Stability and vibration analyses of tapered columns resting on one or two-parameter elastic foundations

This paper presents a generalized numerical method to evaluate element stiffness matrices needed for the free vibration and stability analyses of non-prismatic columns resting on one- or two-parameter elastic foundations and subjected to variable axial load. For this purpose, power series approximation is used to solve the fourth–order differential equation of non-prismatic columns with variable geometric parameters. Then, the shape functions are obtained exactly by deriving the deformation shape of the column as power series form. Finally, the element stiffness matrices are determined by means of the principle of virtual work along the columns axis. In order to demonstrate the accuracy and the efficiency of presented method, several numerical examples including in the free-vibration and buckling analysis of non-prismatic columns, portal frame, and gable frame are presented and obtained results compared with the results of other available numerical and theoretical approaches. The method can be applied for the buckling load and natural frequencies computation of uniform members as well as non- prismatic members.