method is then tested in the buckling analysis of many structures to demonstrate its accuracy, rate of convergence and capabilities. A selected set of numerical results are reported and discussed.