Strongest Column by the Finite Element Displacement Method

Abstract

'^J exhaustive search of the existing literature reveals that the problem of optimization of columns with elastic restraints and subjected to a varying axial load distribution has not received any attention. In this paper a finite element displacement formulation is used to determine the distribution of material which will maximize the critical load parameter, Acr, of an Euler-Bernoulli column of specified length and volume, under various boundary conditions (mixed or not—with or without springs) and subject to the constraint that the cross-sectional area is no smaller than a specified value A0. Contents Consideration is restricted to those columns for which the minimum cross-sectional moment of inertia, /(x), and area, A(x are related by I(x) = pA(x) (p and n are positive constants). This assumption is a restriction but with a suitable choice of p and n it covers a large class of structural configurations. For such a column, shown in Fig. 1, the Rayleigh quotient, A (the factor by which the given axial load distribution has to be scaled in order to produce instability in the column), is given by