COMBINATORIAL REPRESENTATIONS IN STRUCTURAL ANALYSIS

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. The work reported in this paper is part of a general research during which mathematical models based on discrete mathe- matics, called combinatorial representations (CR), were de- veloped, the properties in each and the connections between them investigated, and then applied to represent and solve var- ious engineering problems. The representations are based mainly on graph and matroid theories, whereas in the current paper two graph representations and one matroid representa- tion are used. By working in this approach, interesting results have been achieved, a few of which are mentioned below: • A general perspective on different engineering fields was obtained when the same representation was applied to dif- ferent problems. For example, the resistance graph rep- resentation was applied to analyze both mass-spring- damper systems and indeterminate trusses (Shai and Preiss 1999b). • New connections between different engineering fields have been achieved by using the connections between the combinatorial representations. For example, a dualism connection between determinate trusses and mechanisms was derived based on the dualism connection between their corresponding representations of flow and potential graphs. • Known theorems and methods have been derived from the theorems and methods inherent in the representations. For example, based on a theorem inherent in the resis- tance graph representation, called Tellegen's theorem, Betti's Law and the known method for analyzing the dis- placement of truss joints have been derived (Shai 2001b). This paper is a continuation of the approach and it uses CR to give a global perspective on structural analysis and dem- onstrates it on trusses. It also shows that the known methods, displacement and force, can be derived from the two known methods embedded in the resistance graph representation, con- ductance cutset method (CCM) and resistance circuit method (RCM), respectively. Furthermore, the present paper shows that this approach enables revealing of the connections be- 1