Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory

Abstract In the present investigation, the bending, buckling and free vibration responses of Timoshenko microbeams made of functionally graded materials (FGMs) are studied. To take size effect into account, the most general strain gradient elasticity theory is incorporated into the classical Timoshenko beam theory to develop a size-dependent beam model containing five additional material length scale parameters. The model accommodates the beam models based on the strain gradient theory (SGT), the modified strain gradient theory (MSGT), the modified couple stress theory (MCST) and the classical theory (CT) as special cases. By using Hamilton’s principle, the governing equations and corresponding boundary conditions are derived. Afterward, the governing equations and associated boundary conditions are discretized by employing generalized differential quadrature (GDQ) method. Selected numerical results are given to demonstrate the size-dependent mechanical characteristics of FGM microbeams. Moreover, a comparison between the various beam models on the basis of MCST, MSGT and CT are presented. It is observed that the critical buckling loads and natural frequencies predicted by the beam models based on MSGT and CT are the maximum and minimum values, respectively. By increasing the value of length scale parameter, the deflection curve of FGM microbeam tends to the curve obtained by CT.