Free vibration of functionally graded beams with a through-width delamination is investigated. It is assumed that the material property is varied in the thickness direction as power law functions and a single through-width delamination is located parallel to the beam axis. The beam is subdivided into three regions and four elements. Governing equations of the beam segments are derived based on the Timoshenko beam theory and the assumption of ＇constrained mode＇. By using the differential quadrature element method to solve the eigenvalue problem of ordinary differential equations governing the free vibration, numerical re- suits for the natural frequencies of the beam are obtained. Natural frequencies of delaminated FGM beam with clamped ends are presented. Effects of parameters of the material gradients, the size and location of delamination on the natural frequency are examined in detail.
The relationship between the critical buckling loads of functionally graded material （FGM） Levinson beams （LBs） and those of the corresponding homogeneous Euler-Bernoulli beams （HEBBs） is investigated. Properties of the beam are assumed to vary continuously in the depth direction. The governing equations of the FGM beam are derived based on the Levinson beam theory, in which a quadratic variation of the transverse shear strain through the depth is included. By eliminating the axial displacement as well as the rotational angle in the governing equations, an ordinary differential equation in terms of the deflection of the FGM LBs is derived, the form of which is the same as that of HEBBs except for the definition of the load parameter. By solving the eigenvalue problem of ordinary differential equations under different boundary conditions clamped （C）, simply-supported （S）, roller （R） and free （F） edges combined, a uniform analytical formulation of buckling loads of FGM LBs with S-S, C-C, C-F, C-R and S-R edges is presented for those of HEBBs with the same boundary conditions. For the C-S beam the above-mentioned equation does not hold. Instead, a transcendental equation is derived to find the critical buckling load for the FGM LB which is similar to that for HEBB with the same ends. The significance of this work lies in that the solution of the critical buckling load of a FGM LB can be reduced to that of the HEBB and calculation of three constants whose values only depend upon the through- the-depth gradient of the material properties and the geometry of the beam. So, a homogeneous and classical expression for the buckling solution of FGM LBs is accomplished.
The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.
