Maxwell and Rankine reciprocal diagrams via Minkowski sums for two-dimensional and three-dimensional trusses under load

Abstract

A method is presented for computing and displaying the Maxwell (two-dimensional) or Rankine (three-dimensional) diagram reciprocal to a truss under load. Reciprocals are constructed via the two-dimensional Airy stress function or its three-dimensional analogue. A linear combination of the coordinates of each original node and the coordinates of its reciprocal object leads to the finished diagram. This is related to an underlying Minkowski sum of the polyhedral stress functions of the original and reciprocal diagrams. Some regions of the resulting combined diagram have units of length $L,$ and other regions use units of force $F .$ These regions are connected in a consistent manner, in two dimensions by rectangular areas and in three dimensions by prismatic volumes, each having work units of dimension $F L,$ and which give a visual representation of the $F \cdot L$ terms in Maxwell’s load path theorem. A linear combination weighted in favour of the original leads to a diagram where original bars are augmented by a thickness (in two-dimensional) or a polygonal cross-sectional area (in three-dimensional). The diagrams thus give an appealing visual representation of the material required for a constant stress design. Although the method gives a complete description of loaded trusses in two dimensions, in three dimensions, it is currently only applicable to those trusses which possess a Rankine reciprocal.

Keywords

Airy stress function graphic statics Maxwell reciprocal diagrams Minkowski sum projective geometry Rankine reciprocal diagrams

Introduction

Graphic statics has long been popular in architectural engineering, as it gives an appealing visual representation of the forces carried by the members of a loaded two-dimensional (2D) truss. The underlying geometry is that dictated by the rules for constructing Maxwell¹ reciprocal figures. There are two possible constructions for reciprocal diagrams, as forces may be drawn either parallel or perpendicular to their associated bars. In 2D, the difference is of little consequence. The Cremona convention draws bar forces parallel to bars, and the Maxwell convention draws bar forces perpendicular to bars. Cremona and Maxwell reciprocal diagrams are thus identical up to a 90° rotation. For three-dimensional (3D) trusses, the difference is more significant. In 3D, the Cremona² representation again has forces as lines parallel to the bars. In the perpendicular construction of Rankine,³ forces are represented by polygons perpendicular to each bar. This article concerns only the perpendicular constructions, these being Maxwell reciprocals in 2D and Rankine reciprocals in 3D.

A Rankine diagram, if it can be constructed, may be complicated and difficult to understand by visual inspection. Maxwell⁴ noted that ‘the mechanical interest of reciprocal figures in space rapidly diminishes with their complexity’. However, by combining the advantages of modern computer graphics that were unavailable to Maxwell with the Minkowski sum construction of this article, the accessibility and understanding of reciprocal figures in space may be increased.

Given the absence of elasticity from the construction that will be developed here, the method aligns itself more closely with plastic design than with the elastic structural analysis. It is no coincidence that the resulting figures are strongly similar to the strut-and-tie diagrams used in the lower bound plastic design of deep beams, corbels and the like,⁵ and which now underpin much of modern masonry design and analysis. Moreover, the correspondence between the (equilibrium) states of self-stress of a 2D truss and the rotations of (compatible) yield line plastic collapse mechanisms of a slab⁶ suggests that the technique may find applications beyond the design of trusses.

This article draws heavily on the pioneering 19th-century works of Maxwell,^1,4 Rankine,³ Cremona² and many others and on more recent extensions, particularly by Crapo and Whiteley,^7,8 Akbarzadeh et al.^9,10 and Pottman et al.¹¹

The procedure adopted here for constructing 2D reciprocals is that outlined in McRobie et al.¹² and Mitchell et al.¹³ in this volume. It involves taking the normals to the plane faces of polyhedral Airy stress functions.

A simple illustration

Figure 1 (after Figure 1 of Maxwell¹) shows a six-bar truss and its reciprocal, these being 2D projections of dual tetrahedra. (Strictly, duality here refers to a polarity with respect to a paraboloid of revolution, as described in Maxwell.⁴) One figure (the ‘form’ diagram) may represent bar lengths, while lengths in the other figure (the ‘force’ diagram) represent the bar forces of an equilibrium state of self-stress. Via the duality of 3D projective geometry (nodes, lines, areas) of one correspond, respectively, to (areas, lines, nodes) of the other.

Figure 1.

A simple truss and its reciprocal (after Maxwell 1864 Figures 1 and I).

