Out-of-Plane Buckling of Microstructured Beams: Gradient Elasticity Approach

This paper deals with the lateral-torsional buckling of elastic microstructured beams. A linear gradient elasticity approach is presented. The elastic constitutive law is linear for the generalized beam model, expressed with first- and second-order moment variables in a sixth-order dimensional stress-strain space. The boundary conditions (including the higher-order boundary conditions) are derived from application of the variational principle applied to second-grade elastic models. The effect of prebuckling deformation is taken into consideration based on the Kirchhoff-Clebsch theory. Some analytical solutions are presented for a hinged-hinged strip beam. The lateral-torsional buckling moment is sensitive to small length terms inherent in the microstructured constitutive law. It is shown that the gradient parameter tends to increase the critical lateral-torsional buckling moment, a conclusion different from the one obtained with an Eringen-based nonlocal model. This tendency is consistent with that observed for the in-plane stability analysis (based on a second-grade model), for the lateral buckling of a hinged-hinged axially loaded column. The effect of higher-order boundary conditions is also studied with the vanishing of the couple stress components. It is numerically shown, through the resolution of a sixth-order differential equation, that the lateral-torsional buckling moment is not significantly affected by the choice of higher-boundary conditions for sufficiently small ratio between the characteristic length and the total length of the structure. We finally notice that the introduction of the warping effect can be understood as a special case of gradient elasticity theory.