![]() ![]() Section 4 introduces the notion of self-reciprocity in Graphic Statics with several examples to illustrate the key concepts. Next, relationships between paired optimal frames, with emphasis placed on their corresponding reciprocal diagrams, are described in Section 3. ![]() The remainder of this paper is organized as follows: Section 2 provides a review of graphical methods and the notation used throughout this paper. 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. Examples of structures designed using graphical methods include the work of the brilliant structural designer Maillart (Zastavni 2008, 2010). Unfortunately, graphical methods progressively lost popularity with the development of other mathematical approaches. Graphical calculation methods were a common method to calculate the equilibrium of structures since the fundamental work by Culmann ( 1864) for the engineers of the late 19th century and early 20th century. Historically, graphical solutions for truss systems, cables and arches have been used for a variety of design problems. The parallel mapping of reciprocal lines has generally been used in practice.Īlthough Graphic Statics was a leading method of analyzing trusses in the late 19th and early 20th centuries, it is not commonly employed in the 21st century. Maxwell used a paraboloid of revolution reciprocal mapping which resulted in reciprocal lines that were perpendicular to each other in projection, while Cremona used a hyperboloid reciprocal mapping which resulted in reciprocal lines being parallel to one another. It is of interest that Maxwell and Cremona considered two-dimensional form diagrams (trusses) and their two-dimensional reciprocal force diagrams as the projection of three-dimensional polyhedra (one polyhedron for members and one polyhedron for forces). Maxwell’s work was interpreted and expanded by many others including Jenkin ( 1869), Culmann ( 1864), Cremona ( 1890), Wolfe ( 1921), and others as noted in Kurrer ( 2008). This approach had a major impact on the field of structural engineering. 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. The terms “frames”, “structures”, and “trusses” are used interchangeably within this paper and denote what are currently known as trusses. It should be noted that terminology has changed over the 150 years of work which form the basis of this paper. The first equation in Michell’s paper (stating that the difference between the total tension load paths and the total compression load paths is equal to a constant is inferred from Maxwell’s paper, which is also cited as the origin of reciprocal diagrams and Graphics Statics. Michell is referring to James Clerk Maxwell’s paper “On Reciprocal Figures, Frames, and Diagrams of Forces” (Maxwell 1870). Michell’s seminal paper “The Limits of Economy of Material in Frame-structures” (Michell 1904). ![]() 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. ![]() 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. 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. 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. This paper presents some remarkable relationships between discrete Michell frames and their corresponding reciprocal force polygons using Graphic Statics. 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. ![]()
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