Explore the trade-off between relative cost and relative mass for the air cylinder of Exercise E8.7, considering the replacement of a mild steel tank with one made, first, of low alloy steel, and, second, one made of filament-wound CFRP. The relevant material properties in the table below. Define a relative penalty function

\[Z^{*}=\alpha^{*} \frac{m}{m_{o}}+\frac{C}{C_{o}}\]

where \(\alpha^{*}\) is a relative exchange constant, and evaluate \(Z^{*}\) for \(\alpha^{*}=1\) and for \(\alpha^{*}=100\).

Data From Exercise E8.7

The air is stored in one or a cluster of cylindrical pressure tanks like that shown here (length L, diameter 2R, hemispherical ends). Most are made of low carbon steel, and they are heavy. The task: to explore the potential of alternative materials for lighter air tanks, recognizing that there must be a trade-off between mass and cost – if it is too expensive, the truck owner will not want it even if it is lighter. The table summarizes the design requirements.

 

Material Density p (kg/m) Yield strength (MPa) Cm Mild steel 7850 314 Low alloy steel 7850 775 CFRP 1550 760 Price per/kg C, ($/kg) 0.55 0.85 42.1