Nonlinear Dynamics of Two-Directional Functionally Graded Beam Under a Moving Load with Influence of Homogenization Scheme

A third-order shear deformation beam model for nonlinear dynamic analysis of a two-directional functionally graded (2D-FG) beam under a moving load is presented. Distinct from the existing higher-order beam models, the present model employs the transverse shear rotation rather than the cross-sectional rotation as an independent variable. The material properties are continuously varied in both the axial and thickness directions by power distribution laws, and they are estimated by four homogenization schemes, namely the schemes due to Voigt, Mori–Tanaka, Hashin–Shtrikman and Reuss. A beam element is derived and used to transfer the nonlinear differential equation to a discretized form. Nonlinear dynamic response of the beam is predicted by the Newmark method in conjunction with Newton–Raphson iterative procedure. Numerical results reveal that the derived beam element which ensures a quadratic variation of the moment in axial direction is capable of furnishing accurate nonlinear dynamic characteristics with a smaller number of elements as compared to the one based on the conventional cross-sectional rotation. It is shown that the difference in the nonlinear dynamic responses predicted by different homogenization schemes decreases by increasing the power-law indices, and the variation of the material properties along the beam length in the 2D-FG beam mitigates this difference. The effects of the material gradation, the moving load velocity and the homogenization scheme on the nonlinear dynamic behaviour are studied in detail. The difference in dynamic responses obtained from linear and nonlinear dynamic analyses is also examined and discussed.