Page 230 - Materials Science and Engineering An Introduction

202 • Chapter 6 / Mechanical Properties of Metals

We now incorporate into this equation values of N, r i , ≤p, and t given in the problem state-
ment and solve for s y . Alloys in Table 6.8 that have yield strengths greater than this value
are suitable candidates for the tubing. Therefore,
-3
(4.0)(50 * 10 m)(2.027 MPa - 0.057 MPa)
s y = -3 = 197 MPa
(2 * 10 m)
Four of the six alloys in Table 6.8 have
yield strengths greater than 197 MPa and
satisfy the design criterion for this tube—
that is, steel, copper, brass, and titanium.
(b) To determine the tube cost for each alloy, t
it is i rst necessary to compute the tube r i
volume V, which is equal to the product of r
o
cross-sectional area A and length L—that is,
L
V = AL
2
2
= p(r o - r i )L (6.27)
are, respectively, the tube
Here, r o and r i
inside and inside radii. From Figure 6.21, Figure 6.21 Schematic representation of a cy-
it may be observed that r o = r i + t, or that lindrical tube, the subject of Design Example 6.2.
2
2
2
2
V = p(r o - r i )L = p[(r i + t) - r i ]L
2
2
2
= p(r i + 2r i t + t - r i )L
2
= p(2r i t + t )L (6.28)

Because the tube length L has not been speciied, for the sake of convenience, we assume a
value of 1.0 m. Incorporating values for r i and t, provided in the problem statement leads to
the following value for V:
-3
2
-3
-3
V = p[(2)(50 * 10 m)(2 * 10 m) + (2 * 10 m) ](1 m)
3
-4
= 6.28 * 10 m = 628 cm 3
Next, it is necessary to determine the mass of each alloy (in kilograms) by multiplying this value of
V by the alloy’s density, r (Table 6.8) and then dividing by 1000, which is a unit-conversion factor
because 1000 mm = 1 m. Finally, cost of each alloy (in $US) is computed from the product of this
mass and the unit mass cost (c) (Table 6.8). This procedure is expressed in equation form as follows:
Vr
Cost = a b(c) (6.29)
1000
For example, for steel,
3
3
(628 cm )(7.8 g>cm )
Cost (steel) = c d (1.75 $US>kg) = $8.60
(1000 g>kg)
Cost values for steel and the other three alloys, as determined in the same manner are tabu-
lated below.
Alloy Cost ($US)
Steel 8.60
Copper 41.90
Brass 53.40
Titanium 240.20
Hence, steel is by far the least expensive alloy to use for the pressurized tube.