Each diagram may be cut apart into its component areas (see Figure 2). This gives four triangles for each figure (the three smaller triangles and the overall surrounding triangle). These eight pieces may be rearranged as follows. At each corner of the large original triangle $p q r,$ we place the triangle that is reciprocal to that corner node, that is, at node $2 (= q a r)$ , we add the reciprocal triangle $Q A R$ (and similarly for nodes $3 (= r b p)$ and $1 (= p c q)$ ).

Figure 2.

A Maxwell–Minkowski diagram that combines the reciprocal pair of Figure 1.

Consider bar $p$ . At each end, it now has a line $P$ perpendicular to $p$ , these being lines on the edges of the reciprocal triangles. A copy of $p$ can connect the far ends of the two $P$ lines, giving a rectangle $p P p P$ (which we abbreviate to $p P$ ) connecting the reciprocal triangles $R B P$ and $P C Q$ that now lie at either end of $p$ . Similarly, rectangles $q Q$ and $r R$ connect the reciprocal triangles at the ends of $q$ and $r$ . We thus obtain a hexagon bounded by $p C q A r B$ , containing the original large base triangle $p q r$ , the three smaller reciprocal triangles ( $R B P$ , $P C Q$ , $Q A R)$ and three rectangles ( $p P$ , $q Q$ , $r R$ ).

The construction may be repeated with the remaining polygons, but with the form or force roles reversed. The large reciprocal triangle $A B C$ is placed at the centre of a new diagram, and the smaller original triangles $a q c$ , $b r a$ and $c p b$ are attached to its corners, connected by three rectangles $a A,$ $b B$ and $c C$ . This new diagram is also a hexagon with boundary $p C q A r B$ . The two diagrams can be placed one on top of the other and joined along their common boundary, giving a Maxwell–Minkowski diagram (Figure 2).

Remarks

This simple construction leads immediately to a number of rather novel perspectives:

The diagram has the unusual property that some regions (e.g. the original triangle $p q r$ ) use dimensions of length L, $L$ while other regions (e.g. the reciprocal triangle $A B C$ ) use dimensions of force $F .$ Dimensional consistency is, however, maintained via the rectangles ( $p P$ , etc.) that join them, these having work units of force times length $(F L)$ .

The areas of the rectangles correspond to the $F \cdot L$ terms in Maxwell’s load path theorem,¹⁴ which states that $\sum F_{C} . L = \sum F_{T} . L$ (where $F_{C}$ and $F_{T}$ are the compression and tension forces, respectively). The validity of the theorem is readily apparent from the diagram. The area on each sheet of the diagram is that of the surrounding hexagon. Each hexagon contains versions of the original and reciprocal diagrams. These may occur as the single larger triangle or as the sum of the three smaller triangles that would cover it. Trivially then, the total area of the rectangles on each sheet must equal the area of the hexagon minus the areas of the original and reciprocal diagrams. That is, the total area of the rectangles is the same on each sheet. In symbols $a A + b B + c C = p P + q Q + r R$ , this being the statement of Maxwell’s theorem for this case.

The aspect ratio of each rectangle (e.g. $A / a$ ) gives a diagrammatic representation of the (force divided by length) tension coefficients of rigidity theory.

The construction can be generalised to combine diagrams which are only topologically (rather than geometrically) reciprocal. For example, corresponding lines in the two diagrams need not be perpendicular. In that case, the rectangles of the Maxwell–Minkowski combination become parallelograms. These provide a visual representation of the various statements of virtual work. For example, one figure may represent bar forces $F$ , while the second figure shows infinitesimal nodal displacements $δ x$ (bar forces $F$ should be drawn perpendicular to original bars $L$ ). In that case, the areas of the parallelograms that glue together the pieces of the equilibrium and compatibility diagrams represent the $F \cdot δ x$ terms of virtual work. Given the centrality of virtual work to structural engineering, this novel ability to visualise its components appears to have numerous potential applications, although these are not pursued here.

It is apparent that if the original and reciprocal diagrams are arbitrarily and separately rescaled, a Maxwell–Minkowski diagram can still be constructed. If the pieces of one diagram are rescaled, the connecting work rectangles simply rescale in one direction such that all the pieces are still sensibly connected. This leads to the construction of an interesting higher dimensional object, which can be loosely described as $α$ original $+ (1 - α)$ reciprocal, where $α$ varies from 0 to 1. The $α = 0$ end of this object is thus the reciprocal diagram, while the $α = 1$ end is the original diagram. Any intermediate value of $α$ gives a Maxwell–Minkowski diagram with rectangles whose aspect ratios scale according to the choice of $α$ . Animations that vary $α$ thus reveal how the original and reciprocal diagrams may be transformed into one another.

