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Maxwell's reciprocal diagrams and discrete Michell frames

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. © 2013 The Author(s).