$3-($a$)$ A porous structure fabricated using additive manufacturing is modeled as a simply

supported beam with a span L subjected to an applied force F$=500$N distributed

uniformly along its length. The structure has a density $\backslash$rho $\_($p$)=0\mathrm{.}6\backslash$rho and a mass m in grams.

To ensure sufficient stiffness, the deflection $\backslash$delta under the applied force must be less than

$(1)/(500)$ of the span length L$.$ This leads to the stiffness requirement:

S$\_($B$)>=(500$F$)/($L$)$

Take the density of the material $\backslash$rho $=7900$k$($g$)/($m$^(3)),$ its modulus E$=205$GPa, and the constant

C$\_(1)=76\mathrm{.}8.$

Using this information, calculate the minimum shape factor $\backslash$phi $\_($B$)^($e$)$ for the porous structures

with the spans and masses listed below.

$($b$)$ Repeat the solution for a titanium alloy with $\backslash$rho $=4500$k$($g$)/($m$^(3))$ and E$=110$GPa, and an

aluminum alloy with $\backslash$rho $=2700$k$($g$)/($m$^(3))$ and E$=70$GPa. Which of these three materials gives

the best shape factor?