The role of partial elastic foundations on the bending and vibration behaviors of bi-directional hybrid functionally graded nanobeams using FEM

In this paper, for the first time, the static bending and free vibration of bi-directional functionally graded (2D-FG) nanobeams partially resting on an elastic foundation are investigated. The nanobeams are composed of four material components and exhibit mechanical characteristics that vary smoothly and continuously along the thickness and length of the beam according to a power law. For the purpose of analysis, a finite element model is established using a two-node beam element, with each node having five degrees of freedom, combining Lagrangian and Hermitian shape functions. Based on a higher-order shear deformation theory and nonlocal theory, the governing equations of the 2D-FG nanobeams are derived using Hamilton's principle. Through the comparisons of the results obtained from the model with published results in the open literature, the accuracy and reliability of the present model are confirmed. Therefore, the proposed algorithm is compatible for predicting mechanical behaviors of nanobeams with arbitrary material distribution, various boundary conditions and complex loads. New numerical results are conducted to assess the influence of parameters such as volume fraction indexes, nonlocal parameters, foundation coefficients, length-to-height ratio and boundary conditions on the bending static and free vibration behaviors of the 2D-FG nanobeams. Especially, the role of the partial elastic foundations on the bending and vibration of the 2D-FG nanobeams is extensively investigated.