mech3301 2023 tutorial 4 free variables wall thickness t choice of mat

Question

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