Investigation of the free vibrations of bi-directional functionally graded material beams using differential quadrature method

In this study, we investigated the natural dynamic characteristics of bi-directional (2D) functionally graded material (FGM) beams using differential quadrature method (DQM). The higher order shear deformation theory was used to reconstruct the displacement and strain fields at any point on the beam. The resulting equilibrium equation of elasticity was expressed by stress is equivalently integrated along the height direction of a rectangular beam, which helps derive the ordinary differential characteristic equations of the free vibration problem of 2D FGM beams, including the natural frequency and the triple-coupled mode functions. Then, using the DQM, the eigenvalue problems of the derived differential equation for free vibration of 2D FGM beams were transformed into a set of algebraic equations representing the eigenvalue problems. The obtained algebraic equations were solved using the orthogonal triangular decomposition method (QR). First, several orders of natural frequencies of the free vibrations of 2D FGM beams were calculated sequentially, and then, the triple-coupled mode functions were obtained together. Numerical examples were used to illustrate the effects of the thickness ratio, grading indexes, and the supported edges on the natural frequency in detail. Numerical results validated the feasibility and accuracy of the developed DQM for solving the problem of free vibrations in 2D FGM beams.