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 free vibrations of the axially functionally graded (AFG) horseshoe arch. The modulus of elasticity and the mass density of AFG material of arch are chosen as a univariate quadratic function. The differential equations with the boundary conditions that govern the free vibration of such arch are derived and numerically solved to calculate natural frequencies and mode shapes. Natural frequencies of this study agree well with those of the finite element ADINA. Parametric studies of the geometrical and mechanical properties of the arch on frequencies and mode shapes are performed and extensively discussed.

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 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 presents a numerical analysis of free vibration of thin circular and annular plate using finite element method. The first five natural frequencies are presented for uniform annular plates of various inner-to-outer radius ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of plates. The accuracy of the method is established by comparing the results available in the literature. Results show that natural frequency parameter increases as the inner-to-outer radius ratio increases except in case of free boundary condition, for which it decreases with the inner-to-outer radius ratio. This result provides benchmark values that can be used to validate result obtained by other approximate approaches such as finite difference method, differential quadrature method and boundary element method.