A Note on the Optimum Distribution of Material in a Beam for Stiffness

Abstract

T HAS BEEN STATED by some authors1 that the maximum stiff­ ness of a beam for a given weight is attained when the strain energy is a minimum or when the stress is constant. It can be shown that this condition does not, generally, result in maximum stiffness; however, it closely approximates the optimum condition in some cases. The optimum distribution of material for torsional stiffness of tubular beams is such that the thickness is constant around any cross section, and, for positions along the axis of the tube, the wall thickness should be proportional to the square root of the torsional moment and inversely proportional to the enclosed area. For maximum bending stiffness, the effective flange thick­ ness should be proportional to the square root of the bending moment or the square root of the product of the moment and the axial length on the beam, depending on whether angular or linear deflections are being considered. For the case of torsion, stiffness is measured by the amount of angular rotation, and, if t is independent of s, is given by: