Two extensions towards practical applications of a paradox in curved beams

For particular sections of a rectilinear beam subjected to bending it is possible to simultaneously lower the bending stress and reduce the beam mass, by wisely removing material from zones far from the neutral axis. It has recently been shown that an analogous paradoxical behaviour occurs in a curved beam subjected to bending, by laterally removing material from section zones close to the neutral axis. The bending stress diminution is often of the order of a few %, whereas the mass diminution may reach 10 %. To get practically more interesting results, in this paper the demanding achievement of a concurrent stress and mass reduction is relaxed in favour of two weaker requests: a) for a general section, the intrados stress is assumed as the reference stress, and the maximum mass reduction achieved by laterally removing material is sought under the condition that the intrados stress equals such reference stress; b) the intrados stress of a particular section is assumed as the reference stress, and material is laterally removed from the adjacent sections until their intrados stress equals the above reference value. Analytical applications of the two above approaches to a crane hook are carried out and compared to FE forecasts.