Optimal design of a beam subject to bending: a basic application

The minimisation of both the mass and deflection of a beam in bending is addressed in the paper. To solve the minimisation problem, a multi-objective approach is adopted by imposing the Fritz John conditions for Pareto-optimality. Constraints on the maximum stress and elastic stability (buckling) of the structure are taken into account. Additional constraints are set on the beam cross section dimensions. Three different cross sections of the beam are analysed and compared, namely the hollow square, the I-shaped and the hollow rectangular cross sections. The analytical expressions of the Pareto-optimal sets are derived. As expected, the I-shaped beam exhibits the best compromise in structural performance, which is related on the particular loading considered.