The traditional interpretation of the force diagram is that lengths of lines denote magnitudes of forces. An equivalent perspective is to consider those lengths to be the bar thicknesses that would be required in a constant stress design. (In 2D, ‘stress’ means the tension force per unit distance transverse to the bar.) The usefulness of this notion becomes apparent when applied to a Maxwell–Minkowski diagram where the scaled original diagram dominates (such as $α \approx 0.95$ ). The work rectangles then become visual representations of the bar cross-sectional areas that would be required for a uniform stress design.

The final point is perhaps the key to the usefulness of the Minkowski construction when applied to Rankine reciprocals of 3D trusses. The Rankine diagram, where areas represent forces, is often complicated, and the physical meaning of the many shapes is difficult to discern visually. However, for the $α \approx 0.95$ Rankine–Minkowski diagram, the polygonal areas representing forces manifest themselves as bar cross-sectional areas of the uniform stress solution. We shall explore this in detail later.

Construction details

In cases more general than the simple example, construction is readily implemented by addition of (scaled) original and reciprocal coordinates. That is, at original node $i,$ we extract the set of reciprocal nodes $j (i)$ that define the cell reciprocal to the original node $i$ . For 2D trusses, the cell is a polygon, and for 3D trusses, it is a polyhedron. The set of coordinates

α x_{i} + (1 - α) x_{j (i)}^{*}

(1)

then defines a cell. Nodes on corresponding sides of such cells may then be connected by lines parallel to the original members.

As an alternative to the last step, the cells may be connected by reversing the process, taking a reciprocal node $j$ and finding the original nodes $i (j)$ on the original cell reciprocal to node $j$ . The set of coordinates

α x_{i (j)} + (1 - α) x_{j}^{*}

(2)

then defines a cell which connects the cells of the first construction, thereby completing the diagram (including the rectangles or prisms).

Alternatively again, one may choose to draw only the rectangles or prisms that connect the cells, and this implicitly draws the scaled original and reciprocal cells.

Reciprocal diagrams were constructed via the method described in Mitchell et al.¹³ and McRobie et al.¹² That is, drawings in 2D $(x, y)$ were lifted to polyhedra in $(x, y, h),$ this corresponding to an Airy stress function $h$ . Normals to faces were then based at $(0, 0, 0)$ and intersected with the reciprocal 2D plane $h = 1$ to define reciprocal vertices. This creates the Maxwell reciprocal.

This procedure readily generalises to the 3D Rankine construction. That is, drawings in 3D $(x, y, z)$ are lifted to polytopes in four-dimensional (4D) $(x, y, z, h)$ , this corresponding to a stress function $h$ . Normals to cell volumes are then based at $(0, 0, 0, 0)$ and intersected with the reciprocal 3D space $h = 1$ to define reciprocal vertices. This creates the Rankine reciprocal.

The Minkowski sum

Given two sets $A$ and $B$ of vectors, their Minkowski sum is defined as the set $A + B = {a + b | a \in A, b \in B}$ . To create the Maxwell–Minkowski diagram of the simple example of section ‘A simple illustration’, we take the Minkowski sum, not of the 2D polygons in the diagrams but of their 3D polyhedral stress functions. Let there be an upward-pointing tetrahedral stress function over the original diagram of Figure 1, whose dual is a downward-pointing tetrahedron over the reciprocal diagram (see Figure 3). The Minkowski sum of these two tetrahedra is a polyhedron, swept out by one tetrahedron when a chosen base point is moved to all points in the other tetrahedron. The edges of this larger polyhedron, when projected back to 2D, define the Maxwell–Minkowski diagram of Figure 2.

Figure 3.

The Minkowski sum of two tetrahedra.

The combined polyhedron is a cantellated tetrahedron, which is topologically a cuboctahedron. Indeed, the overall procedure may be viewed as cantellation of the original polyhedral stress function. Loosely speaking, a cantellation of a polyhedron involves bevelling its edges and truncating its vertices. A vertex is truncated such as to expose the polygonal face reciprocal to that vertex, and the bevelling is such as to match the edges of the reciprocal polygons that were created by the truncation. Cantellation of the polyhedral stress function thus gives another geometric method of creating reciprocal diagrams since it leads directly to the combined Maxwell–Minkowski polyhedron, from which the reciprocal diagram may be extracted.

