Equal curvature and equal constraint cantilevers: Extensions of Euler and Clebsch formulas

The flexure of cantilevers is one of the early problems, if not the first, to have been studied by the elasticity theoreticians. One considers axisymmetrical rods and rectangular section beams. This investigation concerns the case where the maximum stress is constant (Galilei-Euler-Clebsch problem) as well as the case where the curvature of the medium fiber is constant. For both cases, it is shown that the equations to solve belong to the same class. The research was into thickness distributions replying to those conditions under various loading cases. At the free end, the distributions obtained degenerate into a family for which the thickness is null, but contrary to a widely held opinion, they also and naturally give forms showing a finite thickness at this end. The proposed distributions have a general form which has not been found in the literature treating elasticity theory or strength of materials [1-9]. They are extensions of Euler and Clebsch formulas.