Calculation Procedure:
- Compute the radial pressure caused by prestressing
Use the relation;? = 2&D/{L>^2 [l/(taEa) + l/fe£ 5 )]}, where/? = radial pressure resulting
from prestressing, lb/in^2 (kPa), with other symbols the same as in the previous calculation
procedure and the subscripts a and s referring to aluminum and steel, respectively. Thus,
p = 2(0.01)/{9^2 [1/(0.5 xlOx 106 ) + 1/(0.25 >( 30 x 106 )]) = 741 lb/in^2 (5109.2 kPa).
- Compute the corresponding prestresses
Using the subscripts 1 and 2 to denote the stresses caused by prestressing and internal
pressure, respectively, we find sal = pD/(2ta), where the symbols are the same as in the
previous calculation procedure. Thus, sal = 741(9)/[2(0.5)] = 6670-lb/in^2 (45,989.7-kPa)
compression. Likewise, ^ 1 = 741(9)/[2(0.25)] = 13,340-lb/in^2 (91,979-kPa) tension.
- Compute the stresses caused by internal pressure
Use the relation ss2lsa2 = EJEa or, for this cylinder, ss2lsa2 = (30 x 106 )/(10 x 106 ) = 3.
Next, compute sa2 from t^a2 I 5 S 82 = pD/2, or sa2 = 800(9)/[2(0.5 + 0.25 x 3)] = 2880-
lb/in^2 (19,857.6-kPa) tension. Also, ss2 = 3(2880) - 8640-lb/in^2 (59,572.8-kPa) tension.
- Compute the final stresses
Sum the results in steps 2 and 3 to obtain the final stresses: sa3 = 6670 - 2880 = 3790-
lb/in^2 (26,132.1-kPa) compression; ss3 = 13,340 + 8640 = 21,980-lb/in^2 (151,552.1-kPa)
tension.
- Check the accuracy of the results
Ascertain whether the final diameters of the steel ring and aluminum cylinder are equal.
Thus, setting s' = O in A£> = (DIE)(s - vs'\ we find AD 0 = -3790(9)7(10 x 10
6
) = -0.0034
in (-0.0864 mm), D 0 = 9.0000 - 0.0034 = 8.9966 in (228.51 mm). Likewise, AZ) 5 =
21,980(9)7(30 x io^6 ) = 0.0066 in (0.1676 mm), D 5 = 8.99 + 0.0066 = 8.9966 in (228.51
mm). Since the computed diameters are equal, the results are valid.
HOOP STRESS IN THICK-WALLED CYLINDER
A cylinder having an internal diameter of 20 in (508 mm) and an external diameter of 36
in (914 mm) is subjected to an internal pressure of 10,000 lb/in^2 (68,950 kPa) and an ex-
ternal pressure of 2000 lb/in^2 (13,790 kPa) as shown in Fig. 22. Determine the hoop stress
at the inner and outer surfaces of the cylinder.
Calculation Procedure:
- Compute the hoop stress at the inner surface of the cylinder
Use the relation S 1 = [/?i(r^2 + r^2 ) - 2/? 2 r^2 ]/Oi - r^2 ), where st = hoop stress at inner surface,
lb/in^2 (kPa);/>! = internal pressure, lb/in^2 (kPa); ^ 1 = internal radius, in (mm); r 2 = external
radius, in (mm);/? 2 = external pressure, lb/in^2 (kPa). Substituting gives st = [10,000(100 +
- 2(2000)(324)]/(324 - 100) - 13,100-lb/in^2 (90,324.5-kPa) tension.
- Compute the hoop stress at the outer cylinder surface
Use the relation S 0 = [2/?ir^2 -p 2 (r\ + r 2 }]l(r 2 - r^2 ), where the symbols are as before. Sub-
stituting gives S 0 = [2(10,00O)(IOO) - 2000(100 + 324)]/(324 - 100) = 5100-lb/in^2
(35,164.5-kPa) tension.
