Dynamic Analyses of Two-Dimensional Functionally Graded Timoshenko Beam using Finite element Method

Abstract

In this work, dynamic analyses of a functionally graded beam are presented. The governing equations of the beam is found based on the displacement field defined by Timoshenko beam theory, then solved by using finite element method based on Hamilton’s principle. The beam is assumed to be free-clambed boundary condition (F-C). The PL index is used for describing the distribution of the beam properties in both transverse and longitudinal directions. A parametric study is accomplished to investigate effect of several parameters on the natural frequency, mode shapes and transient response of the beam., such as the PL indexes (nx and nz) for x and z axis, respectively, and the elasticity modulus ratio (Eratio). To valid the present results and current mathematical formulation, some of the findings are compared with another research. A good agreement is noticed. It is noted that the response of the beam is more sensitive to the variations of the PI in the longitudinal axes than that corresponding in the transverse one. For specific design requirements, the dynamic response of the beam can be adjusted by chose a proposal indexes and modulus ratios.