For a shaft subjected to pure torsion, as in Fig. 26, the angular deflection due to the load may be carried to a certain point before the limit of working fiber stress is exceeded. The equation worked out from mechanics for this condition, is: ao = 584 TL / Gd4, (45) which at once gives the number of degrees of angular deflection for a shaft whose modulus of elasticity, torsional moment, and length are known.
The shearing modulus of elasticity of ordinary shaft steel runs from 10,000,000 to 13,000,000, giving as an average about 12,000,000.
A / 366 = SL / πGd; or S = AπGd /360 L (46)
A twist of one degree in a length of twenty diameters is a usual allowance. Substituting A = 1, L = 20d,and G = 12,000, 000, we have:
S = 5,240 (nearly). (47)
This in a safe value for shearing fiber stress in steel. In fact, in calculations for strength, even for reversing stresses, the usual figure is 8,000 (lbs. per square inch), thus indicating that the relation of one degree to twenty diameters is well within the limit of strength.
For a hollow shaft the above formula becomes :
A0 = 584 TL / G(d04 - d14) (48)
Transverse deflection occurs when the shaft is subjected to a bending moment. It may therefore exist alone or in conjunction with angular deflection. Transverse deflection of shafts, however, rarely exists up to the point of limiting fiber stress, because before that point is readied the alignment of the shaft is so disturbed that it is not practicable as a device for transmitting power. A transverse deflection of .01 inch per foot of length is a common allowance ; but it is impossible to fix any general limit, as in many case this figure, if exceeded, would do no harm, while in others - such as heavily loaded or high-speed bearings - even the figure given might be fatal to good operation.
The formula for transverse deflection, deduced from mechanics, varies with the system of loading. The three most common conditions only are given below, reference to the handbook being necessary if other conditions must be satisfied:
Supported at ends, loaded in middle, e = WL3 / 48 EI. (50)
Supported at ends, loaded uniformly, e = 5WL3 / 384 EI. (51)
For transverse deflection the direct modulus of elasticity must be used, for the fibers are stretched or compressed, instead of being subjected to a shearing action. The most usual value of the direct modulus of elasticity for ordinary steel is 30,000,000, and is denoted in most books by the symbol E. Both the shearing and direct moduli of elasticity are really nothing but the ratio of the stress to the strain produced by that stress, it being assumed that the given material is perfectly elastic. A material is supposed to be perfectly elastic up to a certain limit of stress, and it is within this limit that the relation as above holds good.
Expressed in the form of an equation this would be:
E = S /e/L = Sl / e (52)