A rod of pine 48" × 1" × 1"; resting freely on
supports 40" apart; and laden in the middle.
+-------------+-----------+-------------+
| Number of | Magnitude | Deflection. |
| Experiment. | of load. | |
+-------------+-----------+-------------+
| 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.
370. I take two pine rods, each 48" × 1" × 1", perfectly similar in all
respects, cut from the same piece of timber, and therefore probably of
very nearly identical strength. With a fine tenon saw I cut each of the
rods half through at its middle point. I now place one of these beams
on the supports 40" apart, with the cut side of the beam upwards. I
suspend from it the tray, which I gradually load with weights until the
beam breaks, which it does when the total weight is 81 lbs.
If I were to place the second beam on the same supports with the cut
upwards, then there can be no doubt that it would require as nearly as
possible the same weight to break it. I place it, however, with the cut
downwards, I suspend the tray, and find that the beam breaks with a
load of 31 lbs. This is less than half the weight that would have been
required if the cut had been upwards.
371. What is the cause of this difference? The fibres being compressed
together on the upper surface, a cut has no tendency to open there;
and if the cut could be made with an extremely fine saw, so as to
remove but little material, the beam would be substantially the same
as if it had not been tampered with. On the other hand, the fibres
at the lower surface are in a state of tension; therefore when the
cut is below it yawns open, and the beam is greatly weakened. It is,
in fact, no stronger than a beam of 48" × 0"·5 × 1", placed with its
shortest dimension vertical. If we remember that an entire beam of the
same size required about 140 lbs. to break it (Art. 366), we see that
the strength of a beam is reduced to one-fourth by being cut half-way
through and having the cut underneath.
372. We may learn from this the practical consequence that the sounder
side of a beam should always be placed downwards. Any flaw on the lower
surface will seriously weaken the beam: so that the most knotty face
of the wood should certainly be placed uppermost. If a portion of the
actual substance of a beam be removed—for example, if a notch be cut
out of it—this will be almost equally injurious on either side of the
beam.
373. We may illustrate the condition of the upper surface of the beam
by a further experiment. I make two cuts 0"·5 deep in the middle of
a pine rod 48" × 1" × 1". These cuts are 0"·5 apart, and slightly
inclined; the piece between them being removed, a wedge is shaped to
fit tightly into the space; the wedge is long enough to project a
little on one side. If the wedge be uppermost when the beam is placed
on the supports, the beam will be in the same condition as if it had
two fine cuts on the upper surface. I now load the beam with the
tray in the usual manner, and I find it to bear 70 lbs. securely. On
examining the beam, which has curved down considerably, I find that the
wedge is held in very tightly by the pressure of the fibres upon it,
but, by a sharp tap at the end, I knock out the wedge, and instantly
the load of 70 lbs. breaks the beam; the reason is simple—the piece
being removed, there is no longer any resistance to the compressive
strain of the upper fibres, and consequently the beam gives way.
374. The collapse of a beam by a transverse strain commences by
fracture of the fibres on the lower surface, followed by a rupture of
all fibres up to a considerable depth. Here we see that by a transverse
force the fibres in a beam of 48" × 1" × 1" have been broken by a
strain of 140 lbs. (Art. 366); but we have already stated (Art. 353)
that to tear such a rod across by a direct pull at each end a force of
about four tons is necessary. The breaking strain of the fibres must be
a certain definite quantity, yet we find that to overcome it in one way
four tons is necessary, while by another mode of applying the strain
140 lbs. is sufficient.
375. To explain this discrepancy we may refer to the experiment of
Art. 28, wherein a piece of string was broken by the transverse pull
of a piece of thread in illustration of the fact that one force may be
resolved into two others, each of them very much greater than itself.
A similar resolution of force occurs in the transverse deflection of
the beam, and the force of 140 lbs. is changed into two other forces,
each of them enormously greater and sufficiently strong to rupture
the fibres. We need not suppose that the force thus developed is so
great as four tons, because that is the amount required to tear across
a square inch of fibres simultaneously, whereas in the transverse
fracture the fibres appear to be broken row after row; the fracture is
thus only gradual, nor does it extend through the entire depth of the
beam.
376. We shall conclude this lecture with one more remark, on the
condition of a beam when strained by a transverse force. We have seen
that the fibres on the upper surface are compressed, while those on the
lower surface are extended; but what is the condition of the fibres in
the interior? There can be no doubt that the following is the state
of the case:—The fibres immediately beneath the upper surface are in
compression; at a greater depth the amount of compression diminishes
until at the middle of the beam the fibres are in their natural
condition; on approaching the lower surface the fibres commence to
be strained in extension, and the amount of the extension gradually
increases until it reaches a maximum at the lower surface.
LECTURE XII.
_THE STRENGTH OF A BEAM._
A Beam free at the Ends and loaded in the Middle.—A
Beam uniformly loaded.—A Beam loaded in the
Middle, whose Ends are secured.—A Beam supported
at one end and loaded at the other.