Flexural - Torsional Buckling Analysis of Thin Walled Columns Using the Fourier series Method

In this work, the governing differential equations of elastic column buckling represented by a system of three coupled differential equations in the three unknown displacement functions, v(x), w(x) and (x) are solved using the method of Fourier series. The column was pinned at both ends x = 0, x = l. The unknown displacements were assumed to be a Fourier sine series of infinite terms, which was found to satisfy apriori the pinned conditions at the ends and substituted into the governing equations. The governing equations were found to reduce to a system of algebraic eigenvalue – eigenvector problem. The buckling equation was found to be a cubic polynomial for the general asymmetric sectioned column. The buckling modes were found as flexural torsional buckling modes. For columns with monosymmetric sections, it was found that the buckling mode could be flexural or flexural – torsional depending on the root of the cubic polynomial buckling equation which is the smallest. For columns with bisymmetric sections, it was found that the buckling modes are uncoupled and bisymmetric columns could fail by pure flexural buckling about the axes of symmetry or pore torsional buckling. The findings are in excellent agreement with Timoshenko’s solutions.