449. We shall first consider the application of cast iron to girders, and show what form they should assume.
450. A beam of cast iron, supposing its section to be rectangular, has its strength determined by the same laws as the beams of pine. Thus, supposing the section of two beams to be the same, their strengths are inversely proportional to their lengths, and the strength of a beam placed edgewise is to its strength placed flatwise in the proportion of the greater dimension of its section to the less dimension. These laws determine the strength of every rectangular beam of cast iron when that of one beam is known, and we must perform an experiment in order to find the breaking load in a particular case.
451. I take here a piece of cast iron, which is 2' long, and 0"·5 × 0"·5 in section. I support this beam at each end upon a frame; the distance between the supports is 20". I attach the tray to the centre of the beam and load it with weights. The ends of the beam rest freely upon the supports, but I have taken the precaution of tying each end by a piece of wire, so that they may not fly about when the fracture occurs. Loading the tray, I find that with 280 lbs. the crash comes.
452. Let us compare this result with No. 8 of [Table XXIV. (p. 190)]. There we find that a piece of pine, the same size as the cast iron, was broken with 36 lbs.: the ratio of 280 to 36 is nearly 8, so that the beam of cast iron is about 8 times as strong as the piece of pine of the same size. This result is a little larger than we would have inferred from an examination of tables of the strength of large bars of cast iron; the reason may be that a very small casting, such as this bar, is stronger in proportion than a larger one, owing to the iron not being so uniform throughout the larger mass.
453. I hold here a bar of cast iron 12" long and 1" × 1" in section. I have not sufficient weights at hand to break it, but we can compute how much would be necessary by our former experiment.
454. In the first place a bar 12" long, and 0"·5 × 0"·5 of section, would require 20 × 280 ÷ 12 = 467 lbs. by the law that the strength is inversely as the length. We also know that one beam 12" × 1" × 1" is just as strong as two beams 12" × 1" × 0"·5, each placed edgewise; each of these latter beams is twice as strong as 12" × 1" × 0"·5 placed flatwise, because the strength when placed edgewise is to the strength when placed flatwise, as the depth to the breadth, that is as 2 to 1: hence the original beam is four times as strong as one beam 12" × 1" × 0"·5 placed flatwise: but this last beam is twice as strong as a beam 12" × 0"·5 × 0"·5, and hence we see that a beam 12" × 1" × 1" must be 8 times as strong as a beam of 12" × 0"·5 × 0"·5, but this last beam would require a load of 467 lbs. to break it, and hence the beam of 12" × 1" × 1" would require 467 × 8 = 3736 lbs. to produce fracture. This amounts to more than a ton and a half.
455. It is a rule sometimes useful to practical men that a cast iron bar one foot long by one inch square would break with about a ton weight. If the iron be of the same quality as that which we have used, this result is too small, but the error is on the safe side; the real strength will then be generally a little greater than the strength calculated from this rule. What we have said ([Art. 403]) with reference to the precaution for safety in bars of wood applies also to cast iron. The load which the beam has to bear in ordinary practice should only be a small fraction of that which would break it.
456. In making any description of girder it is desirable on very special grounds that as little material as possible be uselessly employed. It will of course be remembered that a girder has to support its own weight, besides whatever may be placed upon it: and if the girder be massive, its own weight is a serious item. Of two girders, each capable of bearing the same total load, the lighter, besides employing less material, will be able to bear a greater weight placed upon it. It is therefore for a double reason desirable to diminish the weight. This remark applies especially to such a material as cast iron, which can be at once given the form in which it shall be capable of offering the greatest resistance.
457. The principles which will guide us in ascertaining the proper form to give a cast iron girder, are easily deduced from what we have laid down in [Lectures XI]. and [XII]. We have seen that depth is very desirable for a strong beam. If therefore we strive to attain great depth in a light beam, the beam must be very thin. Now an extremely thin beam will not be safe. In the first place it would be flexible and liable to displacement sideways; and, in the second place, there is a still more fatal difficulty. We have shown that when a beam of wood is supporting a weight, the fibres at the bottom of the beam are extended, the tendency being to tear them ([Art. 376]). The fibres on the top of the beam are compressed, while the centre of the beam is in its natural state. The condition of strain in a cast iron beam is precisely similar; the bottom portions are in a state of extension, while the top is compressed. If therefore a beam be very thin, the material at the lower part may not be sufficient to withstand the forces of extension, and fracture is produced. To obviate this, we strengthen the bottom of the beam by placing extra material there. Thus we are led to the idea of a thin beam with an excess of iron at the bottom.
