Trucks rely on compressed air for braking and other power-actuated systems. The air is stored in one or a cluster of cylindrical pressure tanks like that shown here (length \(L\), diameter \(2 R\), 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.

Show that the mass and material cost of the tank described in relative to one made of low carbon steel are given by \[\frac{m}{m_{o}}=\left(\frac{ho}{\sigma_{y}}\right)\left(\frac{\sigma_{y, o}}{ho_{o}}\right) \text { and } \quad \frac{C}{C_{o}}=\left(\frac{C_{m} ho}{\sigma_{y}}\right)\left(\frac{\sigma_{y, ho}}{C_{m, ho} ho_{o}}\right)\]
where \(ho\) is the density, \(\sigma_{y}\) the yield strength and \(C_{m}\) the cost per \(\mathrm{kg}\) of the material, and the subscript ' \(o\) ' indicates values for mild steel.

Transcribed Image Text:

Function Constraints Objectives Air cylinder for truck Must not fail by yielding Diameter 2R and length L specifieda so the ratio Q-2R/L is fixed. Minimize mass m Minimize material cost C Free variables Wall thickness, t Choice of material