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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.


(a) The cell connectivity on a section transverse to an edge beam. (b) The gauche quadrilateral reciprocal to the edge beam (in three dimensions, with e0 information omitted), and the projections of its oriented area that give the forces.
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RSOS160759F14: (a) The cell connectivity on a section transverse to an edge beam. (b) The gauche quadrilateral reciprocal to the edge beam (in three dimensions, with e0 information omitted), and the projections of its oriented area that give the forces.

Mentions: All stress function values have now been determined, and it remains only to evaluate the stress resultant in the edge beams. Although the procedure works even if the roof beam inclinations θ and ϕ are different (and the roof panels are thus gauche), we proceed for brevity with the case θ=ϕ. The cell connectivity on a section transverse to an edge beam is shown in figure 14a. To calculate the forces we plot and connect the gradients of the four cells in three dimensions, temporarily suppressing the e0 information. This gives a gauche quad (figure 14b) whose projections give the force components.Figure 14.


The geometry of structural equilibrium
(a) The cell connectivity on a section transverse to an edge beam. (b) The gauche quadrilateral reciprocal to the edge beam (in three dimensions, with e0 information omitted), and the projections of its oriented area that give the forces.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
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getmorefigures.php?uid=PMC5383818&req=5

RSOS160759F14: (a) The cell connectivity on a section transverse to an edge beam. (b) The gauche quadrilateral reciprocal to the edge beam (in three dimensions, with e0 information omitted), and the projections of its oriented area that give the forces.
Mentions: All stress function values have now been determined, and it remains only to evaluate the stress resultant in the edge beams. Although the procedure works even if the roof beam inclinations θ and ϕ are different (and the roof panels are thus gauche), we proceed for brevity with the case θ=ϕ. The cell connectivity on a section transverse to an edge beam is shown in figure 14a. To calculate the forces we plot and connect the gradients of the four cells in three dimensions, temporarily suppressing the e0 information. This gives a gauche quad (figure 14b) whose projections give the force components.Figure 14.

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.