Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory

A size-dependent nonlinear Euler–Bernoulli beam is considered in the framework of the nonlocal strain gradient theory. The geometric nonlinearity due to the stretching effect of the mid-plane of the size–dependent beam is considered here. The governing equations and boundary conditions are derived by employing the Hamilton principle. The post-buckling deflections and critical buckling forces of simply supported size-dependent beams are analytically derived. The derived results are compared with those of strain gradient theory, nonlocal elasticity theory and classical elasticity theory. It is found that the post-buckling deflections can be increased by increasing the nonlocal parameter or decreasing the material characteristic parameter. The high-order buckling deflections are more sensitive to size-dependent parameters than the low-order buckling deflections. Furthermore, the critical buckling force can be increased by decreasing the nonlocal parameter when the nonlocal parameter is larger than the material characteristic parameter, or increasing the nonlocal parameter when the nonlocal parameter is smaller than the material characteristic parameter.