For Rankine–Minkowski diagrams, the procedure is similar. The 3D original and reciprocal polyhedra are lifted to stress functions in 4D where the Minkowski sum is performed and then the combined polytope is projected back to 3D.

Whether the diagrams that we are calling Maxwell–Minkowski and Rankine–Minkowski diagrams are indeed the projection of Minkowski sums in more complicated cases (such as those involving non-convex polyhedra or polytopes) is a matter for further discussion. It does not affect the validity of the construction, which is defined by the summation of original and reciprocal coordinates (equations (1) and (2)).

Examples

In each of the following examples, Maxwell–Minkowski diagrams are given for the five cases of $α = {0, 0.05, 0.5, 0.95, 1} .$ Diagrams are presented with $α = 1$ at top left, moving clockwise to $α = 0$ at bottom left. Although form and force are interchangeable, we shall tend to consider the first diagram $α = 1$ as the form diagram (‘the original’) and the final diagram as the force diagram (‘the reciprocal’). Similarly, the definitions of tension and compression are interchangeable, but we shall tend to adopt a convention of red for compression and yellow for tension. Video animations of the transition from original to reciprocal are instructive.

2D Maxwell–Minkowski

Figure 4 is based on Maxwell 1864 Figure 1 and illustrates the example considered in section ‘Introduction’, where the form and force diagrams each are the projection of a tetrahedron. In each subfigure, the sum of red and yellow areas is equal, illustrating Maxwell’s load path theorem. The upper right subdiagram thus indicates a triangular compression hoop stressed by tension spokes, while the lower right shows a triangular tension hoop stressed by compression spokes.

Figure 4.

Maxwell 1864, Figures 1 and I (tetrahedron). This is the example already considered in section ‘Introduction’, where form and force diagrams each are the projection of a tetrahedron. In each diagram, the sum of red and yellow areas is equal.

Figure 5 is based on Maxwell 1864, Figures 4 and IV. The reciprocal diagram shown in the lower left subfigure is in the classic Desargues configuration, requiring the radial lines to meet at a point. This is the same topology as the classic yield line mechanism for plastic collapse of a quadrilateral slab, the collapse mechanism resembling a downward-pointing hipped roof. The original diagram (upper left) has 2 degrees of freedom, in that independent stress functions may be generated by independently lifting the two central nodes (see McRobie et al.¹² and Mitchell et al.¹³ for further discussion). The stress function applied here is chosen such as to match Maxwell’s original drawings. The Maxwell–Minkowski representation makes it evident that the original structure has a triangular tension hoop stressed by three compression spokes, while a second set of spokes provides some tensile resistance.

Figure 5.

Maxwell 1864, Figures 4 and IV (Desargues 2D). The lower left subfigure is in the classic Desargues configuration, requiring the radials to meet at a point.

Figure 6 corresponds to Maxwell 1864, Figures 5 and V. The original (upper left) is the projection of an octahedron, and the reciprocal (lower left) is the projection of a topological cube or hexahedron. The stress function over the original has 3 degrees of freedom which may be mobilised by independently lifting the three inner nodes.¹² The particular reciprocal drawn corresponds to that chosen by Maxwell and which is also explored in Mitchell et al.¹³ The dual stress function over the reciprocal has only a single degree of freedom.¹²

Figure 6.

Maxwell 1864, Figures 5 and V (octahedron or hexahedron). The original (upper left) is the projection of an octahedron, and the reciprocal (lower left) is the projection of a topological cube or hexahedron. The original has 3 degrees of freedom which may explored by lifting the three inner nodes. The particular reciprocal drawn is that of Maxwell,⁴ which is revisited in McRobie et al.¹² and Mitchell et al.¹³ The reciprocal has only a single degree of freedom.

Figure 7 is based on a bridge optimisation exercise.¹⁵ The form diagram is shown in the top left subfigure. The upper part of this subfigure is the bridge, and the lower part is a funicular system through which vertical loads can be applied to the bridge. (The use of a funicular in this manner is described in McRobie et al.,¹² such that structures subject to external loads can be treated by the same techniques as were developed for self-stresses alone.) The $α = 0.95$ Maxwell–Minkowski combination shows the arch in compression, with hangers and deck in tension, stressed by vertical tension forces beneath the deck. From Maxwell’s load path theorem, it follows that the minimum weight, uniform stress optimisation problem reduces to adjusting the bridge geometry to minimise the area of the red rectangles that make up one side of the arch.

