Topological structural analysis

Topological structural analysis is a terminology proposed for studying the behaviour of structures when the constitutive law is reduced to a least formulation. In addition to the equilibrium and deformation compatibility equations we only require that the extensional and contractile strains correspond, respectivley, to tensile and compressive stresses, without further specifying the nature of the constitutive law. The only analysis parameter in these equations is the equilibrium matrix which incorporates pure topological information such as node positions and bar connectivities. It is shown that in a topological context the internal forces, which can be realized by a structure, are bounded by a convex combination of the internal forces of its embedded statically determinate substructures. It is also shown that this "structural" equilibrium space is nonconvex. Consequently, the internal forces are bounded, component by component, by the internal forces in the statically determinate solutions. Having established that the structural equilibrium space is a small subset of the equilibrium space it is shown that pure equilibrium solutions, such as are obtained in plastic analysis and design, are not always feasible. It is conjectured that topological design of structures may benefit from using topological analysis rather than a pure equilibrium analysis.