Cantilever beams of uniform strength
1. The object of the paper, its methods and results. The problem of shaping a beam from a given amount of material in such a manner as to obtain maximum strength requires that the maximum stress of each cross section be constant. In the case of bending, the classical treatment of this problem1 '2'M is based on the theory of beams of constant cross section, the influence of shearing stresses and of the weight of the beam being neglected. A collection of solutions of this elementary problem, for rectangular and circular cross sections, is given in the Hiitte handbook for engineers.5 If the strength of the material is relatively low, the weight W of the beam cannot be neglected. This occurs in certain concrete structures, such as reinforced concrete bridges, and was demonstrated by Gaede6 in his treatment of a cantilever of rectangular cross section and constant width, the external load being a force F at the free end. In the present paper, we shall consider cantilevers of more general cross section but with the same type of loading, except in §6 where more general loading will be considered. Let us denote by x the distance from the free end, by A (x) the area of the cross section and by S(x) the section modulus (5 = M/a, where M is the bending moment and a is the maximum stress). The bending moment M(x) =