Figure 7.

Based on a bridge optimisation exercise,¹⁵ the Maxwell–Minkowski construction gives visual expression to the $F \cdot L$ terms that are central to the optimisation process. As described in McRobie et al.¹² and Mitchell et al.,¹³ applying the Airy stress function over the funicular and its reciprocal allows loaded structures such as this to be treated by the same techniques as were developed for self-stresses alone.

3D Rankine–Minkowski

Figure 8 is a higher dimensional generalisation of the 2D Desargues configuration of Figure 5. In this case, the form diagram (upper left) consists of two tetrahedra whose nodes are connected by four lines, and those four lines meet at a point (not shown), such that each quadrilateral between the two tetrahedra is a planar polygon. This figure demonstrates the visual power of the Rankine–Minkowski procedure. Reciprocal to the 3D Desargues configuration is the force diagram (lower left) which consists of a number of connected triangles whose areas represent the force magnitudes in their dual bars. This diagram is not easy to understand. However, in the $α = 0.95$ diagram (upper right), those triangular areas manifest themselves as bar areas of the equal stress solution – a far more intuitive representation. It can, for example, be readily seen that there is an outer compression structure stressed by a set of inner tension cables.

Figure 8.

Desargues 3D. This higher dimensional version of the Desargues configuration of Figure 5 demonstrates the visual power of the procedure. The force diagram (lower left) – consisting of a number of connected triangles whose areas represent the force magnitudes in their dual bars – is not easy to understand. However, in the $α = 0.95$ diagram (upper right), those triangular areas manifest themselves as bar areas of the equal stress solution – a far more intuitive representation.

Figure 9 shows the projection of a 4D hypercube, a tesseract (upper left), and its reciprocal, a 16-cell reciprocal (lower left). Just as the previous figure had upper and lower tetrahedra stressed against each other, so here there are upper and lower cuboids similarly stressed. The upper cuboid of the tesseract has been extended upwards such that the figure can be interpreted as a column-supported mansard roof carrying vertical loads from the peaks on its upper ridges. The lower part of the tesseract acts as a funicular, and loads can be applied via a stress function which lifts the central cuboid into 4D. While the 16-cell reciprocal is somewhat difficult to understand, the $α = 0.95$ diagram gives a clear representation of the member thicknesses required to support the applied loads. Somewhat remarkably, a roof has been designed by simply tilting it into 4D. Material volumes are given by the product of bar lengths with their polygonal reciprocals, and optimisation can proceed by adjusting the structural geometry. In addition to having designed a roof (upper right), there is a sense in which a second structure (lower right) has been designed simultaneously. The reciprocal diagram (lower left) could be viewed as an umbrella-like structure (or, inverted, as a guyed mast), and the $α = 0.05$ Rankine–Minkowski diagram (lower right) gives the material volumes required for a uniform stress solution.

Figure 9.

Tesseract (mansard roof). The original and reciprocal are the projections of a tesseract and 16-cell, respectively. The original could represent a column-supported mansard roof with vertical forces applied via a funicular below. Loads were applied via the stress function, lifting the central cuboid into 4D. The $α = 0.95$ diagram (upper right) gives a clear representation of the member thicknesses required to support the applied loads. Somewhat remarkably, a roof has been designed by simply tilting it into 4D.

Figure 10 is also based on the projection of a tesseract. This time the central cuboid is contained within the outer cuboid (upper left subfigure), but the upper cuboid has been cross braced by members connecting at a central node. Much as before, a system of stresses may be induced via a stress function which lowers the central cuboid into the fourth dimension, $h$ . Another independent stress function can be mobilised by lifting the node at the centre of the upper cross bracing into the fourth dimension. The effect of the latter is to induce tensile forces in the cross bracing, resisted by compressive actions in the upper cuboid.

Figure 10.

Stansted. This example generalises the previous tesseract by adding a bracing system into the upper cuboid. This gives an additional state of self-stress that can be varied separately. One interpretation is that the central cuboid and the braced upper cuboid constitute one of the tree structures supporting the roof at Stansted Airport, and carrying vertical loads at its outer corners.

