Reinvestigation on Optimization of Clamped-Clamped Columns and Symmetry of Corresponding Eigenfunctions

Abstract

In this paper, the optimization problem of the clamped-clamped column under buckling load, which was previously dealt with by Tadjbakhsh & Keller (Trans. ASME, J. Appl. Mech., Vol.29, No.1(1962), p.159) and Olhoff & Rasmussen(Int. J. Solids Struct., Vol.13, No.7 (1977), p.605), is reinvestigated. Both numerical and analytical methods based on the single and bimodal formulations are used in investigating the behavior of the optimum eigenvalues and the corresponding eigenfunctions with respect to the minimum constraint of cross-sectional area. The conditional optimum column under the single modal formulation is determined. The optimum column together with two mutually symmetrical eigenfunctions under bimodal formulation is obtained. Furthermore, on the basis of that we deduce the orthogonal optimum eigenfunctions, with the 1st-order eigenfunction having a symmetrical configuration and the 2nd-order eigenfunction having an anti-symmetrical one. Finally, using Keller's analytical solution, we derive several analytical solutions for the columns mentioned in previous papers.