Mechanisms and states of self-stress of planar trusses using graphic statics, part II: Applications and extensions. Issue 2 (June 2016) - McRobie, Allan; Baker, William; Mitchell, Toby; Konstantatou, Marina
This article extends the overview relating graphic statics and reciprocal diagrams to linear algebra-based matrix structural analysis. Focus is placed on infinitesimal mechanisms, both in-plane (linkage) and out-of-plane (polyhedral Airy stress functions). Each self-stress in the original diagram corresponds to an out-of-plane polyhedral mechanism. Decomposition into sub-polyhedra leads to a basis set of reciprocal figures which may then be linearly combined. This leads to an intuitively appealing approach to the identification of states of self-stress for use in structural design and to a natural 'structural algebra' for use in structural optimisation. A 90° rotation of the sub-reciprocal generated by any sub-polyhedron leads to the displacement diagram of an in-plane mechanism. Any self-stress in the original thus corresponds to an in-plane mechanism of the reciprocal, summarised by the equation s = M * (where s is the number of states of self-stress in one figure, and M * is the number of in-plane mechanisms, including rigid body rotation, in the other). Since states of self-stress correspond to out-of-plane polyhedral mechanisms, this leads to a form of 'conservation of mechanisms' under reciprocity. It is also shown how external forces may be treated via a triple-layer Airy stress function, consisting of a structural layer, a load layer and a layer formed by coordinate vectors of the structural perimeter.