Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method

Abstract In the present manuscript, a non-classical beam theory is developed for the static and nonlinear vibration analysis of microbeams based on a three-layered nonlinear elastic foundation within the framework of the modified couple stress theory and Euler–Bernoulli beam theory together with the von-Karman’s geometric nonlinearity. This non-classical beam model incorporates the length scale parameter which can account for the small size effect. By using the Hamilton’s principle, the equations of motion and the boundary conditions of the problem are derived. The nonlinear partial differential equation governing the motion of the system is reduced to the nonlinear ordinary differential equation with the help of the Galerkin discretization technique. He’s variational method is then applied for the first time to obtain approximate analytical expressions for the nonlinear frequency of the microbeams with pinned–pinned and clamped–clamped end conditions. Static analysis is also performed for uniformly distributed load. Some illustrative numerical examples are presented in order to investigate the influences of the length scale parameter and the stiffness coefficients of the nonlinear foundation on the static deflection and the ratio of nonlinear frequency to linear frequency (the nonlinear frequency ratio). Comparison studies are also performed to verify the present formulation and solutions. Close agreement is observed.