Symmetry Measure of Truss Structures with Disturbed Higher-order Symmetries

Abstract

This paper covers aspects of quantifying the symmetry of two- and three-dimensional elastic bar-and-joint structures. The concept of symmetry as a quantitative property instead of a binary question of 'yes' or 'no' is widely accepted and thoroughly investigated, for example, in molecular physics but also in engineering sciences, mainly in chemical engineering. Similarly to most of the articles written on this topic, our method is also based on the comparison of specific metrics of the analyzed structure and a reference one, i.e., which possesses the desired (perfect) symmetry. The deviation of the analyzed (imperfect) structure from the reference structure is quantified by one scalar. The novelty in our approach is that we consider not just the relative position of the nodes but also the normal stiffness of the truss members, even for structures with higher-order, i.e., polyhedral symmetries. For both geometric and material properties to be accounted for, the eigenvalues of the stiffness matrix were chosen as metrics. The difficulty lies in finding the reference structure which will be carried out based on energy principles.