Limits...
The geometry of structural equilibrium

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.


(a) A simple gridshell roof. (b) A horizontal section showing cell translations that lead to zero force in the midwall verticals. (c) The resulting stress function gradients. (d) The polygon reciprocal to the corner columns has zero projected area on the e0(e1+e2) and e0(e1−e2) planes.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSOS160759F12: (a) A simple gridshell roof. (b) A horizontal section showing cell translations that lead to zero force in the midwall verticals. (c) The resulting stress function gradients. (d) The polygon reciprocal to the corner columns has zero projected area on the e0(e1+e2) and e0(e1−e2) planes.

Mentions: To simplify the example for clarity, consider a roof with four roof panels (figure 12a) supporting only a central point load which is carried ultimately by four corner columns. The roof panels can be gauche polygons. Again, before looking at the roof members we consider the loading system by taking a horizontal cross-section below the roof. The three-dimensional method of the previous section can readily ensure that midwall verticals carry no axial load. However, we also require that these carry no moment.Figure 12.


The geometry of structural equilibrium
(a) A simple gridshell roof. (b) A horizontal section showing cell translations that lead to zero force in the midwall verticals. (c) The resulting stress function gradients. (d) The polygon reciprocal to the corner columns has zero projected area on the e0(e1+e2) and e0(e1−e2) planes.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSOS160759F12: (a) A simple gridshell roof. (b) A horizontal section showing cell translations that lead to zero force in the midwall verticals. (c) The resulting stress function gradients. (d) The polygon reciprocal to the corner columns has zero projected area on the e0(e1+e2) and e0(e1−e2) planes.
Mentions: To simplify the example for clarity, consider a roof with four roof panels (figure 12a) supporting only a central point load which is carried ultimately by four corner columns. The roof panels can be gauche polygons. Again, before looking at the roof members we consider the loading system by taking a horizontal cross-section below the roof. The three-dimensional method of the previous section can readily ensure that midwall verticals carry no axial load. However, we also require that these carry no moment.Figure 12.

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.