Appendix from The role of symmetry in the post-buckling behaviour of structures

Symmetry plays an integral role in the post-buckling analysis of elastic structures. We show that the post-buckling response of engineering systems with given symmetry properties can be described using a preselected set of buckling modes. Therefore, the main original contribution of this paper is to prove the existence of these influential buckling modes and reveal some insights about them. From an engineering point of view, this study leads to the possibility of reducing computational effort in the analysis of large-scale systems. Firstly, symmetry groups for nonlinear elastic structural problems are discussed. Then, we invoke Curie's principle and describe the relationship between these groups and related pre-buckling and linear buckling deformation patterns. Then, for structural systems belonging to a given symmetry group, we re-invoke Curie's principle for describing the relationship between linear buckling modes and post-buckled deformation of the structure. Subsequently, we furnish a simplified asymptotic description which is obtained by projecting the equilibrium equations onto the subset of the most representative modes. As examples, classic bifurcation problems including isotropic and composite laminate panels under compression loading are investigated. Finally, accuracy and computational advantages given by this new approach are discussed.