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Graphic Statics and Symmetry

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Article number112492
<mark>Journal publication date</mark>1/11/2023
<mark>Journal</mark>International Journal of Solids and Structures
Number of pages16
Publication StatusPublished
Early online date25/09/23
<mark>Original language</mark>English


Reciprocal diagrams are a geometric construction dating back to Maxwell and Cremona in which a self-stressed plane framework with a planar graph is paired with another self-stressed reciprocal framework on the dual graph. Either one of the reciprocal frameworks is the form diagram of a self-stressable structure and the other is the force diagram of the corresponding axial forces. This geometric technique offers insights into the self-stresses and infinitesimal motions (mechanisms) of both frameworks in the reciprocal pair. For a symmetric framework with a fully-symmetric self-stress, we obtain an equi-symmetric reciprocal pair of plane frameworks, as well as the associated symmetric discrete dual Airy stress function polyhedra. In this paper we exploit symmetry to refine the Maxwell–Cremona correspondence by considering the decomposition of the self-stress and motion spaces into invariant subspaces corresponding to the irreducible representations of the symmetry group. As such, the familiar
relationship for the number of self-stresses of a framework,
, and the number of mechanisms of the reciprocal,
, is reworked into a symmetry adapted version which provides greater insights into the properties of the reciprocal framework pair. We also show how the quotient graph of a symmetric framework and its reciprocal can be used to efficiently detect infinitesimal motions, self-stresses and polyhedral liftings of different symmetry types. This allows for symmetry-adapted simplified structural analyses of symmetric structures.