E$4\mathrm{.}13$ Material index for a light, strong beam. In stiffnesslimited applications, it is elastic deflection thas is the active constraint: it limits performance. In strength$-$limited applications, deflection is acceptable provided the component does not fail; strength is the active constraint. Derive the material index for selecting materials for a beam of length $L,$ specified strength and minimum weight. For simplicity, assume the beam to have a solid square crosssection $tt.$ You will need the equation for the failure load of a beam $($Appendix B$,$ Section B$4).$ It is

$F_f=\frac{I{}_f}{y_{m}L}$

where $y_m$ is the distance between the neutral axis of the beam and its outer filament and $I=\frac{t^4}{12}=\frac{A^2}{12}$ is the second moment of the cross$-$section. The table itemizes the design requirements.

Plot the criterion on a copy of the $({}_f-)$ chart of Figure $3\mathrm{.}4$ and use it to identify promising candidate materials.