448. A horizontal beam supported at each end, and perhaps at
intermediate points, and designed to support a heavy load is called
a _girder_. Those rods upon which we have performed experiments, the
results of which have been given in Table XXIV., are small girders; but
the term is generally understood to relate to structures of iron: the
greatest girders for railway bridges are made of bars or plates of iron
riveted together.
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.
[Illustration: FIG. 64.]
458. E F (Fig. 64) is the thin iron beam along the bottom of which is
the stout flange shown at C D; rupture cannot commence at the bottom
unless this flange be torn asunder; for until this happens it is clear
that fracture cannot begin to attack the upper and slender part of the
beam E F.
459. But the beam is in a state of compression along its upper side,
just as in the wooden beams which we have already considered. If
therefore the upper parts were not powerful enough to resist this
compression, they would be crushed, and the beam would give way. The
remedy for this source of weakness is obvious; a second flange runs
along the top of the beam, as shown at A B. If this be strong enough to
resist the compression, the stability of the beam is ensured.
460. The upper flange is made very much smaller than the lower one, in
consequence of a property of cast iron. This metal is more capable of
resisting forces of compression than forces of extension, and it is
only necessary to use one-sixth of the iron on the upper flange that is
required for the lower. When the section has been thus proportioned,
the beam is equally strong at both top and bottom; adding material to
either flange without strengthening the other, will not benefit the
girder, but will rather prove a source of weakness, by increasing the
weight which has to be supported.
461. I have here a small girder made of what we are familiar with under
the name of “tin,” but which is of course sheet iron thinly covered
over with tin. It has the shape shown in Fig. 64, and it is 12" long. I
support it at each end, and you see it bears two hundred weight without
apparent deflection.