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Strength of Shafts

shaft, moment, section, twisting, shearing, stress, pounds and cross-section

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A shaft is a part of a machine or system of machines, and is used to transmit power by virtue of its torsional strength, or resist ance .to twisting. Shafts are almost always made of metal and are usually circular in cross-section, being sometimes made hollow.

90. Twisting Moment. Let AF, Fig. 54, represent a shaft with four pulleys on it. Suppose that P is the driving pulley and that B, C and E are pulleys from which power is taken off to drive machines. The portions of the shafts between the pulleys are twisted when it is transmitting power; and by the twisting moment at any cross-section of the shaft is meant the algebraic sum of the moments of all the forces acting on the shaft on either side of the section, the moments being taken with respect to the axis of the shaft. Thus, if the forces acting on the shaft (at the pulleys) are and as shown, and if the arms of the forces or radii of the pulleys are a„ a„ and a, respectively, then the twisting moment at any section in Like bending moments, twisting moments are usually ex pressed in inch-pounds.

Example. Let ct,.= = = 15 inches, = 30 inches, P, = 400 pounds, P, = 500 pounds, P, = 750 pounds, and = 600 pounds.* What is the value of the greatest twisting moment in the shaft ? At any section between the first and second pulleys, the twisting moment is 400 X 15 = 6,000 inch-pounds; at any section between the second and third it is 400 X 15 + 500 X 15 = 13,500 inch-pounds; and at any section between the third and fourth it is 400 x 15 + 500 x 15 — 750 X 30 = — 9,000 inch-pounds. Hence the greatest value is 13,500 inch-pounds.

91. Torsional Stress. The stresses in a twisted shaft are called "torsional" stresses. The torsional stress on a cross-section of a shaft is a shearing stress, as in the case illustrated by Fig. 55, which represents a flange coupling in a shaft. Were it not for the bolts, one flange would slip over the other when either part of the shaft is turned; but the bolts prevent the slipping. Obvi ously there is a tendency to shear the bolts off unless they are screwed up very tight; that is, the material of the bolts is sub jected to shearing stress.

Just so, at any section of the solid shaft there is a tendency for one part to slip past the other, and to prevent the•slipping or *Note. These numbers were so chosen that the moment of P (driving moment) equals the sum of the moments of the other forces. This is always

the case in a shaft rotating at constant speed; that is, the power given the shaft equals the power taken off.

shearing of the shaft, there arise shearing stresses at all parts of the cross-section. The shearing stress on the cross-section of a shaft is not a uniform stress, its value per unit-area being zero at the center of the section, and increasing toward the circumference. In circular sections, solid or hollow, the shearing stress per unit area (unit-stress) varies directly as the distance from the center of the section, provided the elastic limit is not exceeded. Thus, if the shearing unit-stress at the circumference of a section is Fig. 55.

1,000 pounds per square inch, and the diameter of the shaft is 2 inches, then, at inch from the center, the unit-stress is 500 pounds per square inch; and at inch from the center it is 250 pounds per square inch. In Fig. 55 the arrows indicate the values and the directions of the shearing stresses on very small portions of the cross-section of a shaft there represented.

92. Resisting Moment. By "resisting moment" at a sec tion of a shaft is meant the sum of the moments of the shearing stresses on the cross-section about the axis of the shaft.

Let denote the value of the shearing stress per unit-area (unit-stress) at the outer points of a section of a shaft; d the diameter of the section (outside diameter if the shaft is hollow); and the inside diameter. Then it can be shown that the re sisting moment is: For a solid section, 0.1963 0.1963 (d' — For a hollow section, 93. Formula for the Strength of a Shaft. As in the case of beams, the resisting moment equals the twisting moment at any section. If T be used to denote twisting moment, then we have the formulas : For solid circular shafts, 0.1963 c/' = T; For hollow circular shafts, 0'1963 d Ss (d di') =T. (15) In any portion of a shaft of constant diameter, the unit shearing stress is greatest where the twisting moment is greatest. Hence, to compute the greatest unit-shearing stress in a shaft, we first determine the value of the greatest twisting moment, substitute its value in the first or second equation above, as the case may be, and solve for S. It is customary to express T in inch-pounds and the diameter in inches, then being in pounds per square inch.

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