MECH3301 2023, Tutorial 4 Free variables: wall thickness t; choice of material a. Show that the mass and material cost of the tank relative to one made of low carbon steel are given by m m₂ 60-4GD) and b. Explore the trade-off between relative cost and relative mass, considering the replacement of a mile steel tank with one made, first, of low alloy steel and, second, one made of CFRP, using the material properties in the table below. Define a relative penalty function: Z* = a* m m + Material Mild steel Low alloy steel CFRP Density (kg/m³) 7850 7850 1550 Yield Strength (MPa) 314 775 760 Price per kg Cm (S/kg) 0.66 0.85 42.1 c. Figure E.18, and the chart below it, shows the axes of m/mo and C/Co. Mild steel (here labeled "Low carbon steel" lies at the coordinates (1,1). Sketch a trade-off surface and plot contours of Z" that are approximately tangent to the trade- off surface for a = 1 and for a = 100. What sections do these suggest? Plot with y axis → cost, x axis → mass a-Solve it as in the lecture notes for three exchange constants, a = 18/kg; a = 20$ kg (upper bound to the exchange constant for trucks) and a = 100 $/kg (close to the upper bound for airplanes). Adjust the Z*-value to ensure that the penalty functions pass through the center of the low carbon steel bubble. b-Use the line for a = 20 $/kg to decide which materials are too expensive and which ones are OK (affordable) for a lighter tank. c-Plot three or four lines with a = 18 kg but decreasing Z values to see the effect of selecting inside the win-win corner. Plot with y axis → mass, x axis cost Follow steps a-c shown above to solve this exercise. Note that the Z* values in the figure (axes y= mass and x= cost) relate to the Z values of the figure (y= cost and x= mass) by Z*-Z/a* (prove it!)/nQ3 A truck's air cylinder Trucks rely on compressed air for braking and other powder actuated systems. The air is stored in one or a cluster of cylindrical pressure tanks like that shown in the figure below (length L, diameter 2R, hemispherical ends). Most are made of low carbon steel and 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. Pressure p 2R L Aspect ratio Q-2R/L The design requirements: Function: air cylinder for truck Constraints: must not fail by yielding; diameter 2R and length L specified, so the ratio Q=2R/L is fixed Objectives: minimize mass m; minimize material cost C

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