363. While I look through the telescope my assistant suspends the tray from the beam. Instantly I see the cross descend in the field of view. I lower the telescope until the spider’s web again passes through the image of the intersection of the lines, and then by looking at the scale I see that the telescope has been moved down 0"·19, that is, about one-fifth of an inch: this is, therefore, the distance by which the cross lines on the beam, and therefore the centre of the beam itself, must have descended. Indeed, even a simpler apparatus would be competent to measure the amount of deflection with some degree of precision. By placing successively one stone after another upon the tray, the beam is seen to deflect more and more, until even without the telescope you see the beam has deviated from the horizontal.
364. By carefully observing with the telescope, and measuring in the way already described, the deflections shown in [Table XXIII]. were determined. The scale along the vertical pillar was read after the spider’s web had been adjusted for each increase in the weight. The movement from the original position is recorded as the deflection for each load.
Table XXIII.—Deflection of a Beam.
A rod of pine 48" × 1" × 1"; resting freely on supports 40" apart; and laden in the middle.
| Number of Experiment. | Magnitude of load. | Deflection. |
|---|---|---|
| 1 | 14 | 0"·19 |
| 2 | 28 | 0"·37 |
| 3 | 42 | 0"·55 |
| 4 | 56 | 0"·74 |
| 5 | 70 | 0"·94 |
| 6 | 84 | 1"·13 |
| 7 | 98 | 1"·35 |
| 8 | 112 | 1"·61 |
| 9 | 126 | 1"·95 |
| 10 | 140 | 2"·37 |
365. The first column records the number of the experiment. The second represents the load, and the third contains the corresponding deflections. It will be seen that up to 98 lbs. the deflection is about 0"·2 for every stone weight, but afterwards the deflection increases more rapidly. When the weight reaches 140 lbs. the deflection at first indicated is 2"·37; but gradually the cross lines are seen to descend in the field of the telescope, showing that the beam is yielding and finally it breaks across. This experiment teaches us that a beam is at first deflected by an amount proportional to the weight it supports; but that when two-thirds of the breaking weight is reached, the beam is deflected more rapidly.
366. It is a question of the utmost importance to ascertain the greatest load a beam can sustain without injury to its strength. This subject is to be studied by examining the effect of different deflections upon the fibres of a beam. A beam is always deflected whatever be the load it supports; thus by looking through the telescope of the cathetometer I can detect an increase of deflection when a single pound is placed in the tray: hence whenever a beam is loaded we must have some deflection. An experiment will show what amount of deflection may be experienced without producing any permanently injurious effect.
367. A pine rod 40" × 1" × 1" is freely supported at each end, the distances between the supports being 38", and the tray is suspended from its middle point. A fine pair of cross lines is marked upon the beam, and the telescope of the cathetometer is adjusted so that the spider’s line exactly passes through the image of the intersection. 14 lbs. being placed in the tray, the cross is seen to descend; the weight being removed, the cross returns precisely to its original position with reference to the spider’s line: hence, after this amount of deflection, the beam has clearly returned to its initial condition, and is evidently just as good as it was before. The tray next received 56 lbs.; the beam was, of course, considerably deflected, but when the weight was removed the cross again returned,—at all events, to within 0"·01 of where the spider’s line was left to indicate its former position. We may consider that the beam is in this case also restored to its original condition, even though it has borne a strain which, including the tray, amounted to 70 lbs. But when the beam has been made to carry 84 lbs. for a few seconds, the cross does not completely return on the removal of the load from the tray, but it shows that the beam has now received a permanent deflection of 0"·03. This is still more apparent after the beam has carried 98 lbs., for when this load is removed the centre of the beam is permanently deflected by 0"·13. Here, then, we may infer that the fibres of the beam are beginning to be strained beyond their powers of resistance, and this is verified when we find that with 28 additional pounds in the tray a collapse ensues.
368. Reasoning from this experiment, we might infer that the elasticity of a beam is not affected by a weight which is less than half that which would break it, and that, therefore, it may bear without injury a weight not exceeding this amount. As, however, in our experiments the weight was only applied once, and then but for a short time, we cannot be sure that a longer-continued or more frequent application of the same load might not prove injurious; hence, to be on the safe side, we assume that one-third of the breaking weight of a beam is the greatest load it should be made to bear in any structure. In many cases it is found desirable to make the beam much stronger than this ratio would indicate.
369. We next consider the condition of the fibres of a beam when strained by a transverse force. It is evident that since the fracture commences at the lower surface of the beam, the fibres there must be in a state of tension, while those at the concave upper surface of the beam are compressed together. This condition of the fibres may be proved by the following experiment.