The answer in this case is 76, whereas the observed value is 68, or 8 lbs. less; this does not agree very well with the theory, but still the difference, though 8 lbs., is only about 11 or 12 per cent. of the whole, and we shall still retain the law, for certainly there is no other that can express the result as well.
388. But the table will supply another verification. In experiment No. 3 a 40" bar, 1" broad, and 0"·5 deep, broke with 38 lbs.; and in experiment No. 7 a 20" bar of the same section broke with 74 lbs.; but this is so nearly double the breaking weight of the 40" bar, as to be an additional illustration of the law, that for a given section the breaking load varies inversely as the span.
389. We next inquire as to the effect of the breadth of the beam upon its strength? For this purpose we compare experiments Nos. 3 and 4: we there find that a bar 40" × 1" × 0"·5 is broken by a load of 38 lbs., while a bar just half the breadth is broken by 19 lbs. We might have anticipated this result, for it is evident that the bar of No. 3 must have the same strength as two bars similar to that of No. 4 placed side by side.
390. This view is confirmed by a comparison of Nos. 7 and 8, where we find that a 20" bar takes twice the load to break it that is required for a bar of half its breadth. The law is not quite so well verified by Nos. 5 and 6, for half the breaking weight of No. 5, namely 29·5 lbs., is more than 25, the observed breaking weight of No. 6: a similar remark may be made about Nos. 9 and 10.
391. Supposing we had a beam of 40" span, 2" broad, and 0"·5 deep, we can easily see that it is equivalent to two bars like that of No. 3 placed side by side; and we infer generally that the strength of a bar is proportional to its breadth; or to speak-more definitely, if two beams have the same span and depth, the ratio of their breaking loads is the same as the ratio of their breadths.
392. We next examine the effect of the depth of a beam upon its strength. In experimenting upon a beam placed edgewise, a precaution must be observed, which would not be necessary if the same beam were to be broken flatwise. When the load is suspended, the beam, if merely laid edgewise on the supports, would almost certainly turn over; it is therefore necessary to place its extremities in recesses in the supports, which will obviate the possibility of this occurrence; at the same time the ends must not be prevented from bending upwards, for we are at present discussing a beam free at each end, and the case where the ends are not free will be subsequently considered.
393. Let us first compare together experiments Nos. 2 and 3; here we have two bars of the same dimensions, the section in each being 1"·0 × 0"·5, but the first bar is broken edgewise, and the second flatwise. The first breaks with 77 lbs., and the second with 38 lbs.; hence the same bar is twice as strong placed edgewise as flatwise when one dimension of the section is twice as great as the other. We may generalize this law, and assert that the strength of a rectangular beam broken edgewise is to the strength of a beam of like span and section broken flatwise, as the greater dimension of the section is to the lesser dimension.
394. The strength of a beam 40" × 0"·5 × 1" is four times as great as the strength of 40" × 0"·5 × 0"·5, though the quantity of wood is only twice as great in one as in the other. In general we may state that if a beam were bisected by a longitudinal cut, the strength of the beam would be halved when the cut was horizontal, and unaltered when the cut was vertical; thus, for example, two beams of experiment No. 4, placed one on the top of the other, would break with about 40 lbs., whereas if the same rods were in one piece, the breaking load would be nearly 80 lbs.
395. This may be illustrated in a different manner. I have here two beams of 40" × 1" × 0"·5 superposed; they form one beam, equivalent to that of No. 1 in bulk, but I find that they break with 80 lbs., thus showing that the two are only twice as strong as one.
396. I take two similar bars, and, instead of laying them loosely one on the other, I unite them tightly with iron clamps like those represented in [Fig. 56]. I now find that the bars thus fastened together require 104 lbs. for fracture. We can readily understand this increase of strength. As soon as the bars begin to bend under the action of the weight, the surfaces which are in contact move slightly one upon the other in order to accommodate themselves to the change of form. By clamping I greatly impede this motion hence the beams deflect less, and require a greater load before they collapse; the case is therefore to some extent approximated to the state of things when the two rods form one solid piece, in which case a load of 152 lbs. would be required to produce fracture.