If the stresses in the portion of the shaft considered do not exceed the elastic limit, and if the twisting moment is the same for all sections of the portion, then the angle of torsion a (in degrees) can be computed from the following: For solid circular shafts, Here T, 1, d, H, and n have the same meanings as in Arts. 93 and 94, and should be expressed in the units there used. The letter E' stands for a quantity called coefficient of elasticity for shear; it is analogous to the coefficient of elasticity for tension and compression (E), Art. 95. The values of E' for a few materials average about as follows (roughly EI = - E) : For Steel, 11,000,000 pounds per square inch.
For Wrought iron, 10,000,000 " " " " For Cast iron, 6,000,000 " " " Example. What is the value of the angle of torsion of a steel shaft 60 feet long when transmitting 6,000 horse-power at 50 revolutions per minute, if the shaft is hollow and its outer and inner diameters are 16 and 8 inches respectively I Here / = 720 inches; hence, substituting in the appropriate formula (19), we find that 36,800,000 X 6,000 x 720 a = 4.7 degrees.
11,000,000 X — 50 Suppose that the first two pulleys in Fig. 54 are 12 feet apart; that the diameter of the shaft is 2 inches; and that P, = 400 pounds, and a, = 15 inches. If the shaft is of wrought iron,
what is the value of the angle of torsion for the portion between the first two pulleys? Ans. 3.15 degrees.
99. Deformation. The preceding formulas for elongation, deflection, and twist hold only so long as the greatest unit-stress does not exceed the elastic limit. There is no theory, and no formula, for non-elastic deformations, those corresponding to stresses which exceed the elastic limit. It is well known, how ever, that non-elastic deformations are not proportional to the forces producing them, but increase much faster than the loads. The value of the ultimate elongation of a rod or bar (that is, the amount of elongation at rupture), is quite well known for many materials. This elongation, for eight-inch specimens of various materials (see Art. 16), is : Specimens of ductile materials (such as wrought iron and structural steel), when pulled to destruction, neck down, that is, diminish very considerably in cross-section at some place along the length of the specimen. The decrease in cross-sectional area is known as reduction of area, and its value for wrought iron and steel may be as much as 50 per cent.