Buckling of variable cross-section columns. Integral-equation approach

A semianalytical procedure is presented for the axial buckling of elastic columns with step-varying profiles. Profiles with continuous variations can be approxiamed, to any desired degree of accuracy, by a series of step variations. The formulation leads to a general procedure that can be applied to any continuous or discontinuous profile regardless of the number of steps. Since the step changes of the profile are represented by distributions, the differential equation cannot be solved in the ordinary sense. The differential equation is therefore converted to an integral equation. The solution of the integral equation is obtained by polynomial functions. The formulation involves a significant amount of algebraic manipulations. This problem is alleviated by using a symbolic manipulation system to carry out the algebraic manipulations. The integral kernels are obtained for the different boundary conditions. Formulas for buckling loads for members with variable profiles and different boundary conditions can be obtained in terms of the section and profile parameters. The example problems present the solution of some common columns with variable cross sections. The procedure can be used to derive the elastic and geometric stiffness matrices for beam columns with variable cross sections as well as for other elements.