Asymptotic lower and upper bound solutions for lateral torsional buckling of elastic thin-walled beams with mono-symmetric open sections

Abstract

Asymptotic lower and upper bounds to the critical load provide a powerful means of ensuring the stability design with clear margin of safety. However, potential energy-based approaches always lead to an upper bound. Although continuing efforts are devoted to predicting a lower limit, there evidently lacks a consistent approach for predicting the lower bound estimates that are asymptotically close to ever-improving upper bound estimates. This paper is intended to propose a novel method of successive approximations for arriving at asymptotic lower and upper bounds to elastic lateral torsional buckling load of thin-walled beams with mono-symmetric open sections. A deformation-field-independent statically admissible system as indicated by the stress field due to disturbing torque acted at preset stations is employed in the complementary energy formulation. Based on Castigliano’s second theorem and Betti’s reciprocal theorem, a compatibility equation in terms of given location of disturbing torque is derived to determine the critical load. The minimum and maximum values obtained represent the absolute lower and upper bounds. The compatibility equation based on the absolute upper bound is iteratively used for updating the buckling modes at all stations, which are interpolated by using Lagrange polynomials to approximate the buckling mode shapes. Examples demonstrate that the proposed method provides asymptotic lower and upper bounds with minimal approximations. The lower bound with a relative difference within 5 % to the upper bound provides an accurate and safe alternative to the exact critical load under typical loading and constraint conditions.