In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al₂O₃) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e₀a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

Size-dependent stability analysis of nanobeams under axial loading is investigated by strain gradient elasticity. An effective general model is proposed for different deformable boundary conditions by combining Fourier sine series and Stokes’ transformation. The theoretical model presented here bridges the gap between clamped and pinned-pinned supporting conditions, which is of great significance for the application of the strain gradient small scale effects to nanobeam. Analytical solutions are obtained and comparisons with results obtained by the researchers for rigid boundary conditions are carried out. The outcomes can be useful for the optimization and design of nano-sized beams and provide new benchmarks for nanomechanical analyses.