On the prediction of topology and local properties for optimal trussed structures

1 I n t r o d u c t i o n The object of this paper is to demonstrate a somewhat generalized formulation for problems in the optimal design of trusses. The analytical model for this formulation is stated as a constrained, nonlinear extremum problem, one which differs from familiar statements for the truss design problem mainly by the form of the argument in its isoperimetric constraint. This argument has total cost evaluated in terms of an independently specified vector of member unit relative cost factors. As would be expected, the consequence for optimal design solutions of change in the unit relative cost field is a first-order effect and, given sufficient insight into the interdependence between unit cost and design, this feature in the model may be exploited to advantage. In addition to the implementation of generalized cost in the model, the design objective is expressed in a form representing generalized compliance, i.e. the objective is evaluated as the sum over the truss of independently weighted nodal displacements (the usual minimum compliance objective is imbedded within this model as the case where the specified unit weights are taken to be proportional to the nodal loads). The generalized compliance feature leads naturally to a non-selfadjoint design problem, and a variational formulation is introduced that proves to be convenient for working with such problems. The design space for trussed structures is defined in terms of designated sets of support nodes and of interior nodes, and \ i