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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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The representation of a simple structure and its stress resultants as dual abstract polytopes. The constituent elements of the structural polytope PX are arranged in the form of a Hasse diagram (right), the up-down symmetry of which shows the polytope to be topologically self-dual. However, the dual here is not sufficiently rich to obtain meaningful states of self-stress and it will be necessary to add more cells (face cushions) to the original, thereby adding reciprocal nodes along the reciprocal bars (see Example 2 later).
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RSOS160759F2: The representation of a simple structure and its stress resultants as dual abstract polytopes. The constituent elements of the structural polytope PX are arranged in the form of a Hasse diagram (right), the up-down symmetry of which shows the polytope to be topologically self-dual. However, the dual here is not sufficiently rich to obtain meaningful states of self-stress and it will be necessary to add more cells (face cushions) to the original, thereby adding reciprocal nodes along the reciprocal bars (see Example 2 later).

Mentions: The construction then begins with the definition of a pair of topologically dual abstract 4-polytopes (partially ordered sets), with duality defined by reading the Hasse diagrams in opposite order. An example is given in figure 2. This first step is purely topological.Figure 2.


The geometry of structural equilibrium
The representation of a simple structure and its stress resultants as dual abstract polytopes. The constituent elements of the structural polytope PX are arranged in the form of a Hasse diagram (right), the up-down symmetry of which shows the polytope to be topologically self-dual. However, the dual here is not sufficiently rich to obtain meaningful states of self-stress and it will be necessary to add more cells (face cushions) to the original, thereby adding reciprocal nodes along the reciprocal bars (see Example 2 later).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSOS160759F2: The representation of a simple structure and its stress resultants as dual abstract polytopes. The constituent elements of the structural polytope PX are arranged in the form of a Hasse diagram (right), the up-down symmetry of which shows the polytope to be topologically self-dual. However, the dual here is not sufficiently rich to obtain meaningful states of self-stress and it will be necessary to add more cells (face cushions) to the original, thereby adding reciprocal nodes along the reciprocal bars (see Example 2 later).
Mentions: The construction then begins with the definition of a pair of topologically dual abstract 4-polytopes (partially ordered sets), with duality defined by reading the Hasse diagrams in opposite order. An example is given in figure 2. This first step is purely topological.Figure 2.

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