A beam of length \(L\), loaded in bending, must support a specified bending moment \(M\) without failing and be as light as possible. Show that to minimize the mass of the beam per unit length, \(m / L\), one should select a material and a section-shape to maximize the quantity

\[M=\frac{\left(\phi_{B}^{f} \sigma_{f}\right)^{2 / 3}}{ho}\]

where \(\sigma_{f}\) is the failure stress and \(ho\) the density of the material of the beam, and \(\phi_{B}^{f}\) is the shape-efficiency factor for failure in bending.