Keywords: combinatorial representations, graph theory, skeletal structures, conjugate frames.

The use of graph theory in structural mechanics is well known in literature, such as Fenves [1], Spillers [2], Kaveh [3] and others. The idea behind these works was to represent the graph of a structure by a topological matrix, such as an incidence or neighbors matrix, and to proceed by means of matrix manipulations. Typical example for such an approach are the methods based on Roth's diagram [4] allowing to select sequences of manipulations to establish the required result.

Although based on the graph theory as well, the approach presented in this paper is different [7]. The graph corresponding to an engineering system is treated as a discrete mathematical model, the behavior of which is isomorphic to the behavior of the system. The initial effort of the presented approach has been entirely dedicated to develop these discrete mathematical models, referred to as Combinatorial Representations (CR), and to study thoroughly their embedded properties and interrelations [5,6]. Only after completion of this stage, the combinatorial representations were applied to represent different engineering systems.

The paper introduces the application of a special combinatorial representation, called Resistance Graph Representation (RGR), to structural systems. Accordingly, all the analysis process including the systematic derivation of the static equations is done upon the corresponding graph.

Two of the immediate applications of the approach are outlined in the current paper:

1. Derivation of known methods from methods embedded in the graph representation. The known method called the Mixed Variable Method is extended to deal with multidimensional systems and applied to analyze the graph corresponding to frames, beams and trusses, using its direct topological interpretation.

2. Derivation of known theorems in structural mechanics from theorems in graph theory – In this paper, the concept of dualism between graph representations is applied to systematically derive the known conjugate structure method, as is depicted in the following figure 8.1.

Many other results have been derived with in the research framework. Among them a systematic derivation of known energy methods and theorems, like the unit-force method and Betti's Law, through the fundamental Tellegen's theorem that was extended to multi-dimensional graph representations. On the basis of these results it is concluded that the new representation introduced in this paper paves a new avenue of insights and applications in structural engineering through the variety of methods and theorems embedded in graph theory.

1

S.J. Fenves, "Structural Analysis by Networks, Matrices and Computers", Journal of the Structural Division, ASCE, 92, 199-221, 1966.

2

W.R. Spillers, "Network Analogy for Linear Structures", Engineering Mechanics Division, proc. ASCE, 89(EM4), 21-29, 1963.

3

A. Kaveh, "Graphs and Structures", Computers and Structures, 40, 893-901, 1991. doi:10.1016/0045-7949(91)90319-H

4

J.P. Roth, "An Application of Algebraic Topology to Numerical Analysis: On the Existence of a Solution and the Network Problem", Proc. National Academy of Science, 41(7), 518-521, 1955. doi:10.1073/pnas.41.7.518

5

O. Shai, "The Multidisciplinary Combinatorial Approach and its Applications in Engineering", AIEDAM - AI for Engineering Design, Analysis and Manufacturing, Vol 15, No. 2, pp.109-144, April, 2001. doi:10.1017/S0890060401152030

6

O. Shai, "Combinatorial Representations in Structural Analysis", Computing in Civil Engineering, Vol. 15, No. 3, pp. 193-207, July, 2001. doi:10.1061/(ASCE)0887-3801(2001)15:3(193)

7

N. Ta'aseh, "Structural Analysis through Combinatorial Representations", M.Sc. thesis, Tel-Aviv University, Israel, 2002.

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