A compliance based design problem of structures under multiple load cases

There are two popular methods concerning the optimal design of structures. The first is the minimization of the volume of the structure under stress constraints. The second is the minimization of the compliance for a given volume. For multiple load cases an arising issue is which energy quantity should be the objective function. Regarding the sizing optimization of trusses, Rozvany proved that the solution of the established compliance based problems leads to results which are awkward and not equivalent to the solutions of minimization of the volume under stress constraints, unlike under single loading (the layouts would be the same if in the compliance problem the volume is set equal to the result of the first problem). In this paper, we introduce the envelope strain energy problem where we minimize the volume integral of the worst case strain energy of each point of the structure. We also prove that in the case of sizing optimization of statically non-indeterminate (the term non-indeterminate includes both statically determinate trusses and mechanisms) trusses, this compliance method gives the same optimal design as the stress based design method.