Analysis of Symmetric Structures using Group Theory and the Force Method

by R D Kangwai

A symmetric structure is a structure that can be brought into a new position which is mechanically and geometrically identical to its original position. Symmetric structures commonly occur in nature and in engineering design due to their optimal load carrying abilities and their aesthetic appeal.

The symmetry properties of a structure allow any structural problem to be simplified into smaller subproblems using group theory. Group theory is the mathematical language best suited to the description of the symmetry properties of a structure. Using group theory the symmetry properties of even the most complex structure can be fully exploited.

Previous work applied group theory to the stiffness method of structural analysis in order to exploit symmetry properties of the structure and hence simplify the analysis of the induced displacements of the structure. This thesis provides a new application of group theory to the force method of structural analysis in order to simplify the full analysis of a symmetric structure. The equilibrium equations can be decomposed into a number of independent subsystems of equations, each with differing symmetry properties of the structure. The induced bar-forces, bar-elongations and displacements of the structure can be found by solving these subsystems of equilibrium equations. In particular, states of self-stress and inextensional mechanisms in each of these subsystems of equilibrium equations can be found which have the particular symmetry properties of the subsystem.