The theories of graphs and matroids as a generalization for analysis of structures

This paper presents a general approach for structural analysis, based on graph and matroid theories, where the implementation to trusses is reported in this paper. Two general methods from graph network theory are introduced with their mathematical proofs, the resistance circuit and conductance cutset methods. These are proved to be dual. In addition, it is shown that when the conductance cutset method is applied to analysis of trusses, a method similar to the stiffness method is derived. After an introduction to matroids, it is shown that by using the duality property from matroid theory, the displacement and force methods are dual. The results reported in the paper are not only intellectually interesting, but have practical applications, some of which are mentioned in the paper. These include developing a new type of reasoning by analogy in Artificial Intelligence, based on the connections and properties of the graph and matroid representations. and a new direction for engineering education in which the students are first taught graph representations and then structural mechanics. This approach enables the students to understand structural mechanics in a more general prospective.