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The Scholar and the State: It is the various projections of this loop that will define the stress resultant at the point X s. Figure 3 is schematic, and somewhat misleading. Strictly the construction occurs in four dimensions, but the behaviour in the fourth dimension is difficult to illustrate.
The three-dimensional projection to e 1 , e 2 , e 3 of the full construction is exactly as drawn, such that the shaded face is essentially extruded along the bar IJ. That this face does not change along the bar is a reflection of the fact that the force vectors axial plus two shear do not change along the bar.
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However, the behaviour in the e 0 direction is given by the dot product X. A and this has not been shown. This quantity may vary along the bar, reflecting the fact that moments may vary along the bar. Each point on each line is raised by X. A in the fourth dimension e 0 , but this is not shown.
For present purposes, we restrict attention to the cases where the bars in P A are straight otherwise the wedge formula that follows would need to become some more complicated integral. We may thus say that the polygon R is reciprocal to the bar IJ at the point X s. In the new four-dimensional description, there is no discontinuity, in the sense that the stress function is completely encapsulated by the reciprocal diagram, which is the dual polytope P A , a continuous, connected object in four dimensions.
X where A K is the stress function over any of the cells C K of which the bar is a common edge. That is, earlier descriptions have considered objects of the form A. X , x , thereby portraying the stress function A. The new four-dimensional description A. The four-dimensional object containing items of the form A. This new object similarly combines both, but is a continuous four-dimensional object which includes all six components of the stress resultant.
There are a number of ways to generalize the Corsican sum. This merely makes explicit the scale factor between length and force objects that is already implicit in the fundamental definition. Given that four-dimensional information is difficult to display graphically, a number of potential visualization methods—including the use of colour—are described in the examples that follow.
This corresponds to previous methods of formulating the problem, representing the stress function as a discontinuous function over the original three-dimensional structure. We may similarly obtain terms of the form X. A , a , this object being the generalization of the Rankine reciprocal. The difference is that it is a four-dimensional object consisting of polygonal faces, these being one-dimensional loops in four dimensions which need not be orthogonal to the bars to which they are reciprocal, and which may be gauche polygons. Again, this is a function over an object with discontinuities at bars.
We have restricted attention to the case where the bars in P A are straight such that we may use the wedge sum formula. However, as with Rankine—Minkowski diagrams [ 33 ], this is not a problem, and all items of physical interest emerge with the correct units. The scaling we choose for the vector part of the Corsican sum is of far less importance than the fact that we must use X.
It is use of the X. A term that allows us to correctly represent moment equilibrium. If X is allowed to move along a bar, there is no change in the e 1 , e 2 and e 3 coordinates of the reciprocal polygon vertices. The force in the bar given by the oriented area in these three dimensions is thus constant. The moments, however, depend on the e 0 behaviour and thus the moments may vary along the bar. When summed cyclically over i and halved, the first term gives the constant force and constant bending moment.
That is, the Corsican sum ensures moment equilibrium via the X. These are cells possessing only a single polygonal loop, and will be an important element of how the theory endeavours to overcome issues of incompleteness that have hampered previous progress with the Rankine construction. This was the case for the Maxwell—Minkowski diagrams for two-dimensional trusses, two-dimensional frames and two-dimensional grillages and Rankine—Minkowski diagrams for three-dimensional trusses. The essential idea was that the two-dimensional polygons or three-dimensional polyhedra provided a double cover of a region of two- or three-dimensional space.
We do not pursue these ideas in this paper.
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Instead we now demonstrate how the overall geometric picture may be applied to a number of examples. A triangular bicycle wheel with six spokes three either side connecting to the ends of a central hub is a three-dimensional structural manifestation of one of the simplest 4-polytopes, the 5-cell. Consisting of five conjoined tetrahedra, it provides the higher dimensional analogue of fig.
The 5-cell has five nodes which may be labelled 1,2,… , 10 bars which may be labelled 12,23,… , 10 triangular faces labelled ,,… and five tetrahedra ,,…. The duality is readily annotated via complementary labelling, by making node 1 reciprocal to cell BCDE , bar 12 reciprocal to face CDE , face reciprocal to bar DE , etc. Geometry is specified by picking the four-dimensional coordinates of the five P X nodes 1,2,… and of the five P A nodes A , B ,…. Since the duality is topological, these coordinates of both can be chosen freely.
