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The pitch should not in general be greater than one-fifth of the effective diameter, and may be considerably less: for example, onetenth and one-twelfth are ordinary proportions.

In order that the resistance of a screw or screw-bolt to rupture by stripping a triangular thread may be at least equal to its resistance to direct tearing asunder, the length of the nut should be at least one-half of the effective diameter of the screw; and it is often in practice considerably greater; for example, once and a half that diameter.

The head of a bolt is usually about twice the diameter of the spindle, and of a thickness which is usually greater than fiveeighths of that diameter.

SECTION IV.-Of Resistance to Twisting and Wrenching.

428. Twisting or Torsion in General.—Torsion is the condition of strain into which a cylindrical or prismatic body is put when a pair of couples of equal and opposite moment, tending to make it rotate about its axis in contrary directions, are applied to its two ends. Such is the condition of shafts which transmit motive power. The moment is called the twisting moment, and at each crosssection of the bar it is resisted by an equal and opposite moment of stress. Each particle of the shaft is in a state of distortion, and exerts shearing stress.

In British measures, twisting moments are expressed in inch-lbs. 429. Strength of a Cylindrical Shaft. (A. M., 321.)—A cylindrical shaft, A B, fig. 267, being subjected to the twisting moment of a

Fig. 267.

B

pair of equal and opposite couples applied to the cross-sections A and B, it is required to find the condition of stress and strain at any intermediate cross-section, such as S, and also the angular displacement of any crosssection relatively to any other.

From the uniformity of the figure of the bar, and the uniformity of the twisting moment, it is evident that the condition of stress and strain of all cross-sections is the same; also, because of the circular figure of each cross-section, the condition of stress and strain of all particles at the same distance from the axis of the cylinder must be alike.

Suppose a circular layer to be included between the cross-section S, and another cross-section at the longitudinal distance d x from it. The twisting moment causes one of those cross-sections to rotate relatively to the other, about the axis of the cylinder, through an angle which may be denoted by d . Then if there be two points at the same distance, r, from the axis of the cylinder, one in

the one cross-section and the other in the other, which points were originally in one straight line parallel to the axis of the cylinder, the twisting moment shifts one of those points laterally, relatively to the other, through the distance r de. Consequently the part of the layer which lies between those points is in a condition of distortion, in a plane perpendicular to the radius r; and the distortion is expressed by the ratio

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which varies proportionally to the distance from the axis. There is therefore a shearing stress at each point of the cross-section, whose direction is perpendicular to the radius drawn from the axis to that point, and whose intensity is proportional to that radius, being represented by

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The STRENGTH of the shaft is determined in the following manner:-Let be the limit of the shearing stress to which the material is to be exposed, being the ultimate resistance to wrenching if it is to be broken, the proof resistance if it is to be tested, and the working resistance if the working moment of torsion is to be determined. Let r1 be the external radius of the axle. Then q, is the value of q at the distance r1 from the axis; and at any other distance, r, the intensity of the shearing stress is

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Conceive the cross-section to be divided into narrow concentric rings, each of the breadth dr. Let r be the mean radius of one of these rings. Then its area is 2 ar dr; the intensity of the shearing stress on it is that given by equation (3), and the leverage of that stress relatively to the axis of the cylinder is r; consequently the moment of the shearing stress of the ring in question, being the product of those three quantities, is

2 = 91.3 dr; Τι

which being integrated for all the rings from the centre to the circumference of the cross-section, gives for the moment of torsion, and of resistance to torsion,

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if h

=

2 r1 be the diameter of the shaft

(1-5708;0-196 nearly).

If the axle is hollow, he being the diameter of the hollow, the moment of torsion becomes

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The following formulæ serve to calculate the diameters of shafts when the twisting moment and stress are given; solid shafts :

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which last formula serves to compute the diameter of a hollow axle, when the ratio ho h1 of its internal and external diameter has been fixed. (See pages 581, 584.)

Values of the ultimate shearing strength of various substances are given in the Tables. As for the working stress, a long series of practical trials has shown that wrought-iron axles bear a stress of 9,000 lbs. per square inch, or 6-3 kilogrammes on the square millimètre, for any length of time, if well manufactured of good material, the factor of safety being about 6. If the ultimate shearing stress of cast iron, 27,000 lbs. on the square inch, is divided by the same factor, the modulus of working stress is found to be 4,500 lbs. on the square inch, or nearly 3-2 kilogrammes on the square millimètre.

It is chiefly in the shafting of mills that those large apparent factors of safety are met with, referred to in Article 414, page 490. 430. Angle of Torsion.—Suppose a pair of diameters, originally parallel, to be drawn across the two circular ends, A and B, fig. 267, page 500, of a cylindrical shaft, solid or hollow; it is proposed to find the angle which the directions of those lines make with each other when the shaft is twisted, either by the working moment of torsion, or by any other moment.

This question is solved by means of equation (2) of Article 429, page 501, which gives for the angle of torsion per unit of length,

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The condition of the shaft being uniform at all points of its length, the above quantity is constant; and if x be the length of the shaft, and the angle of torsion sought, expressed in length of arc to 9 do

radius 1, we have

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and therefore,

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I. Let the moment of torsion be the working moment, for which

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the value taken for the modulus, 91, being the safe working stress. Then the angle of working torsion is

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This formula

and is the same whether the shaft is solid or hollow. gives the angle in circular measure; that is, in arc to radius unity; so that if at each end of the shaft there is an arm of the length y, the displacement of the end of one of those arms relatively to the other will be y e.

Values of C, the co-efficient of transverse elasticity, are given in the tables. In calculating the working torsion of wrought-iron shafts, we may make

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II. The proof torsion, to which a shaft may be twisted by a gradually applied load when testing it, may be made double the working torsion.

III. Let the moment of torsion have any amount, M, consistent with safety. Then for we have to put its value in terms of M and h1; and the results are as follows:—

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An example of the application of equation (4) has already been given in Article 344.

431. The Resilience of a Cylindrical Shaft is the product of onehalf of the moment of proof torsion into the corresponding angle of torsion; and it is given by the following equation:

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432. Shafts not Circular in Section.-When the cross-section of

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of the shearing

a shaft is not circular, it is certain that the ratio stress at a given point to the distance of that point from the axis of the shaft is not a constant quantity at different points of the cross-section, and that in many cases it is not even approximately constant; so that formulæ founded on the assumption of its being constant are erroneous. The mathematical investigations of M. de St. Venant have shown how the intensity of the shearing stress is distributed in certain cases.

The most important case in practice to which M. de St. Venant's method has been applied is that of a square shaft; and it appears that its moment of torsion is given by the formula,

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in which h is one side of the square cross-section.

SECTION V.-Of Resistance to Bending and Cross-Breaking. 433. Resistance to Bending in General.—In explaining the principles of the resistance which bodies oppose to bending and crossbreaking, it is convenient to use the word beam as a general term to denote the body under consideration; but those principles are applicable not only to beams for supporting weights, but to levers, cross-heads, cross-tails, shafts, journals, cranks, and all pieces in machinery or framework to which forces are applied tending to bend them and to break them across; that is to say, forces transverse to the axis of the piece.

Conceive a beam which is acted upon by a combination of parallel transverse forces that balance each other, to be divided into two parts by an imaginary transverse section; and consider separately the conditions of equilibrium of one of those parts. The

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