Effect of Hygrothermal Loads on the Nonlinear Vibration of Porous Bi-Directional Functionally Graded Beams Placed on an Elastic Foundation

In this study, we investigated the effect of hygrothermal loads on the nonlinear free vibrations of a porous bi-directional functionally graded Timoshenko beam with a Winkler-Pasternak foundation. The corresponding governing equations were established according to Hamilton's principle and von Karman nonlinearity assumption. Using the differential quadrature and iterative methods, the nonlinear fundamental frequencies were determined. A numerical example was presented, and we showed that the discretization of nonlinear items considerably decreased the number of iterations. A series of parametric studies were performed, and we determined that (1) all the vibration mode shapes were significantly affected by the axial functionally graded index, while they were almost unchanged by the thickness-wise functionally graded index. Moreover, compared to the thickness-wise graded index, the axial functionally graded index played a more important role in the higher-order mode shapes. (2) The geometric nonlinearity considerably influenced the critical temperature rise value of the beams. Increasing the maximum deflection value increased the critical temperature rise value. (3) The influence of increase in temperature and moisture on the nonlinear fundamental frequency became stronger with increase in the functionally graded index, but it weakened with increase in the maximum deflection value.