TY - JOUR

T1 - Graphic Statics and Symmetry

AU - Schulze, Bernd

AU - Millar, Cameron

PY - 2023/11/1

Y1 - 2023/11/1

N2 - 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.

AB - 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.

KW - Graphic statics

KW - Reciprocal diagram

KW - Symmetry

KW - Equilibrium stress

KW - Discrete Airy stress function polyhedron

U2 - 10.1016/j.ijsolstr.2023.112492

DO - 10.1016/j.ijsolstr.2023.112492

M3 - Journal article

VL - 283

JO - International Journal of Solids and Structures

JF - International Journal of Solids and Structures

SN - 0020-7683

M1 - 112492

ER -