Page 52 - Handbook of Civil Engineering Calculations, Second Edition

STATICS, STRESS AND STRAIN, AND FLEXURAL ANALYSIS 1.35

Calculation Procedure:

1. Compute the radial pressure caused by prestressing
2
Use the relation p 2 D/{D [1/(t a E a ) 1/(t s E s )]}, where p radial pressure resulting from
prestressing, lb/sq.in. (kPa), with other symbols the same as in the previous calculation pro-
cedure and the subscripts a and s referring to aluminum and steel, respectively. Thus, p
2
6
6
2(0.01)/{9 [1/(0.5 10 10 ) 1/(0.25 >( 30 10 )]} 741 lb/sq.in. (5109.2 kPa).
2. Compute the corresponding prestresses
Using the subscripts 1 and 2 to denote the stresses caused by prestressing and internal pres-
sure, respectively, we find s a1 pD/(2t a ), where the symbols are the same as in the previ-
ous calculation procedure. Thus, s a1 741(9)/[2(0.5)] 6670-lb/sq.in. (45,989.7-kPa)
compression. Likewise, s s1 741(9)/[2(0.25)] 13,340-lb/sq.in. (91,979-kPa) tension.
3. Compute the stresses caused by internal pressure
6
6
Use the relation s s2 /s a2 E s /E a or, for this cylinder, s s2 /s a2 (30 10 )/(10 10 ) 3.
Next, compute s a2 from t a s a2 t s s s2 pD/2, or s a2 800(9)/[2(0.5 0.25 3)] 2880-
lb/sq.in. (19,857.6-kPa) tension. Also, s s2 3(2880) 8640-lb/sq.in. (59,572.8-kPa)
tension.
4. Compute the final stresses
Sum the results in steps 2 and 3 to obtain the final stresses: s a3 6670 2880 3790-
lb/sq.in. (26,132.1-kPa) compression; s s3 13,340 8640 21,980-lb/sq.in.
(151,552.1-kPa) tension.
5. Check the accuracy of the results
Ascertain whether the final diameters of the steel ring and aluminum cylinder are equal.
6
Thus, setting s
0 in. D (D/E)(s vs
), we find D a 3790(9)/(10 10 )
0.0034 in. ( 0.0864 mm), D a 9.0000 0.0034 8.9966 in. (228.51 mm). Like-
6
wise, D s 21,980(9)/(30 10 ) 0.0066 in. (0.1676 mm), D s 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/sq.in. (68,950 kPa) and an
external pressure of 2000 lb/sq.in. (13,790 kPa) as shown in Fig. 22. Determine the hoop
stress at the inner and outer surfaces of the cylinder.

Calculation Procedure:
1. Compute the hoop stress at the inner surface of the cylinder
2
2
2
2
2
Use the relation s i [p 1 (r 1 r 2 ) 2p 2 r 2 ]/(r 2 r 1 ), where s i hoop stress at inner sur-
face, lb/sq.in. (kPa); p 1 internal pressure, lb/sq.in. (kPa); r 1 internal radius, in. (mm);
r 2 external radius, in. (mm); p 2 external pressure, lb/sq.in. (kPa). Substituting gives
s i [10,000(100 324) 2(2000)(324)]/(324 100) 13,100-lb/sq.in. (90,324.5-
kPa) tension.
2. Compute the hoop stress at the outer cylinder surface
2
2
2
2
2
Use the relation s 0 [2p 1 r 1 p 2 (r 1 r 2 )]/(r 2 r 1 ), where the symbols are as before.
Substituting gives s 0 [2(10,000)(100) 2000(100 324)]/(324 100) 5100-
lb/sq.in. (35,164.5-kPa) tension.