Nondimensional Shape Optimization of Nonprismatic Beams with Sinusoidal Lateral Profile

The present paper deals with the optimal design of nonprismatic beams, i.e., beams with variable cross section. To set the optimization problem, Euler-Bernoulli unshearable beam theory is considered and the elastica equation expressing the transverse displacement as a function of the applied loads is reformulated into a system of four differential equations involving kinematic components and internal forces. The optimal solution (in terms of volume) must satisfy two constraints: the maximum Von Mises equivalent stress must not exceed an (ideal) strength, and the maximum vertical displacement is limited to a fraction of beam length. To evaluate the maximum equivalent stress in the beam, normal and shear stresses have been considered. The former was evaluated through the Navier formula, and the latter through a formula derived from Jourawsky and holding for straight and untwisted beams with bisymmetric variable cross sections. The optimal solutions as function of material unit weight, maximum strength, and applied load are presented and discussed. It is shown that the binding constraint is usually represented by the maximum stress in the beam, and that applied load and strength affect the solution more than material unit weight. To maintain the generality of the solution, the nondimensionalization according to Buckingham pi-theorem is implemented and a design abacus is proposed.