Aiming at the performance-enhancement in coarse mesh modeling, we utilize a number of closed form solutions of a class of torsionally loaded thin-walled bars to formulate a two-noded element for spatial buckling analysis. The key in this relates to the use of the "exact" solution for the displacement fields (as oppose to the more conventional finite element approach where polynomial/Lagrangian-type interpolation is employed). That is, in addition to the well known "exact" solution for the coupled flexure/transverse-shear problem, we utilize a new "exact" solution for the more difficult case of coupled system of differential equations governing a torsionally loaded thin-walled beam using the higher-order theories of non-uniform twist/bi-moment with coupled warping-shear deformations. For the linear analysis, convergence and accuracy study indicated that the proposed model to be rapidly convergent, stable and computationally efficient; i.e. one element is sufficient to exactly represent an end loaded part of the beam. Such model has been extended to account for nonlinear analysis, in particular, the flexural torsional buckling of thin-walled structures. To this end, the effect of finite rotations in space is accounted for as per the modern theories of spatial buckling, resulting in second-order accurate geometric stiffness matrices. Compared with the classical theory of thin-walled structures, the present approach is more general in that all significant modes of stretching, bending, shear (due to both flexure and torsional/warping), torsion, and warping are accounted for. The inclusion of non-uniform torsion is accomplished through adoption of the principle sectorial area. This requires incorporation of a warping degree of freedom in addition to the conventional six degrees of freedom at each node. The element is derived for general cross sections including the Wagner-effect contributions. The model's properties and performance, particularly with regard to the resulting (significant) improvements in mesh accuracy, are assessed in a fairly complete set of numerical simulations.
Flexural - Torsional Buckling Analysis of Thin Walled Columns Using the Fourier series Method