Coupled instability of thin-walled columns based on multi-scale continuum mechanics

The authors have proposed a concept of micro-mode materials from the standpoint of multi-scale continuum mechanics, which is suitable to express warping deformation of thin-walled beams. In the present study, the concept of micro-mode materials is extended to express bifurcation buckling. Using the second Piola–Kirhhoff stress and hierarchical multi-scales for integration, the authors derive the balance equations for buckling. The obtained equations, including the newly discovered local-bimoment, are presented herein. These equations may enable the derivation of the bucking equations with every bucking mode. The application of the derivation of Euler buckling is also demonstrated using the concept of micro-mode materials.