This study investigates the generalized stiffness of laterally functionally graded materials (LFGMs) and applies these findings to dynamic beam elements. The generalized stiffnesses of LFGM, coupled with material and cross-sectional properties such as flexural and axial rigidity, mass per unit length, and mass-moment of inertia, are explicitly formulated. In the context of LFGM, material properties depend on an asymmetrical power law function with respect to cross-sectional depth. An example of the generalized numerical stiffness of a circular cross-section is provided for various material properties. To illustrate the application of generalized stiffness to dynamic beam elements, free vibration of LFGM beams with rotary inertia is considered. The dimensionless differential equation governing the free vibration of such beams is derived and numerically solved to obtain natural frequencies and corresponding mode shapes. Numerical results demonstrate a good consistency with the finite element method.

This paper deals with buckling lengths of the heavy column with various end conditions, where both top and bottom ends are either free or hinged or clamped. Based on equilibrium equations of the buckled column element, the differential equation governing the buckled mode shape is derived. For solving the buckling length, the differential equation is integrated by the direct integration method and the buckling length is calculated by the determinant search method. The buckling lengths of this study agree well with those of references. The buckling lengths with various end conditions, buckled mode shapes and buckling stresses are presented.

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