Note how this differs from earlier approaches to three-dimensional graphic statics where, given a structure a form diagram the analyst then goes to some trouble to find the geometry of an admissible force diagram. Figure 4 a , b shows a structure P X and with a dual P A albeit that in this configuration, the structure P X does not look like a triangular wheel. The dual polytope P A , however, is fully four-dimensional and as ever, there is difficulty in plotting a four-dimensional object on a two-dimensional image. In the visualization shown in figure 4 b , the reciprocal bars have been artificially thickened and coloured to indicate the coordinate values in the e 0 direction.
There is no artificial thickening in the subfigures that follow, the apparent thickening of the bars being a natural consequence of the Corsican sum. The skeletal diagram c illustrates how original nodes are embellished by their reciprocal cells, and d shows the dual case. In figure 4 e , f , some faces are coloured to indicate the stress function values X.
A in the e 0 direction. Values of a 0 in the e 0 direction are indicated by colouring the artificially thickened bars. Copies of reciprocal cells are evident at original nodes, and vice versa. A indicated by colouring. Figure 4 e is perhaps the most important of the subfigures. However, since reciprocal polygons are no longer orthogonal to bars, we have lost the Lower Bound Plasticity interpretation that was possible for Rankine—Minkowski diagrams.
A bar carrying a large shear has a reciprocal polygon inclined relative to the bar. The projection of this polygon perpendicular to the bar direction can indeed represent the amount of material necessary to carry the axial forces, but additional material would be needed to carry the shear forces and the moments.
The representation of the stress function components associated with the moments by means of colouring is also far from ideal. While it leads to an interesting graphic, it is difficult to interpret visually, and later examples will look at alternative graphical procedures for conveying the moment information.
However, it nevertheless illustrates some important points. First, it reminds us that the Corsican sum contains all the information about the stress resultants. Second, it can be seen that the continuity of the colouring reflects how the Corsican sum is a continuous object.
Despite the difficulties of creating an intuitive visual presentation, all stress resultant information can nevertheless be extracted algebraically to obtain the equilibrium state of stress in the original structure P X that is defined by this choice of reciprocal structure P A. Finally, we note that not all possible equilibrium states of self-stress of this structure can be captured by its representation via dual 5-cells. In the 5-cell, however, each bar is common to only three cells, and one consequence is that at any point the forces will be necessarily orthogonal to moments when each is represented by a three-dimensional vector.
Figure 5 a shows a simple structure P X.
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It consists of two curved beams with a tie. Given that much of the literature on polytopes concerns regular polytopes with plane faces and straight edges such as the tesseract it may not be immediately apparent that the simple structure has a dual polytope representation. However, the abstract polytope of figure 2 earlier corresponds to exactly this case. The dual polytope there, though, is not sufficiently rich to represent any interesting states of self-stress.
However, given the generality of the formulation here, this problem can be readily overcome by simply adding more cells to the original. These are cells which have only two surfaces, each spanning the same single polygonal loop of bars. That is, they resemble a cushion padding the face in question. A face cushion is a cell, thus adding a face cushion means introducing an additional reciprocal node along the reciprocal bar that was dual to the original face. This freedom to add face cushions and corresponding reciprocal nodes almost anywhere is key in providing the richness of the solution space which thereby enables a wide set of equilibrium states to be represented by the new description.
It is even permissible and sometimes necessary to add more than one face cushion to a face. Cushions have been added to horizontal and vertical faces of the structure. In the tied beam example here, we choose to add two face cushions, one each on the horizontal and vertical faces, such that the original, P X , now has four cells figure 5 b.
The abstract polytope representation is shown in figure 5 c. In this new approach, it is no longer necessary to calculate an admissible reciprocal. Rather, we are now free to choose any geometry for the dual polytope P A. For simplicity of explanation, we choose the simple arrangement shown in figure 5 d. By way of further clarification, the Hasse diagrams showing the topological duality of the polytopes P X and P A are shown in figure 6. With the inclusion of the face cushions, the structure P X has four cells, thus the dual has four nodes.
Similarly, the structure has two nodes such that the dual has two cells.
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From a geometrical perspective, these two cells may be considered to have zero thickness, since all face polygons lie within the e 1 , e 2 plane. Nevertheless, the generality of the previous definitions admits the possibility of such three-face polyhedra of arbitrary or even zero thickness, and the two polyhedra together provide the double cover necessary for the definition of the polytope P A. Hasse diagrams of the structure P X and its topological dual P A. The structure with the inclusion of the face cushions has four cells, thus the dual has four nodes.
