testing to apply loads gradually and not suddenly. Also we see that if we do not wish a material to be strained beyond the elastic limit, we must not apply suddenly to it loads which would produce, if applied gradually, more than half the proof stress.

In many cases we can simplify our preceding work by omitting from the energy exerted the term W. BB', for BB' is a small quantity. The equation then becomes

simply

Wh=Rx (putting x for BB').

Also the impact may be effected on a bar lying horizontally, the weight W being by some means given a velocity V f.s. say.

In this case

a form which is often used for such cases as a blow from a hammer, even where the rod is vertical, because x is negligible. The equation is of course identical with

WV2/2g is called the Energy of the Blow.

In chap. xiii. we have calculated the resilience of a bar, or the greatest amount of work which can be done on it without exceeding the proof stress, and from our present work we see that we may now define the resilience as the greatest amount of energy which can be applied to a bar without injuring it, or at least altering its properties.

A very convenient way of stating the resilience of a material is by giving the height from which a piece may fall without injury.

This will be found on calculation to be for most materials very small, e.g. for wrought-iron it is about 20 ins. only. We see then that when such materials withstand blows they will generally be strained beyond the limit of elasticity. We proceed then to examine

M

M'

Case II. The resistance now can only be shown graphically, since it follows no law which can be expressed algebraically. AB is the bar, broken in the figure, since its length must be considerable compared with the extensions. Produce AB, and draw the load-strain diagram BCM'M. BC is very nearly vertical if drawn to any ordinary scale. The slope which we have given it in former figures was for clearness. Let the blow be such as to cause the extension BB'. Then B'M is the equivalent steady load which produces the same stress as the actual blow, and we have from the principle of work

Energy of blow=work done,
=area BCMB'.

Now of this area the resistance inside the elastic limit only provided the very small portion BCN, and hence we see that materials having curves, such as we have here drawn, resist impact chiefly by being strained above the elastic limit.

Ductility. The materials of which we have just spoken have the property of extending considerably before breaking, to which property, when combined with tenacity, the name Ductility is applied, and the materials are called Ductile Materials. Wrought-iron, mild steel,

and copper are the chief examples.

In

Effect of raising Limit of Elasticity on Resistance to Blows. We can now explain the statement on page 423, relative to the effect of stress beyond the elastic limit having in many cases a weakening effect. For in Fig. 319 we have stress beyond the elastic limit, and consequently the load-strain curve of the bar will now be, as shown, dotted. Let now a second blow of equal energy be struck, then the bar must extend to B" such that MM'B"B = BCMB', and the effect produced is equivalent to the application of a steady load M'B". the figure we have taken such dimensions that M' is the point of maximum load and development of local contraction, and it consequently follows that the second blow has practically destroyed the bar, owing to the raising of the elastic limit caused by the first blow. The effect here shown also explains how by continued blows a piece of material will be finally broken, although it may be able to withstand one, two, or even a large number.

Coefficients of Strength-Factors of Safety. -In former times, when the strength of materials had not been so thoroughly investigated, and testing machines were rare, very little about materials beyond their ultimate strength was known, and this strength being given, the working stress allowed was taken less than this in a certain ratio, called a Factor of Safety. This factor was taken

empirically, and depended on the nature of the loads to which the piece was subject, being usually about 6 to 8. The method still holds to a very large extent, but with better knowledge of the strengths its value is not determined so purely empirically, and also the true nature of the ratio is better understood.

The name factor of safety is a bad one, since it gives the idea that, when it is, say, 6 for a given bar, the bar is for safety 6 times stronger than it need be. But now take the case, say, of a wrought-iron bar, ultimate strength 54,000 lbs. per sq. inch (apparent is always understood); then the working strength is 9000 lbs., and a bar which was to be exposed to a working load of 9000 lbs. would be 1 sq. in. in sectional area, but would not be 6 times stronger than necessary. For probably the load would be applied suddenly—as in a piston-rod at each end of the stroke—and will thus stress the bar up to 18,000 lbs. per sq. in.; and this 18,000 will be about the limit of elasticity, which, as we have seen, is the point beyond which the metal is injured. Hence, then, in this case, instead of being six times too strong, the bar is only barely strong enough; because a continuous repetition of suddenly applied loads a little above 9000 lbs. would in time break it. If, however, the load 9000 lbs. were to be applied once for all and to remain constant, then the bar would be twice as big as necessary, and there would be a true factor of safety of 2.

The ratio of working to ultimate strength should then depend on the ratio of proof to ultimate, and to the manner in which the load is applied. It is generally 6 for piston-rods and similar pieces, or perhaps a little greater to allow a small real safety factor. For bridges in which most of the load is constant but some is variable, i.e. the loads which cross, 5 is the usual value; while for the shell plates of a boiler, subject to a gradually applied steady steam pressure, 4 and slightly under is allowed.

It is, however, probable that in the future proof stress will be the determining factor and not ultimate stress.

Compression of Ductile Material. We have taken the case of tension at great length, because it is the case of which most is known, and the results are most clearly defined. All the work as to the effect of impact of course applies equally to all cases, also the effect of stress beyond the elastic limit, so we shall not need to consider other cases so fully.

In the present case, referring to chap. xiii., there is a difference between the results obtained from long pieces and those from short ones. The former being the more

important we will first consider them.

Long Pillars-Gordon's Formula-All Materials. When a long pillar is loaded, as Fig. 320, then, if the load were right in the axis and the pillar perfectly straight and homogeneous, it would remain straight, and the stress on the section would be W/A, A:

section.

= area of

Actually, however, these conditions are never fulfilled, and thus the pillar bends, and as the load increases the bending also increases, and the pillar finally gives way by the combined effects of compression and bending.

We cannot here enter into the analysis of these effects, but must simply give the empirical formula constructed by Gordon and modified by Rankine to represent the results of an extensive series of experiments by Hodgkinson.

This formula gives the breaking load W for a pillar of length / as follows:

A sectional area, f=coefficient of strength.

n is the constant in the formula I=nAh2 (page 368), h is depth of the pillar in the plane of bending, i.e. for a