Computational simplifications and observations on buckling modes of polygonal ring frames under symmetric compressive joint loads

The in-plane buckling of rigidly jointed Cnv-symmetric polygonal ring frames under a symmetric system of self-equilibrating compressive joint loads is considered. It is shown how group theory predicts the lowest modes of deformation without the need for any detailed calculations. Based on the classical slope-deflection formulation, the conventional stiffness matrix of the ring frame is assembled, and group theory then applied to reduce the conventional matrix to a series of independent symmetry-adapted matrices of smaller dimensions, each associated with a specific subspace of the problem. In this way, the calculation of eigenvalues and eigenmodes of the ring is significantly simplified. Analytical results of group theory are compared with the results of finite-element modelling, and excellent agreement achieved. Based on both analytical and numerical results, general observations on the buckling behaviour of regular polygonal ring frames are made. Overall, the effectiveness of group theory in providing insights on buckling behaviour, and in simplifying the actual computation of eigenvalues and mode shapes, is demonstrated.