It is a well known fact that in some mechanical components stresses can be relieved by wisely removing material. As far as beams are concerned, this technique works for straight ones subject to bending, where the zones from which a material removal, from a generic initial cross section, produces a bending stress reduction are farthest from the neutral axis. Recently, it has been shown that curved beams modelled in terms of Winkler theory present the same paradoxical behaviour, provided that the removal occurs close to the neutral axis instead. In this article, the delimiting radii of the grooved zone are analytically determined under mild hypotheses. Finally, an application to the optimization of a crane hook has been developed showing a nontrivial mass reduction and optimal solutions have been confirmed by finite element analysis