The structure has two nodes such that the dual has two cells. Since P X and P A are now defined, topologically and geometrically, the Corsican sum can now be constructed. A section at constant z is shown in figure 7 a. Since e 3 information is not required, we use the third dimension to display e 0 information. Reciprocal to the points on each bar is a polygon whose oriented area defines the stress resultant.
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For this choice of P A , the tie carries an axial force of gh which is resisted equally by the two beams. Since all forces are given by e 1 e 2 bivectors i. The bending moments are given by the e 0 e 1 and e 0 e 2 bivector components of the reciprocal polygons. The example demonstrates that a graphical method exists for determining the equilibrium stress resultants in a rather general three-dimensional frame. Many further complications may be introduced into this example, and the general procedure still works. For example, the beams could be very general three-dimensional space curves, even though it may not then be immediately apparent that the structure has any representation as a polytope.
Nevertheless, the representation as dual abstract polytopes would be identical to that here. In the example here, the curved beams are the common edge of only three cells, but additional face cushions can be readily introduced to create a dual polytope that is rich enough to represent all possible states of self-stress.
Having developed such a general method for three-dimensional frames, we now apply it back to some earlier problems that have plagued the Rankine construction for three-dimensional trusses. For this, we do not even need the full four-dimensional methodology, but can work purely in three dimensions. The new procedure described in this paper appears capable of avoiding any such limitations, and while we do not yet have a full proof of completeness either for frames or for trusses, we give a few examples of how some archetypal problematic configurations may now be readily dealt with.
Throughout, the key features of the new description that appear to surmount previous problems are the ability to deal with gauche polygons, and the freedom to add as many face cushions as desired. A standard problem for Rankine reciprocals concerns a bar that has 4-valent nodes at each end figure 8 a. For each node, there is a reciprocal tetrahedron but they cannot be conjoined to assemble an overall Rankine diagram: The configuration is a three-dimensional variant of the classic two-dimensional Desargues arrangement wherein two triangles are connected by three bars.
In that case, there is a two-dimensional truss state of self-stress if and only if the lines of the three connecting bars meet at a point. The three-dimensional generalization here has tetrahedra G and H connected by four bars. If the lines of these four bars meet at a point, then a three-dimensional truss state of self-stress can be readily modelled with a Rankine reciprocal. However, there also exist truss states of self-stress where the four connecting lines do not meet at a point.
The configuration then contains a number of gauche quad polygons. To encapsulate the problem for the purposes of illustration, we restrict attention to the inner seven truss members, assuming there is some wider outer frame capable of resisting any resulting forces.
As drawn, we have seven bars in the problematic morphology where no three of the seven are coplanar. Initially, there are five cells of interest, the end tetrahedra G and H and three prism-like polyhedra A , B and C that connect them. The incompleteness problem is solved by adding face cushions S , T and U over the faces of the tetrahedron G to separate it from the prism-like cells A , B and C figure 8 b.
The topological connectivity of the cells is shown in figure 8 c. Without face cushions, the topological dual has topological tetrahedra with a common face ABC the topological viewpoint ignoring the possibility that the faces may not match geometrically. The addition of the face cushions results in there being additional reciprocal nodes S , T and U on the bars of the left-hand topological tetrahedron, which then possesses one triangular and three pentagonal faces.
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We now introduce geometry. Moving an object corresponds to applying an additional constant gradient stress function over its original cells. The resulting geometry on the two-dimensional plane orthogonal to bar 12 is shown in figure 8 d. Each pentagonal face can be expressed as the sum of a triangle and a quad, as per figure 8 e , where the triangle GST is that of the original Rankine, and the quad is a Zero Bar.
The force in that member is the oriented area of the pentagon, which thus equals the original Rankine tension GST along the bar, plus zero. Collecting all the elements together creates two conjoined polyhedra whose polygonal faces have oriented areas orthogonal to the bars. As each is a closed polyhedron, we thus have zero total oriented area, this corresponding to a state of equilibrium where each bar carries an axial force equal to the area of the reciprocal polygon.
The result is shown in figure 8 f , and this is a generalized Rankine diagram for the two problem nodes. This is another example of a three-dimensional truss, an octahedron with a central spindle, as was considered by McRobie [ 36 ].