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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.

No MeSH data available.


Edge beam moments displayed as oriented areas. The points of the Corsican sum are plotted on the e0ei(i=1,2,3) subspaces. These are placed such that the oriented area is orthogonal to the ei direction corresponding to the moment vector Mi.
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RSOS160759F15: Edge beam moments displayed as oriented areas. The points of the Corsican sum are plotted on the e0ei(i=1,2,3) subspaces. These are placed such that the oriented area is orthogonal to the ei direction corresponding to the moment vector Mi.

Mentions: To calculate the moments, we need the a0 terms for each cell in order to determine the stress function values adjacent to any point on the beam. These have all been determined in the preceding analysis, and are shown in the legend of figure 15. In particular, at the point (1,x,L,−t(L+x)) on the beam, we have the roof cushion stress function value C=−4gL+gx+gL+(−g/t)(−t(L+x))=−2g(L−x). The figure also demonstrates one possible method for illustrating the moment information. The moments are given by the e0ei bivector areas, but the four-dimensional nature of the problem creates visualization difficulties. Here the bivector areas e0ei have been presented on the ejek planes (i≠j≠k), such that they appear as areas oriented perpendicular to the vector that would usually represent them. The final diagram is not particularly instructive. Nevertheless, it is evident that, whether algebraically or graphically, the moments can be determined. More intuitive graphical methods for displaying them may yet be devised, failing which one can always simply plot the moments via traditional methods.Figure 15.


The geometry of structural equilibrium
Edge beam moments displayed as oriented areas. The points of the Corsican sum are plotted on the e0ei(i=1,2,3) subspaces. These are placed such that the oriented area is orthogonal to the ei direction corresponding to the moment vector Mi.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSOS160759F15: Edge beam moments displayed as oriented areas. The points of the Corsican sum are plotted on the e0ei(i=1,2,3) subspaces. These are placed such that the oriented area is orthogonal to the ei direction corresponding to the moment vector Mi.
Mentions: To calculate the moments, we need the a0 terms for each cell in order to determine the stress function values adjacent to any point on the beam. These have all been determined in the preceding analysis, and are shown in the legend of figure 15. In particular, at the point (1,x,L,−t(L+x)) on the beam, we have the roof cushion stress function value C=−4gL+gx+gL+(−g/t)(−t(L+x))=−2g(L−x). The figure also demonstrates one possible method for illustrating the moment information. The moments are given by the e0ei bivector areas, but the four-dimensional nature of the problem creates visualization difficulties. Here the bivector areas e0ei have been presented on the ejek planes (i≠j≠k), such that they appear as areas oriented perpendicular to the vector that would usually represent them. The final diagram is not particularly instructive. Nevertheless, it is evident that, whether algebraically or graphically, the moments can be determined. More intuitive graphical methods for displaying them may yet be devised, failing which one can always simply plot the moments via traditional methods.Figure 15.

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.