Exact solution of buckling load of axially exponentially graded columns and its approximation

Abstract

This paper studies a generalized Euler problem of inhomogeneous cantilever columns with an end linked to rotational spring and another free end. This paper has two-fold aims. One is to give a characteristic equation to exactly determine buckling loads when axial inhomogeneity is exponential gradient, and the other is to provide an approximate explicit expression for buckling loads. For the former, we make substitution of variables and transform an associated problem to a Bessel equation. Under appropriate boundary conditions, an exact characteristic equation for buckling loads is derived. For the latter, we reduce it to an integral equation, and then apply the moment method to obtain an approximate expression for buckling load. By comparing approximate results with the exact ones, the approximation critical loads provide satisfactory accuracy for a large range of gradient index and rotational spring stiffness. The dependence of buckling loads on the gradient index and spring stiffness is examined. The buckling loads and shapes for an exponential graded column are presented graphically for typical rotational spring stiffness. The derived exact solution can be taken as a benchmark solution to examine the accuracy of other numerical approaches and is of benefit to optimum design of non-homogeneous columns in engineering.