The optimum shape of a bending beam

The problem of minimizing the static deflection of an elastic beam of variable cross-section and fixed volume in the case of free supported and rigidly clamped ends is considered. In the first case it is proved that the solutions obtained earlier, based on the necessary conditions for an extremum, satisfy the sufficient conditions. In the case of clamped ends, which is of the most interest from the point of view of applications, it is proved that the optimum solutions must necessarily have points inside the solution range in which the distribution of the beam thicknesses degenerates to zero ("internal hinges"). A qualitative, analytical and numerical analysis of this phenomenon is given. In particular, in the case of clamped ends for a class of point loads, analytical solutions for which the beam splits into two cantilevers are obtained. (C) 2001 Elsevier Science Ltd. All rights reserved.