Maximal torsional rigidity: some qualitative remarks

Abstract

We give an existence result and some qualitative remarks about the optimisation of the torsional rigidity of a beam.

Optimization of cross-section of hollow prismatic bars in torsion

“…If 1 is a circle then it is a classical result [6] that a solution to the problem is the concentric circle. In a general case it is known [7] that under certain hypotheses upon 1 the problem has a solution. Note that the ÿxed boundary 1 can be either the inner or outer boundary [8].…”

“…It was found that of all doubly connected domains with the prescribed area of the hole and the cross section the ring bonded by two concentric circles has the maximal torsional rigidity. The problem is of great interest for both applied and theoretical mathematics and it is still a subject of the active research [7]. The problem becomes of considerable practical interest if either the inner or outer boundary is ÿxed and the remaining boundary has to be determined such that for the prescribed area of the cross section torsional rigidity of the domain is maximal.…”

On a variational principle for shape optimization and elliptic free boundary problems

“…Using a relaxed problem defined on a ball B R as before, the existence of an optimal domain Ω R has been proved in [14]. In [22], Lederman and in [30] R. Tahraoui prove that the domain Ω R is connected and, for a radius R large enough, it solves the original problem. If Γ 0 is C 2 , C. Lederman [22] has recently proved that the exterior boundary Γ of the optimal set Ω R is locally analytic.…”