Impact of material distribution on the dynamic response of a bidirectional FG beam under general boundary conditions and supported by nonlinear substrate

The present work aims to investigate the dynamic behavior of a bidirectional functionally graded (BDFG) beam lying on nonlinear elastic substrate tacking into account the impact of the material distribution. For the first time, this effect will be studied. The beam characteristics are considered to change in the axial and thickness directions as a function of a specific function with four distinct distribution models. The BDFG beam is supposed to be in contact with a nonlinear three-parameter substrate with general boundary conditions. Hamilton’s principle is used to determine the governing equations based on a quasi-3D solution. An analytical solution is then is applied to obtain the free and forced vibration parameters of the BDFG beam. Furthermore, the validation results show excellent agreement between the proposed theory and those given in the literature. On the other hand, a set of numerical results is exposed and discussed to assess the influence of several geometric and mechanical parameters like distribution patterns of material, beam geometry, foundation’s parameters and external force on the forced and the free vibration of a BDFG beam sustained by a nonlinear elastic foundation. The results show that these parameters have a major influence on the dynamic response of BDFG beams supported by nonlinear foundation. These results can be used as reference solutions for future investigations.