Equations governing the vibrations and buckling of multilayered orthotropic graphene sheets can be expressed as a system of n partial differential equations where n refers to the number of sheets. This description is based on the continuum model of the graphene sheets which can also take the small scale effects into account by employing a nonlocal theory. In the present article a variational principle is derived for the nonlocal elastic theory of rectangular graphene sheets embedded in an elastic medium and undergoing transverse vibrations. Moreover the graphene sheets are subject to biaxial compression. Rayleigh quotients are obtained for the frequencies of freely vibrating graphene sheets and for the buckling load. The influence of small scale effects on the frequencies and the buckling load can be observed qualiatively from the expressions of the Rayleigh quotients. Elastic medium is modeled as a combination of Winkler and Pasternak foundations acting on the top and bottom layers of the mutilayered nano-structure. Natural boundary conditions of the problem are derived using the variational principle formulated in the study. It is observed that free boundaries lead to coupled boundary conditions due to nonlocal theory used in the continuum formulation while the local (classical) elasticity theory leads to uncoupled boundary conditions. The mathematical methods used in the study involve calculus of variations and the semi-inverse method for deriving the variational integrals.