Mechanisms and states of self-stress of planar trusses using graphic statics, part II: Applications and extensions
International Journal of Space Structures
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McRobie, A., Baker, W., Mitchell, T., & Konstantatou, M. (2016). Mechanisms and states of self-stress of planar trusses using graphic statics, part II: Applications and extensions. International Journal of Space Structures, 31 (2-4), 102-111. https://doi.org/10.1177/0266351116660791
This paper extends the overview (Mitchell et al. ) 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.
External DOI: https://doi.org/10.1177/0266351116660791
This record's URL: https://www.repository.cam.ac.uk/handle/1810/262178