Decomposition of the degenerate subspace of C3v-symmetric structural configurations

Physical systems exhibiting symmetry properties may be conveniently studied using the mathematics of group theory. In structural mechanics, group theory has been successfully employed to simplify problems of the bifurcation, stability, statics, kinematics and vibration of symmetric configurations of space frames, space truss domes, double-layer and triple-layer space grids, plates and cable-net systems. Besides significantly reducing computational effort, group theory affords deeper insights on structural behaviour, and a better understanding of complex structural phenomena. The key to group-theoretic simplification is the decomposition of the space of the problem into independent subspaces that are spanned by symmetry-adapted variables obtained by applying idempotents of the symmetry group on the normal variables of the problem. However, for degenerate subspaces of a symmetry group (i.e. subspaces associated with repeating solutions), the associated idempotents do not sufficiently decompose the problem. The aim of this paper is to present, for the C3v symmetry group describing the symmetry of a regular 3-sided polygon, a pair of algebraic operators that fully decompose such subspaces. Compared with existing group-theoretic formulations, these operators offer an alternative approach that is simpler and more suited to practical engineering computations, and that affords clearer insights on the physical characteristics of the structural system (such as type of symmetries within the degenerate subspaces). The validity of the operators is confirmed through comparisons with results of eigenvalue vibration and stability problems drawn from the literature.