Maxwell’s reciprocal diagrams and discrete Michell frames

“Maxwell” is the first word in A. G. M. Michell’s seminal paper “The Limits of Economy of Material in Frame-structures” (Michell 1904). Michell is referring to James Clerk Maxwell’s paper “On Reciprocal Figures, Frames, and Diagrams of Forces” (Maxwell 1870). 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.

It should be noted that terminology has changed over the 150 years of work which form the basis of this paper. The terms “frames”, “structures”, and “trusses” are used interchangeably within this paper and denote what are currently known as trusses.

Maxwell starts with concepts from W. 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. This approach had a major impact on the field of structural engineering.

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). 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 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. The parallel mapping of reciprocal lines has generally been used in practice.

Although 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. Historically, graphical solutions for truss systems, cables and arches have been used for a variety of design problems. 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. Unfortunately, graphical methods progressively lost popularity with the development of other mathematical approaches. Examples of structures designed using graphical methods include the work of the brilliant structural designer Maillart (Zastavni 2008, 2010).

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. The remainder of this paper is organized as follows: Section 2 provides a review of graphical methods and the notation used throughout this paper. Next, relationships between paired optimal frames, with emphasis placed on their corresponding reciprocal diagrams, are described in Section 3. Section 4 introduces the notion of self-reciprocity in Graphic Statics with several examples to illustrate the key concepts. Generalized observations on discrete cantilever Michell frames are provided in Section 5 followed by some concluding remarks and extensions of this work.

Graphical methods have been used for centuries to analyze and design a variety of structures. In this section, we give a brief historical review of such methods and introduce the methodology and notation used throughout this work.

As shown in Stromberg et al. (2012), the minimal volume problem for a given set of balanced forces and nodal connectivity can be written as follows: where x is a vector of design variables containing the nodal coordinates of the unloaded nodes, σ is a constant that represents the allowable stress, L represents the lengths of the members and P the internal forces of the members. We note that if the structure is optimal, the assumption of constant stress is valid. The quantity, \(\sum |P|\cdot L\), represents the total load path of the structure; therefore, minimizing the total load path of a structure corresponds directly to minimizing its volume, or weight. For a given connectivity of nodes, Graphic Statics has all of the information needed to determine this quantity: The form diagram gives the length, L, of each member and the force diagram gives the force, P, in each member.

A minimal load path structure can be found by changing the design variables, or in this case, the nodal locations in the force diagram subject to the rules of reciprocal diagrams such that \(\sum |P|\cdot L\) is minimized. In Graphic Statics, the form diagram and the force diagram are, as Maxwell noted, reciprocal. This leads to the remarkable observation that the force diagram could also represent the geometry of another optimal truss with its own external loads. In the course of finding one minimum load path structure, a second minimum load path structure is also found. In addition, the forces in the second minimum load path structure are represented by the length of the lines in the original structure. These two trusses are both discrete optimal trusses for their external loads in the sense that they are solutions to (1).

In Figs. 2 and 3, the structures (form diagrams) given on the left are both different discrete optimal structures (Hemp 1973; Mazurek et al. 2011). The supposition that the force diagram in Fig. 2 is also the geometry of a second discrete optimal truss with its own loading is explored in Fig. 3. The geometry of the truss in Fig. 3 (the form diagram) is taken from the force diagram for Fig. 2.

Now that two optimal truss geometries have been found, the remaining task is to find the external loads that correspond to a truss represented by the geometry of the force diagram. The external forces for this second truss are found by closing the “open” polygons of the original form diagram in a continuous manner. The vector formed by closing an open polygon in the original form diagram is the external force that would be applied to the node in the force diagram that corresponds to the original open polygon. Closing the open polygons provides the proper external forces for the dual structure because it closes the force polygon at the node which corresponds to the open polygon in the original form diagram.

It is clear that the form and force reciprocal diagrams in Figs. 2 and 3 are interchangeable. In order to make the relationships more clear, Bow’s notation in Fig. 3 is reversed.

For the problem in (1), the lengths are taken from the truss on the left and the internal forces from the truss on the right, though the reverse (i.e. internal forces from the left diagram, lengths from the right diagram) would result in the same optimal solution. The advantage to using Graphic Statics for this class of optimal problems is that it provides all of the information about the loads and the pathsin a graphical manner.

Another example of paired or dual discrete trusses is given in Fig. 4, where the discrete truss on the left (form diagram) is the minimum load path structure for a given set of forces and nodal connectivity (Prager 1970). The reciprocal force diagram in the middle not only represents the forces of truss but also the geometry of another optimal truss, where the external forces for this optimal structure can also be found by closing the open polygons in the form diagram. For example, the external force applied at node a in Fig. 4b is given by closing the “open” polygon, A, or drawing a line from joint A-3-C-A to joint A-B-1-A. Figure 4c shows the dual truss with the external loads given by closing all of the polygons in (a).

The Appendix has a chart showing some dual discrete trusses along with the Bow notation. This chart graphically shows both the geometry and the forces for each pair of optimal trusses.

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. An example of a self-reciprocal structure is the optimal discrete Michell truss taken from Chan (1960) and Mazurek et al. (2011) shown in Fig. 5. This self-reciprocity can be observed for an unbounded cantilever with an equal number of members at each support when there is a single applied load acting parallel to a line drawn through the supports and the reactions intersect the tip of the cantilever. Figure 6 shows another example of a self-reciprocal frame for the three-point problem. We note here that the notion of self-reciprocity can also apply for non-symmetric structures. The reasons for this remarkable self-reciprocity for certain discrete Michell Frames is explained in Section 5.

