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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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Hasse diagrams of the structure PX and its topological dual PA. 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.
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RSOS160759F6: Hasse diagrams of the structure PX and its topological dual PA. 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.

Mentions: By way of further clarification, the Hasse diagrams showing the topological duality of the polytopes PX and PA are shown in figure 6. With the inclusion of the face cushions, the structure PX has four cells, thus the dual has four nodes. Similarly, the structure has two nodes such that the dual has two cells. From a geometrical perspective, these two cells may be considered to have zero thickness, since all face polygons lie within the e1, e2 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 PA.Figure 6.


The geometry of structural equilibrium
Hasse diagrams of the structure PX and its topological dual PA. 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.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSOS160759F6: Hasse diagrams of the structure PX and its topological dual PA. 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.
Mentions: By way of further clarification, the Hasse diagrams showing the topological duality of the polytopes PX and PA are shown in figure 6. With the inclusion of the face cushions, the structure PX has four cells, thus the dual has four nodes. Similarly, the structure has two nodes such that the dual has two cells. From a geometrical perspective, these two cells may be considered to have zero thickness, since all face polygons lie within the e1, e2 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 PA.Figure 6.

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