Algebraic Graph Statics

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@article{VanMele2014,
    author  = "Van Mele, T. and Block, P.",
    title   = "Algebraic Graph Statics",
    journal = "Computer-Aided Design",
    year    = "2014",
    volume  = "53",
    number  = "",
    pages   = "104-116",
    month   = "",
    doi     = "10.1016/j.cad.2014.04.004",
    note    = "",
}

Abstract

This paper presents a general, non-procedural, algebraic approach to graphical analysis of structures. Using graph theoretical properties of reciprocal graphs, the geometrical relation between the form and force diagrams used in graphic statics is written algebraically. These formulations have been found to be equivalent to the equilibrium equations used in matrix analysis of planar, self-stressed structural systems. The significance and uses of this general approach are demonstrated through several examples and it is shown that it provides a robust back-end for a real-time, interactive and flexible computational implementation of traditional graphic statics.

Contributions

The paper brings together concepts and techniques from graph theory and matrix analysis of structures and present them in a unified framework for algebraic graphical analysis built around the reciprocal relation between the form and force diagrams of graphic statics.

A general scheme is discussed for a computational implementation of the presented approach that can be used as back-end of a real-time, interactive graphic statics application. Different steps of the implementation are illustrated using a Fink truss, which is a statically determinate structure that cannot be calculated directly with traditional graphic statics, because it contains crossing edges. Relevant algorithms are provided as code snippets.

The use of this framework for non-procedural graphic statics is demonstrated through four examples: a three-hinged trussed frame, an externally statically indeterminate three-bar truss, a geometrically constrained thrust line, defining its funicular loading, and a pre-stressed net. Finally, we briefly discuss the relevance of the presented approach for three-dimensional equilibrium methods, such as Thrust Network Analysis.