A unified relaxed approach easing the practical application of a paradox in curved beams

The paper deals with an arising paradox in curved beams subjected to bending moment and normal force. This paradox consists in the fact that by laterally removing material from section zones close to the neutral axis, not only an obvious reduction of the beam mass can be obtained, but also an unexpected, though technically negligible, reduction of the bending stress. It has recently been shown that the relaxation of the demanding achievement of a concurrent mass and stress reduction may practically lead to interesting results, yet solvable numerically. In this paper we show that, under some mild assumptions, a remarkable simplification of the intrados stress functional is obtained. Hence, a unified approximate mathematical approach based on linearization is developed for the derivation of analytical closed-form solutions for the lateral grooved zones. A practical example of the application of the relaxed paradox to optimize a crane hook subjected to bending and normal force is illustrated and compared to finite element forecasts.