Please answer CSI.D$6$ a$,$b$,$d$,$f$.$ accompanying figure. Derive strength and stiffness performance index expressions analogous to Equations CS$1\mathrm{.}9$ and CS$1\mathrm{.}11$ for this bar.

CS$1.$D$6$ Consider the plate shown in the accompanying figure, supported at its ends and subjected to a force $F$ that is uniformly distributed over the upper face as indicated. The deflection $$ at the $\frac{L}{2}$ position is given by the expression

$=\frac{5F{L}^3}{32Ew{t}^3}$

$($b$)$ From the properties database presented in Appendix B $($of Introduction and Fundamentals$),$ select the metal alloys with stiffness performance indices greater than $26\mathrm{.}0($for $E$ and $$ in units of GPa and $\frac{g}{c}{m}^3,$ respectively$).$

Furthermore, the tensile stress at the underside and also at the $\frac{L}{2}$ location is equal to

$=\frac{3FL}{4w{t}^2}$

Case Study $1$ I Materials Selection for a Torsionally Stressed Cylindrical Shaft

$($a$)$ Derive stiffness and strength performance index expressions analogous to Equations CS$1\mathrm{.}9$ and CS$1\mathrm{.}11$ for this plate. $($Hint: Solve for $t$ in these two equations, and then substitute the resulting expressions into the mass equation, as expressed in terms of density and plate dimensions.$)$

$($b$)$ From the properties database in Appendix B $($of Introduction and Fundamentals$),$ select the metal alloys with stiffness performance indices greater than $1\mathrm{.}40($for $E$ and $$ in units of GPa and $\frac{g}{c}{m}^3,$ respectively$).$

$($c$)$ Also, using the cost database $[$Appendix C $($of Introduction and Fundamentals$)],$ conduct a cost analysis in the same manner as the case study. Relative to this

analysis and that in part $($b$),$ which alloy would you select on a stiffness$-$per$-$mass basis?

$($d$)$ Now select the metal alloys with strength performance indices greater than $5\mathrm{.}0($for ${}_y$ and $$ in units of MPa and $\frac{g}{c}{m}^3,$ respectively$),$ and rank them from highest to lowest $P.$

$($e$)$ Using the cost database, rank the materials in part $($d$)$ from least to most costly$.$ Relative to this analysis and that in part $($d$),$ which alloy would you select on a strength$-$per$-$mass basis?

$($f$)$ Which material would you select if both stiffness and strength are to be considered relative to this application? Justify your choice.