E$14\mathrm{.}12.$ Energy$-$efficient floor joists: embodied energy and shape. Floor joists are beams loaded in bending. They can be made of wood, of steel, or of steel$-$reinforced concrete, with the shape factors listed below. For a given bending stiffness and strength, which of these carries the lowest production$-$energy burden? The relevant data, drawn from the tables of Appendix A$,$ are listed.

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a$.$ Start with stiffness. Locate from Eq$.(10\mathrm{.}20)$ of the text the material index for stiffness$-$limited$,$ shaped beams of minimum mass. Adapt this to make the index for stiffness$-$limited$,$ shaped beams of embodied energy by multiplying density $$ by the embodied energy $\frac{?}{kg},H_p.$ Use the modified index to rank the three beams.

b$.$ Repeat the procedure, this time for strength, creating the appropriate index for strength$-$limited shaped beams at minimum energy content by adapting Eq$.(10\mathrm{.}28).$

What do you conclude about the relative energy$-$penalty of design with wood and with steel?

$\backslash$table$[[$Material$,$Density ${?}^{}(k\frac{g}{m^3}),$Modulus $E($GPa$),$Strength ${}_{f(MPa)},$Energy $H_p(M\frac{J}{k}g),$Shape Factor ${}_B^e,$Shape Factor ${}_B^f$