Analysis Of The Bending And Vibration Of Functionally Graded Material Timoshenko Beams

Functionally graded material (FGM) is a new type of non-uniform composite materials, which is produced according to a specific technique from two or more kind materials with different properties through certain design procedure so that it can make the material components change continuously in the gradient direction and eliminate the non-continuity of the physical material properties, or avoid the internal interfaces. The study on the static and dynamic responses of functionally graded material beams is still one of the interesting and focused topics in the theory of structural mechanics as well as in engineering application. In this dissertation, linear transition relationship between static bending solutions of the FGM beams, with the material properties varying continuously in the thickness direction, and of the corresponding homogenous beam were studied based on both the first and higher-order shear deformation beam theory by analytical approach. Furthermore, by using numerical shooting technique, free vibration response of FGM beams was studied based on the first order shear deformation theory. Main work of this dissertation is formed of the following three parts:1)、Based on the first-order shear deformation theory, linear transformative relation between the static bending solutions of FGM beams and the corresponding homogenous beams is analyzed. Through comparing non-dimensional governing equations of FGM beams with those of the corresponding homogenous beams, linearly dependent relation between their bending solutions was found. Therefore, solution of the bending problem of functionally graded materials beams can be transformed to that of uniform beam and calculating of the transition factors under the same loadings and boundary conditions. The method can provide of simple and convenient approach in the analysis and solution of non-uniform beams.2)、Analysis of free vibration of FGM Timoshenko beams were performed by using a numerical shooting method. In the derivation of dynamic governing equations, the influences of all the axial, lateral, rotational, tension-bending coupling inertia forces and the first order shear deformation were considered. The first three natural frequencies and the corresponding vibration modes were obtained for the FGM beams under three common boundary conditions of FGM beams. Effects of the parameters of the material gradient, the slenderness and the boundary conditions on the natural frequencies were examined in detail. The numerical results show that effects of the shear deformation and the rotational and tension-banding inertial forces on the natural frequencies become significant with the increase of the relative thickness of the beam.3)、Based on the higher-order shear deformation beam theory, static bending of FGM beams is further investigated. Through studying and examining the similarity and the difference between the governing equations for static bending of FGM and the homogenous beam and by using of the basic theory of differential equations, we can get the linear transformation relationship between bending solutions of FGM beams under the higher-order shear deformation beam theory and those of the corresponding homogenous ones based on the classical beam theory. As a result, the solution of static bending of a FGM beam based on higher-order shear deformation theory can be finally reduced into the calculation of the transition parameters related to the regulation of the variation of the material gradient if the classical solution of the uniform beams is known.