Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories

In this paper, fundamental frequency analysis of functionally graded (FG) beams having different boundary conditions is analyzed within the framework of the classical, the first-order and different higher-order shear deformation beam theories. The material properties of the beams vary continuously in the thickness direction according to the power-law form. Two types of formulation are developed. In the first formulation, total bending rotation measured on the beam middle surface is taken as unknown function whereas the shear rotation measured on the beam middle surface is taken as unknown function in the second formulation. The frequency equation is obtained by using Lagrange's equations and the boundary conditions of beams are satisfied with Lagrange multipliers. The unknown functions denoting the axial and the transverse deflections, the bending and the shear rotations of the cross-section of the beam are expressed in the polynomial form. In this study, the effects of slenderness ratio, material variations, the different formulations and the beam theories on the fundamental frequencies are examined. It is believed that the tabulated results will be a reference with which other researchers can compare their results.