Show that the index for selecting materials for a strong panel with the dimensions shown in the figure, loaded in bending, with minimum embodied energy content is that with the largest value of \[M=\frac{\sigma_{y^{\prime}}^{1 / 2}}{H_{p} ho}\]

where \(H_{p}\) is the embodied energy of the material, \(ho\) its density and \(\sigma_{y}\) its yield strength. To do so, rework the panel derivation in Chapter 4, Materials Selection - The Basics (Eq. 4.9) replacing the stiffness constraint with a constraint on failure load \(F\) requiring that it exceed a chosen value \(F^{*}\) where

\[F=C_{2} \frac{I \sigma_{y}}{h L}>F^{*}\]

where \(C_{2}\) is a constant and \(I\) is the second moment of area of the panel, \(I=\frac{b h^{3}}{12}\).