Analysis Of Bending, Buckling And Free Vibration Of Functionally Graded Materials Beam

As a new kind of inhomogeneous complex material, functionally graded material has the unique capability to prevent delamination and alleviate thermal stress, which has more advantages than other traditional complex material. It has become a kind of selected material to be used in new structures. So, more attentions have been paid on the macroscopic mechanical behaviors of FGM structures. In this paper, we choose FGM beams with the material properties varying continuously in the thickness direction as investigating a mechanical model. Based on Euler beam theory and Timoshenko beam theory, respectively, static bending, buckling and free vibration responses of the FGM structure were investigated and some analytical results which can be conveniently used in engineering were obtained. The main research including the following three parts:1. Through the analyzing and solving the bending, buckling and vibration problem of FGM Euler-Bernoulli beams, it can be found that there exists comparability between the governing equations of the uniform beam and the FGM beam. They can be transformed each other through a transition parameter which including the non-homogeneity characteristics. So, finding a solution of FGM beam can changed to find the solution of a uniform beam and to compute the non-homogenous transition parameter, which provides a convenient means for the solution of the non-homogenous beam. The above mentioned theory and method also be extended to the analysis of bending of FGM Timoshenko beam.2. The problem of free vibration of FGM Timoshenko beams, in which the transverse shear and the rotation of inertial force is considered, it is not easy to find similar transformation as FGM Euler beam, found the similar relationship to the vibration of homogeneous beams. So, by using a shooting method the boundary value problem of ordinary differential equations governing the free vibration of the FGM Timoshenko beams was solved and numerical solution of the natural frequencies of the beams with fixed-fixed and fixed-free ends are obtained. Effects of the parameter characterizing the graded variation of material properties and the slenderness ratio on the natural frequencies are analyzed. The results indicate that non-dimensional natural frequency decreases monotonously with the increase of the material parameters and increase with the increment in the value of the slenderness ratio.3. Bending solution of longitudinally non-uniform Euler beams under static load was obtained by using differential quadrature method (DQM). The cases that both the cross-section and the Young's modulus vary continuously with the axial coordinate was considered. Differential quadrature method(DQM) was used to transform the fourth-order differential equations with variable coefficients and the boundary conditions in term of the deflection to linear algebraic equations in terms of the deflection at the discrete points so that numerical solution of the problem was obtained. Numerical solution for the deflection of the beam with the Young's modulus and the cross-sections vary continuously in the axial coordinate under uniformly distributed static load was attained. The results show that non-symmetry of the material geometry and in material property deflection distribution exhibits non-symmetry about the middle point of the non-uniform beam.