TABLE 129.- PROVING that RECTANGULAR BEAMS RESIST IMPACT with EQUAL ENERGY when STRUCK on their NARROW or BROAD DIMENSIONS.

result is that necessarily the ratio between the Dynamic Safe and Breaking weights is very much higher than between the like Statical weights. For instance, if the elasticity of a beam is perfect (688), the deflection being in that case exactly proportional to the weights, and say that the ratio of the Statical safe and breaking loads is 1 to 3, then as we have with the latter three times the weight and three times the deflection, we get 3 × 3 9 times the resistance to impact; in fact, the Dynamic ratio is the square of the Statical ratio, and the latter being say 3, 4, 5, &c., the former will be 9, 16, 25, &c.

(826.) When the elasticity is imperfect, and the deflections increase more rapidly than the weights, the ratio is higher still: for instance, for cast iron Table 67 shows that with a statical ratio of 3 to 1, the deflections by cols. 8 and 11 are ⚫0785÷ • 01971 = 3.983, or nearly 4 to 1, and the resistance to impact 3.983 × 3 = 12 to 1 nearly. Table 67 also shows by col. 9 that Timber, with a Statical Ratio of 5 to 1, has a Dynamic ratio varying from 53.0 to 1 in Ash, to 27 1 to 1 in Elm.

(827.) This high ratio, which we have seen to be a result of the nature of the case, is a great practical advantage, because in most cases the dynamic strain is uncertain as to its amount, being frequently the result of accident, such as the failure of a

rope, &c. Thus, if we arrange the proportions of a cast-iron beam to bear safely a falling load that will strain it to the limit of safety, say 3rd of the Statical breaking weight; it will be strained to only th of the Dynamic breaking weight; leaving thus a very large margin for safety: see (786).

(828.) "Stiffness a Source of Weakness.”—In many cases stiffness or rigidity is necessary; for instance, in girders carrying a water-tank where undue flexure would throw a strain on the joints, &c., and would be likely to cause leakage if not rupture: but when an impulsive strain has to be borne, the most flexible beam is the strongest, other things being equal. For instance, if we had two beams whose breaking weights were the same, but one deflecting twice as much as the other, then the latter would bear twice the strain dynamically, or a given weight falling double the height, &c.

(829.) It is shown in (778) that R, or the resistance to Impact, is directly as the length of the Beam, other things being equal, which of course is directly contrary to the case of a dead load, where the strength is inversely as the length. This remarkable result is shown to be experimentally true by Table 130, where bars similar in depth and breadth, but differing in length, are arranged in groups. The Ratios of the lengths are 2 to 1 in all cases, and col. 7 shows that R, or the mean resistance to Impact, is 1.99 to 1.

(830.) "Summary of Remarkable Laws.". -There are several remarkable laws of Impact which it may be interesting and instructive to collect from the foregoing investigation: excluding the effect of Inertia for the moment we find :—

1st. The resistance to Impact, or the Resilience of a Beam, R, is the same whether the blow is struck at the centre or elsewhere in the length: see (809).

2nd. In rectangular beams of unequal dimensions the resistance R is the same whether the bar is struck on its narrow, or broad dimension: see (824).

3rd. With rectangular beams of the same material, the resistance to Impact R is simply proportional to the weight of the beam between supports irrespective of the particular dimensions: (821).

TABLE 130.-PROVING that RESISTANCE to IMPACT is directly as the LENGTH of the BEAM.

Combining 1, 2, 3, we are conducted to this remarkable fact, that the weight of the rectangular beam being the same, it is immaterial whether it be long or short, broad or narrow, deep or shallow. Moreover, whether the beam is struck on edge or on the flat;—in the centre or anywhere out of the centre, the result is the same in all cases.

4th. The power R is directly as the length, instead of inversely as for a dead load': see (829).

5th. The power R of a beam in resisting Impact is increased by loading it with a dead load up to a certain point: (801). Thus by col. 9 of Table 125 it is more than doubled.

6th. With 3rd of the Breaking weight, which is the ratio adopted by most Engineers for the Working dead load, the resistance to Impact is a maximum: (803).

7th. The power required to produce given deflections in any beam by an impulsive strain is proportional to the deflection squared, not as the deflection simply, as with dead loads: (812).

8th. The stiffest beams are the weakest, and vice versa, other things being equal (828).

9th. The Ratio between the Breaking and Safe strains by Impact, or between R and r, is exceedingly high, being as the square of the Ratios with dead loads, as shown by (825).

IMPACT FROM ROLLING LOAD.

(831.) "Rolling Load at High Velocity."-When the load on a horizontal beam rolls over it at a high velocity, the strain becomes more or less a dynamic one, but under certain limitations as governed by the speed of the transit. Let A in Fig. 193 be an unloaded beam, W, a weight, which as a dead or statical load deflects the beam to B. But by (775), and Table 119, it is shown that if that same weight were laid quietly on the centre of the beam A and suddenly released, it would deflect it to C, producing double the deflection and thereby double the strain, the weight really falling as the beam deflects, and acting therefore as an impulsive load.

Now let the weight W, roll horizontally upon the beam with a high velocity, such, that in travelling half the length of the beam, or from w to W, it would, if free, follow the line d, e, f, and fall by gravity from W to W1, and deflect the beam as before, or as when it fell vertically the same height. The horizontal velocity necessary to effect this, is easily calculated :— for instance, let the beam be 10 feet long between bearings, and the dynamic deflection A C, = 4 inches. Then by the laws of falling bodies:

=

in which h= the height fallen in inches, and t = time in seconds, we obtain in our case (4 ÷ 193) √ = · 144 second, in which time the weight must travel half the length of the beam, or 5 feet, hence its horizontal velocity must be 5÷144 = 34 feet per second, or 34 × 3600 ÷ 5280 23.18 miles per hour. In this case then, a load passing over this beam at a velocity of 23.18 miles per hour, will deflect that beam, and thereby strain it to the same extent as a double load acting as a dead weight; or, in other words, the strain with any load is doubled on this particular beam by a horizontal velocity of 23.18 miles per hour. The deflection is a maximum with this velocity, that is to say, with a higher or a lower velocity the deflection would be less. With a higher velocity the weight W would not have time to fall the height A, C, or 4 inches:-for instance, with

46 36 miles per hour, it would

2072 second, the

double velocity, or 23 18 x 2 =
reach the centre of the beam in 144
fall due to which time by the laws of gravity or:-

becomes in our case 072 × 193 = 1 inch only, instead of 4 inches, so that with a velocity of 46.36 miles per hour the deflection and therefore the strain would have been 4th only of that due to 23 18 miles per hour, and half only of that with the same weight acting statically or as a dead load in the contre. So that in this case, while a velocity of 23 18 miles per hour doubles the effect of a given statical load, a double velocity or 45 36 miles reduces the effect of a given dead load to half. Again, with velocities lower than 23.18 miles per hour, the deflections would be less than with that velocity, rising with the velocity from A, B, or that with a dead load, to A, C with 23 18 miles per hour.

(834.) These theoretical results are confirmed by the experiments of Captain James, R.E., who found with bars of steel 2 feet long, 2 inches broad, and inch deep, that with velocities of

inches, attaining a maximum with the velocity of 29 feet per second, which was reduced by an increase in velocity.

Similar results were obtained by wrought-iron bars 9 feet long, 3 inches deep, and 1 inch wide:—with a load of 1778 lbs. travelling at velocities of

inches respectively, attaining a maximum with a velocity of 36 feet per second.