In this research, we develop a novel theoretical and numerical multiscale framework on the static, dynamic and buckling analysis of functionally graded porous beams reinforced with grapheme platelets (FGPB-GPLs). The functionally graded beam is composed of layers with different volume fractions of porosities and GPLs which is non-uniformly distributed along the thickness. The variations of mechanical property due to the porosity are based on closed-cell cellular solids under Gaussian Random Field scheme. The mechanical responses are obtained through multiscale analysis of the FGPB-GPLs. First, the elastic moduli of each nanocomposite layer are obtained by using Halpin-Tsai micromechanics model. The structural analysis on the functional gradation on beams is then conducted based on the kinematic assumptions of Timoshenko’s beam theory and Ritz method. This paper formulates the principle of virtual work and reciprocal theorem of work for the FGPB-GPLs. Then the principles of minimum potential energy and minimum complementary energy are derived and proved, the former of which is used for the derivation of the variational principles for the frequency of free vibration and critical load of buckling.

According to the proposed variational principles, the governing equations of static bending, free vibration and buckling can be obtained for the FGPB-GPLs as well as the corresponding boundary conditions. Finally, some numerical examples are presented and compared with the other solutions available in literatures to demonstrate the present theory.