Abstract
This paper unites two major legacies of James Clerk Maxwell’s ground-breaking paper, “On Reciprocal Figures, Frames, and Diagrams of Forces” (Maxwell, Philos Mag 26:250-261, 1864; Edinb Roy Soc Proc 7:160–208, 1870): (i) the fundamental theorem used by Michell (Philos Mag 8(47):589–597, 1904) to derive trusses of least weight and (ii) reciprocal frames. This paper presents some remarkable relationships between discrete Michell frames and their corresponding reciprocal force polygons using Graphic Statics. Several examples are given to illustrate the notions of duality and self-reciprocity in these diagrams, with particular emphasis placed on discrete optimal benchmark structures. For a given connectivity of nodes, Graphic Statics provides all of the information needed to determine the total load path of the structure in the form and force diagrams. Because the form and force diagrams are reciprocal, in the course of finding one minimum load path structure, a second minimum load path structure is also found. These observations between the corresponding form and force diagrams are generalized for discrete cantilever Michell frames, and several comments on the extensions of this work are included.
Highlights
The first equation in Michell’s paper
Each of the reciprocal diagrams in Graphic Statics consists of a set of points, or nodes, straight lines interconnecting all the nodes, and polygons defined by those lines
For some cases of paired trusses, it can be shown that the geometry of the reciprocal force diagram is the exact same as the member diagram; it can be said that these types of structures are self-reciprocal
Summary
Michell is referring to James Clerk Maxwell’s paper “On Reciprocal Figures, Frames, and Diagrams of Forces” (Maxwell 1870). J. Macquorn Rankine’s paper, “Principle of the Equilibrium of Polyhedral Frames,” (Rankine 1864) and extends the work to show how certain trusses have reciprocal diagrams which represent the forces in the trusses. Macquorn Rankine’s paper, “Principle of the Equilibrium of Polyhedral Frames,” (Rankine 1864) and extends the work to show how certain trusses have reciprocal diagrams which represent the forces in the trusses This approach had a major impact on the field of structural engineering. This paper will bring together these two threads of Maxwell’s paper and show that some discrete Michell trusses (frames) have a very remarkable relationship to the reciprocal force polygons as determined by Graphic Statics.
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