Graphene (G) is a two dimensional plate-like nanostructure with remarkable and extraordinary physical properties. The intrinsic mechanical properties make graphene one of the most applicable structures in different nano engineering systems as sensors or reinforcements. In the present study, vibration and buckling analysis of multilayered graphene sheets (MLGSs) surrounded by two-parameter elastic foundation are investigated using the finite element method and the Ritz functions. Since the classical continuum theories cannot take account of the small scale effects on the mechanical behavior of structures at nano scale, these quantum effects are introduced through the modified size dependent Eringen’s nonlocal continuum theory. Employing the nonlocal constitutive equation for planar orthotropic structures and adopting classical plate theory (CLPT) assumptions, the load-displacement relation associated with each graphene layer of a MLGS is obtained. The Lennard-Jones and Winkler-Pasternak models are used to simulate the interlayer interactions and foundation pressure, respectively. The intensity of Van der Waals forces between each two adjacent and non-adjacent layers is calculated by a relation obtained from linearization of the Lennard-Jones interaction equation. Also the springy and shear effects of elastic medium are considered by the two parameter Winkler-Pasternak elastic model. Decoupled forms of governing vibration and buckling equations based on the nonlocal continuum theory are obtained for the current extended model of embedded MLGS. Variational principle is introduced via an inverse approach using the governing equations to make a basis for numerical analysis of MLGSs and other plate like nanostructures. Finite element solutions for natural frequencies and critical in-plane loads of embedded MLGS are obtained through minimizing the introduced nonlocal potential term. There is also a study on the application of the Ritz functions in vibration and buckling analysis of elliptical nanoplates on elastic foundation using a similar approach. The Ritz functions eliminate the need for mesh generation and thus large numbers of degrees of freedom arising in discretization methods such as finite element method (FEM). Small scale effects on natural frequencies and critical buckling loads of MLGS and elliptical nanoplate are investigated for different nonlocal parameters, lengths, aspect ratios, inter-layer interactions, foundation parameters, mode numbers and boundary conditions. It is shown that both natural frequencies and buckling loads depend obviously on the non-locality of the nano-plates, especially when the dimensions decrease. Also, it is seen that classical continuum mechanics overestimates the natural frequencies and buckling loads of MLGS which leads to deficient in precision required for frequency and stability design of these load bearing structures for application in different nano-mechanical systems. Consequently, it is seen that small scale effects play an important role in dynamic behavior of nanostructures and should be considered via size dependent theories such as nonlocal elasticity theory. The nonlocal continuum mechanics and finite element model developed in this work provide designers efficient tools to predict mechanical responses of plate-like nanostructures in different nano devices. Finally, it should be noted that current FE and Ritz methods can be applied for analyzing nanoplates with arbitrary geometries, variations in thickness and boundary conditions. Keywords Multilayered graphene sheet, Elliptical nanoplate, Small scale effects, Nonlocal elasticity theory, ...