Asymptotic lower and upper bound solutions for elastic lateral torsional buckling of thin-walled beams considering the effect of prebuckling deflections

The investigation of prebuckling deflections on the lateral torsional buckling resistance of elastic thin-walled beams is naturally a geometric nonlinear problem. However, it is usually encountered that such deflections are essential for determining the critical load but do not substantially change the prebuckling shape from an initial unstressed configuration. Besides, the pre-buckling effect was mainly considered for I-section beams. This work aims to propose a semi-analytical method for efficiently predicting asymptotic lower and upper bounds to the critical load of pre-deflected beams with I- and rectangular hollow sections, provided the maximum prebuckling deflection-to-span ratio does not surpass a relaxed limit. The governing equations are derived by reserving up to the quadratic order terms, where the nonlinear torsional and flexural curvature terms negligible for I-sections are investigated for necessity if rectangular hollow sections are concerned. Based on Betti's reciprocal theorem, a compatibility equation is derived by choosing the stress field due to a disturbing torque as the statically admissible system. The minimum and maximum critical loads determined from the compatibility equation by varying the acting point of the disturbing torque represent the absolute lower and upper bounds. It is demonstrated that the available closed-form formulas to calculate critical load increment due to prebuckling deflections are valid for doubly symmetric I-beams under specific loadings or in general cases with sufficiently small I y /I x ratio. The nonlinear torsional curvature has a negligible effect on lateral stability, whereas the nonlinear flexural curvature is important for rectangular hollow section beams with moderate I y /I x ratios. The asymptotic approach provides a safe and efficient estimate of exact critical buckling load as long as the maximum prebuckling deflection-to-span ratio does not exceed a relaxed limit.