In this paper, free vibrations of the tapered shear deformable column are studied. The column is clamped at the bottom and is free or hinged or clamped at the top. The column has a tapered cross-section, square and circular, under the condition of constant volume. The axial compressive load is acted to the top. The differential equations of such column were derived, in which the effects of the rotary inertia and shear deformation were included. For computing natural frequencies with mode shapes, the differential equations were solved numerically. The effects of column parameters of the frequencies and mode shapes were performed, and its results were extensively discussed.

This paper deals with the elastica and buckling loads of the nonlinear elastic tapered cantilever columns subjected to an axial load at the free end. The column cross section is rectangular, where the width and depth vary linearly with the column axis. The column material is nonlinear elastic, which obeys the Ludwick’s constitutive law. The ordinary differential equations governing the elastica of buckled column are derived and solved numerically for computing the elastica and buckling loads. Parametric studies for the annealed copper, N.P. 8 aluminum alloy and steel columns are conducted