A well-posed non-selfadjoint layout problem: Least-weight plane truss for one load condition and two displacement constraints

Abstract

Exact optimal plane truss layouts are derived for a vertical support and a concentrated load with two displacement constraints. The latter are imposed at the point of application of the load, in the direction of the load and in another direction. It is shown that for the above class of problems the optimal solution always consists of two symmetrically positioned bars. These solutions are derived analytically by two independent methods: (i) in the first one a two-bar topology is assumed and then the orientations and cross-sectional areas of the bars are optimized; (ii) in the second one, the same optimal solutions are derived from general optimality criteria, which show that the optimum is valid even when we consider all possible topologies. The paper demonstrates the power and versatility of continuum-type optimality criteria and also shows that for two displacement constraints at a loaded point the problem is non-selfadjoint but always well-posed, having a stationary optimum with a finite structural weight. The exact layout solutions given in this paper can be used as test examples for numerical methods in topology optimization.