Optimum Design of Structures of Finite Number of Bars. Single Load Variant

Abstract

The central problem of the present chapter is the minimization of the compliance of bar structures. It turns out that the optimum design of trusses of minimal compliance leads to an auxiliary problem, which naturally appears in the problem of minimization of truss volume under the condition of stresses in bars being bounded from both sides. This auxiliary problem is a problem of linear programming, which makes it possible to construct optimal structures of huge number of bars. The numerical algorithm is based on the ground structure method. These optimum designs of huge number of bars are approximants of Michell's structures to be discussed in the next chapter. The optimal designs of grillages (Sect. 2.5) are governed by more complicated equations. Only additional approximations imposed pave the way towards the linear programming problems thus delivering approximate solutions of Prager-Rozvany grillages, being flexural counterparts of Michell's structures. 2.1 Trusses of Minimal Compliance and Fixed Volume The optimal trusses will be formed from straight bars connecting nodes whose positions are a priori fixed in a design domain. Let e be the maximal number of bars connecting the nodes. The loads are concentrated at the nodes. The subject of the consideration is construction of the trusses of the least compliance and a given cost whose nodes are chosen from the given set and the bars linking selected nodes lie within the given design domain. The cost of the design is assumed in the form e k=1 E A k l k = Λ (2.1)