Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads

The nonlinear hygro-thermal dynamics of a bi-directionally functionally graded beam with coupled transverse and longitudinal displacements are studied. The material properties and hygro-thermal distributions are assumed to gradually change along both thickness and length directions according to the arbitray functions. The nonlinear coupled dynamic equations governing the transverse and longitudinal motions of the beam are derived using Hamilton's principle, the von Korman geometric nonlinearity and Euler-Bernoulli theory, as well as considering uniform, linear and sinusoidal distributed hygro-thermal loads. The generalized differential quadrature method with an iterative technique is applied for the nonlinear analysis. Parametric studies are implemented to explore the impacts of the material gradations, temperature rise and moisture concentration on nonlinear frequencies of the beam for various boundary conditions. Results show that the nonlinear dynamic is highly depended on hygro-thermal effects as well as the material properties, which can be used for designing accurately the multi-directionally functionally graded structures in different environmental conditions.