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sistances to torsion, the sum M of the moments of the exterior forces must be equal to the sum of the moments (3) of the molecular resistances to torsion, so that we shall have the equation

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Such is the general relation which expresses the condition of equilibrium between the molecular resistances to torsion and the exterior forces, for each section of a homogeneous prism.

The corresponding expression for a cylindrical prism or shaft of length L is deduced in the following manner. (See Morin's Work just referred to.)

67. Torsion of a cylindrical beam or shaft.

Let us consider a cylindrical beam or shaft AB (fig. 33), acted upon by a force P which tends

to twist the beam about its axis.

B

Fig. 33.

Then if all the cross sections of the beam between A and B be supposed to be equally fixed, the effects of torsion will be transmitted progressively to the cylindrical beam from the extremity A to the extremity B, so that any one section of the beam will experience the effects of torsion only when the immediately preceding section to this begins to rotate. Consequently the extreme section B will begin to move only when all the sections which precede the section at B have been put in a state of torsion.

Let a cylinder in a state of torsion be projected on two planes perpendicular to one another, the circle ABCDEF (fig. 34) being the projection on one of these planes, and the figure G df the projection on the other plane.

Then a particle A at one extremity of the cylinder has been transmitted by the torsion of the cylinder to a certain point D, at the instant the corresponding point a at the other extremity of the cylinder begins to move. Hence the generating line A a

of the cylinder has been curved into the thread of a screw,

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Applying this result to (4) of last article, we get

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a being the angle of torsion for the whole cylinder, and M the sum of the moments of the exterior forces. The quantities E' and I, are the same as in (4), Art. 66.

Cor. It appears from (2) that the angle of torsion is directly proportional to the sum M of the moments of the exterior forces, and to the distance L between the extreme ends of the cylinder.

Obs. According to General Morin, these results have been verified by the experiments of MM. Duleau and Savart. (Morin's Résistance des Matériaux, p. 455.)

68. The equation, (1) Art. 67, may be put in a more convenient form for use in the following manner :

Since I is the moment of inertia of a circular section of the

cylinder about an axis perpendicular to its plane and passing through its centre of gravity; if r be the radius of the cylinder, we have by Art. 45,

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h1, h2, being respectively the exterior and interior radii; and therefore for hollow cylinders

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69. Values of E' in the formula (1), Art. 67, for different materials in pounds avoirdupois per square inch, from Morin's work on Résistance des Matériaux.

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* The effect of torsion (says Morin) may be often observable to the eye at the instant of putting in motion large masses; and this effect is more appreciable when it transmits and multiplies itself from one beam to another. In some cases, such as in the hydraulic wheel, we see the prime mover in progress long before the extreme parts of the machine begin to

move.

According to the experiments of M. Duleau, a beam of forged-iron of Périgord of 2m-80 long by 0.0142 of diameter, has supported, without

Examples.

1. Find the diameter of a beam of transmission of iron 9 feet 10 inches long, twisted by a force of 13,228 lbs., acting at the extremity of a lever arm of 1 foot 7 inches, and under the condition that the angle of torsion be 1 minute.

Solving the equation (1), Art. 68, we get for the radius of the circular shaft, the value

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Substituting these values in the preceding equation, we get

r=9:15 inches.

2. A water wheel of 9 feet radius, is fixed in a cylindrical shaft of cast-iron, 12 feet long and 10 inches across; find what torsion will be produced in the shaft by a force of 2,000 lbs. acting on the wheel. Ans. Angle of torsion= 31′25′′.

3. Find the angle of torsion in the preceding example, on the supposition that the shaft is a hollow cylinder, the thickness of metal being th of the exterior diameter. Ans. α = 36'.

4. Find the diameter of a cylindrical shaft of wrought-iron, 26 feet long, which is to transmit a power of 2,000 lbs. acting on the circumference of a wheel attached to the shaft; the radius of the wheel is 9 feet, and the twisting is not to exceed half a degree. Ans. 8.24 inches.

breaking, a torsion of 13°4, produced by a load of 10 kilogrammes, acting at the extremity of a lever of Om·32.

It appears also, from experiments made at Mulhouse, that a cast-iron beam (of Bouchot Franche-Comté) having a length of 1m-50, and a diameter of Om-10, acted upon by a load of 1,640 kil., at the extremity of a lever arm of 2 metres, has occasioned an angle of torsion of 15°. Under a load of 2,080 kil. the torsion rose up to 20°.25. Lastly, the beam broke under a load of 2,180 kilogrammes. (See M. Mahistre's Mécanique Appliquée, p. 549.)

CHAPTER X.

RESISTANCE OF COLUMNS.

70. In order to calculate the permanent loads of cast-iron and other columns, Hodgkinson has given the following empirical formulæ, in which

D is the external diameter of the column in inches;

d is the internal diameter in inches;

L is the length of the column in feet ;

W the breaking load in tons.

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When both ends are rounded as in I., the length of the column is supposed to exceed fifteen times its diameter; and in the case of flat ends, the length of the column exceeds thirty times its diameter. The columns, moreover, are supposed to yield wholly by bending.

When the columns are shorter than those indicated above, or when the material yields by crushing and bending, the following

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