Equilibrium stresses in frameworks via symmetric averaging

TL;DR

This paper introduces a symmetric averaging method for frameworks to analyze equilibrium stresses, develops a hierarchy of localized and extensive self-stresses, and presents algorithms for designing symmetric and non-symmetric structures with many self-stresses, aiding engineering applications.

Contribution

It proposes a novel symmetric averaging map for frameworks, explores self-stress behaviors under symmetry, and develops algorithms for designing structures with desired stress properties.

Findings

Symmetric averaging produces frameworks with desired symmetry properties.

Frameworks can have many self-stresses, both symmetric and non-symmetric.

Algorithms enable the design of structures with extensive self-stresses for engineering applications.

Abstract

For a bar-joint framework $(G,p)$, a subgroup $\Gamma$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $\Gamma$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.