A nonlinear Timoshenko beam formulation based on strain gradient theory

Developed herein is a comprehensive geometrically nonlinear size-dependent microscale Timoshenko beam model based on strain gradient and von Kármán theories.The nonlinear governing equations and the corresponding boundary conditions are derived from employing Hamilton's principle.A simply supported microbeam is considered to delineate the nonlinear size-dependent free vibration behavior of the presented model.Utilizing the harmonic balance method, the solution for free vibration is presented analytically.The influence of the geometric parameters, Poisson's ratio, and material length-scale parameters on the linear frequency and nonlinear frequency ratio are thoroughly investigated.The results obtained from the present model are compared, in special cases, with those of the linear strain gradient theory, linear and nonlinear modified couple stress theory, and linear and nonlinear classical models; excellent agreement is found.It is concluded that the nonlinear natural frequency and nonlinear frequency ratio predicted by strain gradient theory are more precise than those from the other theories mentioned, especially for shorter beams.