Microbeam Model and Related Differential Quadrature Finite Elements

Abstract

A size-dependent quasi-3D functionally graded (FG) microbeam model was presented within the combined framework of the modified couple stress theory and a 4-unknown higher-order shear and normal deformation theory. Then the model was applied to analyze the static bending and free vibration of FG microbeams. With the 2nd Lagrange equation, the corresponding motion equations and the appropriate boundary conditions were obtained. A 2-node 16DOF differential quadrature finite element combining the Gauss-Lobatto quadrature rule with the differential quadrature rule was constructed to handle the general static/dynamic boundary value problems of FG microbeams. A comparison study was performed to show the efficacy of the proposed theoretical model and solution method. Finally, the effects of the gradient index, the intrinsic length scale parameter, the geometrical parameters and the boundary conditions on the static and dynamic characteristics of FG microbeams were examined. Numerical results reveal that the developed beam model and element are applicable to the analysis of mechanical behaviors of FG microbeams with various slenderness ratios. Besides, introduction of the couple stress effect can significantly change the static and dynamic characteristics of FG microbeams.