Even though the form and force diagrams of self-reciprocal discrete Michell Frames are the same, the mapping between reciprocal lines is complex. For example, the forces in the outside chords of the Michell cantilever in Fig. 5 are proportional to the lengths of the lines in the fans at the supports. It is noteworthy that self-reciprocal Michell Frames provide both the geometry and the forces in one diagram if one knows how to “read” the structure. As more and more members are added to the discrete trusses, the solution approaches the continuum solutions of Michell, Chan, and others.

Discrete optimal trusses for 3-point or 3-force problems have been obtained in Mazurek et al. (2011) and Mazurek (2012). The 3-point problem can be described by finding the optimal (minimum load path) structure for two points of support and a single point load. The 3-force problem is when the optimal structure is found for three loaded points (the three point loads must be in equilibrium). It has been shown in Mazurek et al. (2011) and Mazurek (2012) that the geometry of the optimal discrete trusses is fairly regular regardless of the number of elements utilized in the solutions. The optimal truss geometry can also be defined using only two parameters (i.e. angles \(\gamma \) and one of \(\psi _{\alpha }\) or \(\psi _{\beta }\)) for a 3-point problem (see Fig. 7a) or three parameters (i.e. angles \(\gamma \), one of \(\psi _{\alpha }\) or \(\psi _{\beta }\) and one of \(\lambda _{\alpha }\) or \(\lambda _{\beta }\)) for the 3-force problem (see Fig. 7b).

The configuration of the members at every node of these trusses is exactly the same and, therefore, they can be constructed through a process of repetition. Even though in the examples shown, the externally loaded or constrained nodes for 3-point and 3-force problems are in the same locations and the tip load is applied in the same magnitude and direction for both problems, the optimal trusses have different geometries (see Fig. 7). This is because in the 3-point problem, the directions of the reactions are not prescribed, and they vary depending on the structure, whereas in the 3-force problem, all of the forces are given. Since the 3-point problem has one less parameter constrained (the direction of the reactions), the volume of the truss is always smaller or equal to the one obtained for the 3-force problem.

In the creation of reciprocal diagrams for a 3-point or 3-force optimal truss (Fig. 8), certain geometrical similarities with its original form diagram can be observed. We note that these geometric rules may not hold if additional constraints are incorporated into the optimization problem, such as a bound on the feasible design space. For unconstrained 3-point or 3-force problems, from the geometrical dependencies of the lines, it can be concluded that, except for the following three differences, the geometry of the force diagram is the same as the corresponding form diagram: With the differences listed above, a certain family of discrete optimal structures can be described where force diagrams are directly proportional to their form diagrams. Namely, if we select a discrete optimal truss with \(n_{\alpha }=n_{\beta }\) for a problem that produces \(\lambda _{\alpha }=\chi _{\alpha }\iff \lambda _{\beta }=\chi _{\beta }\Rightarrow \psi _{\alpha }=\psi _{\beta }=90^{\circ }\), the force diagram will be self-reciprocal. Also, because \(\psi _{\alpha }=\psi _{\beta }\) and \(\lambda _{\alpha }=\chi _{\alpha }\), the acting force is parallel to the line between nodes of the reacting forces. From the direct proportionality, it can be shown that the lines of reaction forces in these trusses intersect at the tip node of the truss, as the reaction at the \(\alpha \)-support (node A-3-6-C-A) can be expressed by vector ca in the force diagram and the reaction at the \(\beta \)-support (node C-8-7-B-C) by vector bc (see Fig. 9).

Even more interesting is the observation that if we select a discrete optimal truss with \(\lambda _{\beta }=\chi _{\alpha }\) and \(\lambda _{\alpha }=\chi _{\beta }\), the geometry of the force diagram will be proportional to a mirror of the form diagram (see Fig. 10). From Mazurek et al. (2011) we also know that the criterion \(\lambda _{\beta }=\chi _{\alpha }\) defines solutions where the acting forces are located along their line of action producing structures of least volume. It is quite remarkable that the two diagrams come together so nicely for these optimal trusses. Thus, the ratio of the force in member 2-4 to the length of member B-3 is the same as the force applied at B-1-A-B to the distance between C-A-2-4-C and C-5-B-C. The same ratio exists between the force in member C-5 and the length of member A-1, and so on.

Maxwell’s work on structural reciprocity and load paths in 1864 and 1870 launched two trajectories in the theory and design of trusses that this paper attempts to reconnect. The first trajectory was that of reciprocal diagrams and Graphic Statics. This aspect was quickly embraced and enlarged by Culmann, Cremona and others and widely used by engineers as a practical method of designing and analyzing trusses. Michell, in his landmark paper Michell (1904), starts with another aspect of Maxwell’s 1870 paper and develops optimal frames of minimal total load path. The total load path of a structure is the result of Graphic Statics with one diagram representing the paths (lines of action) and the reciprocal diagram representing the forces (loads). This paper connects these two legacies of Maxwell through the application of Graphic Statics to discrete optimal trusses. It results in some observations worth noting: As an extension of this work, the design of optimal bridges and cable systems using Graphic Statics and Rankine’s Theorem are currently under investigation by the authors.