Diameter of Shafting to Transmit a given Horse Power.-An engine of given I. H. P. runs at N revolutions per minute; it is required to find the necessary diameter of its shafting. First let

Tm mean twisting moment of engine in tons-inches.

=

Then

Work done per revolution = Tm × 27 inch-tons.

But in chap. ix. we proved that the twisting moment does not remain constant at its mean value, but varies; the ratio of maximum to mean twisting moment depending partly on considerations there discussed, viz. number of cylinders, and connecting-rod crank ratio, and partly on other considerations, into which we cannot The effect is then that if

enter.

we have

T=greatest twisting moment,

T=KTm

where K is a constant greater than 1, depending on the foregoing considerations. A mean value for a pair of cylinders would be about 1 for the propeller shafting and 2 for the crank shaft. The greater value in the latter case includes an allowance for the very severe bending to which a crank shaft is subject.

But the shaft must be designed to be strong enough to withstand T, whence taking a solid shaft

which gives the actual diameter in inches of a solid shaft, or the equivalent solid diameter if we use a hollow one. In the latter case we have

whence, being given the ratio da/d1, we obtain the values of d and d. Usual values of the ratio are from to §. Or we may be given that d2- d is not to be less than a certain thickness, say for large shafts about 3 ins.; then, using this value, we obtain the two diameters.

EXAMPLES.

1. If the greatest shearing stress allowed on the pin of a pin joint be of the tensile stress allowed in the metal of the rods joined, show that the diameter of the pin should very approximately equal the diameter of the rod.

Find

2. A single riveted lap joint in inch plate is subject to a load of 3 tons per square inch of the plate section through the line of rivets. The rivets are in. diameter, pitch 13 in. the shearing stress on the rivets, and the efficiency of the joint. Ans. 3.8 tons; .6.

3. The steel plates of a girder are in. thick, riveted with 1 inch rivets. The joint is treble riveted, double butt strap, and the shearing strength of the rivets is the tensile strength of the plates. Find the pitch. Ans. 6 ins.

4. A square bar of steel is under a tensile pull of 4 tons per sq. inch along its axis, and a compressive stress of 2 tons at right angles to the axis. Find the direction of a plane on which there is a pure shearing stress, and its amount.

5. Find the greatest twisting moment a steel tube 12 ins. mean diameter, inch thick, can withstand, shearing stress allowed 4 tons per sq. inch. If the length be 6 ft. find the angle of torsion. Ans. 113 tons-ins. ; 5.86°.

6. Find the diameter of a solid steel shaft to transmit 6000 H. P. at 116 revolutions per minute, the maximum twisting moment being 1.3 times the mean, and stress allowed 10,000 lbs. per sq. inch. Ans. 12 ins.

7. Find the size of a hollow shaft to replace the preceding, diameter of hole ğ the outside diameter. Estimate the saving in weight in 50 feet of shafting.

Ans. 12 ins. outside, 8 ins. inside diameter; 8170 lbs.

8. The angle of torsion of a cylindrical shaft is required not to exceed one degree for each 5 feet of length, and the stress not to be greater than 12,000 lbs. per sq. inch. Determine the diameter of shaft above which the second condition, and below which the first condition, fixes a limit to the greatest twisting moment which may be applied.

Ans. 8 inches; a T. M. of 540 tons-inches then produces both the limiting torsion and stress. Below this the torsion at 12,000 lbs. stress would be more than allowed, and above this the stress at the given torsion would be more than 12,000 lbs.

9. If the modulus of rigidity be 4800 in ton-inch units, what is the greatest stress to which the material of a shaft should be subjected, in order that the angle of torsion may not exceed one degree for each length of ten diameters.

Ans. 4.2 tons per sq. inch.

10. In renewing the engines of a ship, the speed of revolution is increased by one-third, the horse power is doubled, the ratio of maximum to mean crank effort is altered from 1.5 to 1.25, and the strength of the material used for the shaft is greater by 25 per cent. Show that the size of the shaft is unaltered.

CHAPTER XXI

EXPERIMENTAL FACTS-ELASTICITY-STRENGTH

RESISTANCE TO IMPACT

IN all the preceding chapters we have assumed that materials obey certain laws connecting together the stress in a piece and the alteration of form produced. The laws we have assumed are

and

p=E, or Ee,

q=Cp,

which may both be, in words, stated as follows:

Stress varies directly as the corresponding strain. These laws are the result of experiment, and they are satisfied, within certain limits, by the principal materials we have to deal with, allowing for the small irregularities which we always find in actual practice.

Elastic State. In most materials, if stresses less than a certain amount be applied to a piece, it is found that the laws above are satisfied; and that, when the stress is removed, the piece returns exactly to its original condition. The material is then said to be perfectly elastic or in the elastic state; by elasticity being meant the power of resuming its original shape and size.

Proof Stress. If, however, a certain stress depending on the nature of the material be reached, it will be found that, when released, the piece no longer

resumes its original dimensions, but a permanent alteration of form has taken place, or there is a Permanent Set. If we apply stresses greater than the above, it is found that the simple laws connecting stress and strain no longer hold. We say then that the elastic state is passed; and the limiting stress, above which these changes occur, is called the Elastic Limit or Proof Stress.

It must not be supposed that there is a sharply defined limit always the same in all cases for the same metal or material-in fact in some materials, castiron for instance, there is, as we shall see, no perfectly elastic state; yet in most cases a limiting or proof stress may be found, below which the material is practically perfectly elastic, and above which it deviates entirely from perfect elasticity.

Now in all practical cases we want a piece of material not only not to break, but also to keep its dimensions unchanged, omitting the elastic stretching or change of shape necessarily accompanying stress, and which disappears when the stress is removed. Hence the examination of material in this state is of the most practical importance, which explains why the whole of our preceding work has been confined to this. It is however necessary, for the purpose of determining, for one thing, the elastic limit, and also for investigating the manner of resistance to certain actions not hitherto considered, that we should examine how material behaves under stresses of any magnitude up to those which actually break it. We are now therefore going to describe what is actually found by experiment to occur; the results have been partially anticipated by what we have just been saying, but the preliminary statement which has been made will be found, probably, useful, by showing beforehand the principal points which have to be considered.

Testing Machines.-When a piece of material is tested the load should be applied gradually, for reasons