Page 56 - Handbook of Civil Engineering Calculations, Second Edition

STATICS, STRESS AND STRAIN, AND FLEXURAL ANALYSIS 1.39

Calculation Procedure:

1. Compute the temperature rise required
Use the relation T D/(cD), where T temperature rise required, °F (°C); D
change in cylinder diameter, in. (mm); c coefficient of expansion of the cylinder 6.5
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10 /°F (1.17 10 /°C); D cylinder internal diameter before heating, in. (mm). Thus
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T (3/32)/[6.5 10 (48)] 300°F (167°C).
2. Compute the hoop stress in the cylinder
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Upon cooling, the cylinder has a diameter /32 in. (0.8 mm) larger than originally. Compute
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the hoop stress from s E D/D 30 10 ( /32)/48 19,500 lb/sq.in. (134,452.5 kPa).
3. Compute the associated radial pressure
Use the relation p 2ts/D, where p radial pressure, lb/sq.in. (kPa), with the other sym-
bols as given earlier. Thus p 2(5/16)(19,500)/48 254 lb/sq.in. (1751.3 kPa).

TORSION OF A CYLINDRICAL SHAFT

A torque of 8000 lb·ft (10,840 N·m) is applied at the ends of a 14-ft (4.3-m) long cylindri-
cal shaft having an external diameter of 5 in. (127 mm) and an internal diameter of 3 in.
(76.2 mm). What are the maximum shearing stress and the angle of twist of the shaft if the
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modulus of rigidity of the shaft is 6 10 lb/sq.in. (4.1 10 MPa)?

Calculation Procedure:

1. Compute the polar moment of inertia of the shaft
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For a hollow circular shaft, J ( /32)(D d ), where J polar moment of inertia of a
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transverse section of the shaft with respect to the longitudinal axis, in (cm ); D external
diameter of shaft, in. (mm); d internal diameter of shaft, in. (mm). Substituting gives
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J ( /32)(5 3 ) 53.4 in (2222.6 cm ).
2. Compute the shearing stress in the shaft
Use the relation s s TR/J, where s s shearing stress, lb/sq.in. (MPa); T applied
torque, lb·in. (N·m); H radius of shaft, in. (mm). Thus s s [(8000)(12)(2.5)]/53.4
4500 lb/sq.in. (31,027.5 kPa).
3. Compute the angle of twist of the shaft
Use the relation TL/JG, where angle of twist, rad; L shaft length, in. (mm);
G modulus of rigidity, lb/sq.in. (GPa). Thus (8000)(12)(14)(12)/[53.4(6,000,000)]
0.050 rad, or 2.9°.
ANALYSIS OF A COMPOUND SHAFT
The compound shaft in Fig. 24 was formed by rigidly joining two solid segments. What
torque may be applied at B if the shearing stress is not to exceed 15,000 lb/sq.in. (103.4
MPa) in the steel and 10,000 lb/sq.in. (69.0 MPa) in the bronze? Here G s 12 10 6
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lb/sq.in. (82.7 GPa); G b 6 10 lb/sq.in. (41.4 GPa).