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