Fourier Cosine Series Method for Solving the Generalized Elastic Thin-walled Column Buckling Problem for Dirichlet Boundary Conditions

Abstract

The Fourier cosine series method was used to obtain solutions to the generalized elastic thin-walled column buckling problem for the case of Dirichlet end boundary conditions. The problem is a boundary value problem given by a set of three coupled ordinary differential equations with three unknown displacement functions, and subject to the Dirichlet end boundary conditions. By choosing the origin of the longitudinal coordinate at the middle of the column, the Fourier cosine series was found to be a suitable shape function for the problem and hence the three displacement modal functions were represented using Fourier cosine series, with unknown modal amplitudes. The Fourier cosine series representation of the unknown displacement modal functions simplified the problem to an algebraic eigenvalue problem given by a set of homogenous algebraic equations whose nontrivial solution yielded the characteristic buckling equation. The stability equation was obtained as a third-degree polynomial for the asymmetric cross-sectioned column. For asymmetric cross-sections, the buckling modes were obtained as coupled flexural-torsional modes. For bisymmetric cross-sections, the buckling modes were uncoupled and failure could be flexural or flexural-torsional. For monosymmetric cross-sections, one of the buckling modes is uncoupled while the others are coupled; and failure could be by either Euler bending or bending-torsional buckling. The solutions obtained in this work agree with results obtained by previous researchers.