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.

In this work, torsional vibration of nanorods with torsional elastic boundary conditions is presented via non‐local elasticity theory. The present model developed based on non‐local elasticity theory gives the opportunity to interpret size effect. Two torsional elastic springs are attached to a nanorod at both ends. A mathematical transformation known as ‘Stoke transformation’ is utilised to work out the Fourier series for the nanorods with torsional restraints. A coefficient matrix including torsional coefficients is determined by using non‐local boundary conditions. A comparison is performed to validate numerical simulations with those given in the literature and the results agree with each other exactly. The non‐local effects of torsional end restraints on the free torsional vibration response are investigated for both deformable and rigid boundary conditions.

The buckling of elastically restrained embedded microbeam under axial compression load is researched. The effects of small size, axial compression load and surrounding elastic medium are taken into account at the same time. Winkler elastic foundation approach is used to simulate the interaction between microbeam and elastic medium. Fourier sine series is employed for the simulation of microbeam deflections. A coefficient matrix is obtained with the aid of applying Stokes' transformation to corresponding boundary conditions. The buckling characteristics of elastically restrained embedded microbeams are investigated in some numerical examples. There are very good agreements between this study and the previous results indicating the validity of the presented method.