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The geometry of structural equilibrium

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ABSTRACT

Building on a long tradition from Maxwell, Rankine, Klein and others, this paper puts forward a geometrical description of structural equilibrium which contains a procedure for the graphic analysis of stress resultants within general three-dimensional frames. The method is a natural generalization of Rankine’s reciprocal diagrams for three-dimensional trusses. The vertices and edges of dual abstract 4-polytopes are embedded within dual four-dimensional vector spaces, wherein the oriented area of generalized polygons give all six components (axial and shear forces with torsion and bending moments) of the stress resultants. The relevant quantities may be readily calculated using four-dimensional Clifford algebra. As well as giving access to frame analysis and design, the description resolves a number of long-standing problems with the incompleteness of Rankine’s description of three-dimensional trusses. Examples are given of how the procedure may be applied to structures of engineering interest, including an outline of a two-stage procedure for addressing the equilibrium of loaded gridshell rooves.

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(a) If the seven bars shown are in general position, then this part of a truss has no Rankine reciprocal. (b) Face cushions S, T and U may be introduced to isolate the left-hand tetrahedron G. (c) Additional reciprocal nodes are thus introduced into the topology, creating three pentagonal faces. (d) The new reciprocal nodes may be placed in the plane ABC in such a way as to introduce no additional oriented area. (e) The gauche pentagon GSABT decomposes to a triangle and a Zero Bar quad. (f) The oriented areas of the seven faces are perpendicular to the seven original bars.
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RSOS160759F8: (a) If the seven bars shown are in general position, then this part of a truss has no Rankine reciprocal. (b) Face cushions S, T and U may be introduced to isolate the left-hand tetrahedron G. (c) Additional reciprocal nodes are thus introduced into the topology, creating three pentagonal faces. (d) The new reciprocal nodes may be placed in the plane ABC in such a way as to introduce no additional oriented area. (e) The gauche pentagon GSABT decomposes to a triangle and a Zero Bar quad. (f) The oriented areas of the seven faces are perpendicular to the seven original bars.

Mentions: A standard problem for Rankine reciprocals concerns a bar that has 4-valent nodes at each end (figure 8a). For each node, there is a reciprocal tetrahedron but they cannot be conjoined to assemble an overall Rankine diagram: although the two triangles reciprocal to the connecting bar have the same area, their geometry may differ.Figure 8.


The geometry of structural equilibrium
(a) If the seven bars shown are in general position, then this part of a truss has no Rankine reciprocal. (b) Face cushions S, T and U may be introduced to isolate the left-hand tetrahedron G. (c) Additional reciprocal nodes are thus introduced into the topology, creating three pentagonal faces. (d) The new reciprocal nodes may be placed in the plane ABC in such a way as to introduce no additional oriented area. (e) The gauche pentagon GSABT decomposes to a triangle and a Zero Bar quad. (f) The oriented areas of the seven faces are perpendicular to the seven original bars.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5383818&req=5

RSOS160759F8: (a) If the seven bars shown are in general position, then this part of a truss has no Rankine reciprocal. (b) Face cushions S, T and U may be introduced to isolate the left-hand tetrahedron G. (c) Additional reciprocal nodes are thus introduced into the topology, creating three pentagonal faces. (d) The new reciprocal nodes may be placed in the plane ABC in such a way as to introduce no additional oriented area. (e) The gauche pentagon GSABT decomposes to a triangle and a Zero Bar quad. (f) The oriented areas of the seven faces are perpendicular to the seven original bars.
Mentions: A standard problem for Rankine reciprocals concerns a bar that has 4-valent nodes at each end (figure 8a). For each node, there is a reciprocal tetrahedron but they cannot be conjoined to assemble an overall Rankine diagram: although the two triangles reciprocal to the connecting bar have the same area, their geometry may differ.Figure 8.

View Article: PubMed Central - PubMed

ABSTRACT

Building on a long tradition from Maxwell, Rankine, Klein and others, this paper puts forward a geometrical description of structural equilibrium which contains a procedure for the graphic analysis of stress resultants within general three-dimensional frames. The method is a natural generalization of Rankine’s reciprocal diagrams for three-dimensional trusses. The vertices and edges of dual abstract 4-polytopes are embedded within dual four-dimensional vector spaces, wherein the oriented area of generalized polygons give all six components (axial and shear forces with torsion and bending moments) of the stress resultants. The relevant quantities may be readily calculated using four-dimensional Clifford algebra. As well as giving access to frame analysis and design, the description resolves a number of long-standing problems with the incompleteness of Rankine’s description of three-dimensional trusses. Examples are given of how the procedure may be applied to structures of engineering interest, including an outline of a two-stage procedure for addressing the equilibrium of loaded gridshell rooves.

No MeSH data available.


Related in: MedlinePlus