Buckling Loads of Nonuniform Columns

Using finite-difference method, the buckling problem of a hinged-hinged column is expressed as a linear algebraic eigenvalue problem. An iterative procedure to evaluate accurately the minimum eigenvalue of the involved coefficient matrix, without recourse to the characteristic equation or inversion of the matrix, is presented. By the method of Rayleigh and Kato, the upper and lower bounds of the minimum eigenvalue of the coefficient matrix are evaluated. Iteration is used to bring the upper and lower bounds closer together until they are almost equal. Another feature of the method is its capability to find the buckling loads of columns which have discontinuities in the value of moment of inertia by introducing a new concept of equivalent flexural stiffness. The use of equivalent stiffness gives more exact results with fewer divisions of the beam. Several typical examples are solved and results given.