Combinatorial Representations in Structural Analysis

Abstract

The work presented in this paper is part of a general research work, during which combinatorial representations based on graph and matroid theories were developed and then applied to different engineering fields. The main combinatorial representations used in this paper are flow and resistance graphs, and resistance matroid representations. The first was applied to the analysis of determinate trusses and the last two were applied to the analysis of indeterminate trusses. This paper gives a description of the representations and the methods embedded within them. The principal methods described in this paper are the conductance cutset method and the resistance circuit method that are mutually dual and are defined for both resistance graph and resistance matroid representations. The present paper shows that the known displacement and force methods are dual since they are the derivatives of the conductance cutset method and resistance circuit method, respectively. The importance of using combinatorial representations in structural mechanics is not only due to the intellectual insight provided by it, but also to its practical applicability. Some practical applications of the approach are reported in this paper, among them even a novel pedagogical framework for structural analysis.