Figure 11 shows Jessen’s orthogonal icosahedron, a tensegrity structure which is known to possess a state of self-stress. However, since the structure has only a single three-cell volume, the icosahedron, its 4D projective geometry reciprocal (i.e. the Rankine diagram) consists of only a single point, which is incapable of representing a state of self-stress. A meaningful Rankine reciprocal can, however, be constructed by adding a central node connected to all other nodes by radial bars (a procedure known as ‘coning’). The ‘coned’ icosahedron is thus triangulated internally by 20 tetrahedra. Lifting the central node into the fourth dimension and taking normals lead to a non-trivial Rankine reciprocal (lower left). When combined with the original in an $α = 0.95$ Rankine–Minkowski construction (upper right), the outer icosahedron bars can be seen to have triangular cross sections while the radial bars have crossed-quadrilateral cross sections of zero-oriented area (shown grey). It is easy to misread the α=0.95 $α = 0.95$ diagram (upper right) as showing an outer icosahedron stressed by radial spokes. Instead, the added radials are unstressed, and the icosahedron bar areas show the self-equilibrating bar forces in the tensegrity state of self-stress that exists in the surrounding icosahedron.

Figure 11.

Jessen’s orthogonal icosahedron. This final example considers the icosahedral tensegrity structure (upper left). The tensegrity possesses a state of self-stress, which can be activated in the Rankine construction via ‘coning’. That is, an additional node has been placed at the centre, connected to all other nodes via bars of zero area (grey).

Rankine incompleteness

Unfortunately, not every self-stressed 3D truss has a meaningful Rankine reciprocal. The general issue is covered in Crapo and Whiteley.⁷ For the icosahedral tensegrity above, the problem was solved by adding a further node which connected to the structure with bars of zero area. However, it is not yet clear whether similar procedures will be sufficient in all cases. However, such difficulties concern the Rankine construction rather than the Minkowski construction: if a structure has a meaningful Rankine reciprocal, the two can be combined to create a Rankine–Minkowski diagram.

Indeed, the Minkowski construction is an aid to understanding why Rankine diagrams do not always exist. For example, the force in a single bar may be in equilibrium at one end with the forces in three bars that join there. Around that node, a Rankine tetrahedron may be defined whose face areas are proportional to the four forces meeting there. At the node at the other end, three other bars may meet with the four forces there being in equilibrium, and again, a Rankine tetrahedron may be defined. In general, although the triangles at each end that represent the force in the bar under consideration will have equal areas, they need not be similar triangles. They thus cannot be ‘extruded’ along the bar to create a prismatic Rankine–Minkowski representation and cannot be glued to form a Rankine reciprocal.

The 3D Desargues configuration of Figure 8 is another case in point. In that example, the four lines which connect the nodes of the upper tetrahedron with those of the lower one, when extended, meet at a point, analogous to the 2D case. However, equilibrium configurations exist which have the same bar topology, where the four lines do not meet at a point (although they must, however, all intersect at some line). Some ‘faces’ of these configurations are what Maxwell calls ‘gauche polygons’, meaning that while they are defined by a closed circuit of lines, those lines need not lie in a single plane. Such ‘faces’ are not planes, and thus, the projective geometry duality breaks down.

The incompleteness arises from the use of the single stress function in 3D. For 2D stresses, the Airy stress function gives a complete description. For 3D stresses, Maxwell⁴ described two stress function methods. One method corresponds to that used here to create the Rankine reciprocal, with Maxwell noting that the description was incomplete. Maxwell’s other method involved three separate stress functions, and while that method is complete, it corresponds to the Cremona description where bar forces are represented by vectors parallel to the bars.

It is not the purpose of this article to resolve the incompleteness of the Rankine description. This remains a topic for further research. Instead, the intention here is to show how the Maxwell and Rankine constructions can be augmented to give Maxwell–Minkowski and Rankine–Minkowski diagrams which are conceptually simple and yet which can provide much insight.

Summary and conclusion

The Minkowski sum construction for Maxwell and Rankine diagrams provides insight into the internal forces carried by 2D and 3D trusses under load. In 2D, the Maxwell description is complete, but the Rankine description in 3D is incomplete. Nevertheless, the Minkowski constructions shown here may prove to be of use and not only in truss design but also in various other spheres of structural engineering.

Footnotes

Acknowledgements

The author is grateful for conversations with Bill Baker, Marina Konstantatou, Toby Mitchell, Elissa Ross and Walter Whiteley. Much of the impetus for this work occurred as a result of the workshop on Advances in Combinatorial and Geometric Rigidity at the Banff International Research Station, 12–15 July 2015, organised by Robert Connelly, Steven Gortler, Tibor Jordan, Brigitte Servatius, Meera Sitharam and Walter Whiteley.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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