Exact Solutions for the Generalized Euler’s Problem

Abstract

This paper is concerned with buckling analysis of a nonuniform column with classical∕nonclassical boundary conditions and subjected to a concentrated axial force and distributed variable axial loading, namely, the generalized Euler’s problem. Exact solutions are derived for the buckling problem of nonuniform columns with variable flexural stiffness and under distributed variable axial loading expressed in terms of polynomial functions. Then, more complicated buckling problems are considered such as that the distribution of flexural stiffness of a nonuniform column is an arbitrary function, and the distribution of axial loading acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for such problems is reduced to Bessel equations and other solvable equations for seven cases by means of functional transformations. A class of exact solutions for the generalized Euler’s problem involved a nonuniform column subjected to an axial concentrated force and axially distributed variable loading is obtained herein for the first time in literature.