Slips of pine (cut from the same piece) supported
freely at each end; the length recorded is the
distance between the points of support; the load
is suspended from the centre of the beam, and
gradually increased until the beam breaks;
area of section × depth
Formula, _P_ = 6080 ————————————————————————
span
+-------+---------------------+------------+----------+------------+
| | |Mean of the | P. |Difference |
| | Dimensions. |observations|Calculated| of the |
|No. of +-----+--------+------+ of the | breaking |observed and|
|Experi-| | | | breaking | load | calculated |
|ment. |Span.|Breadth.|Depth.|load in lbs.| in lbs. | values. |
+-------+-----+--------+------+------------+----------+------------+
| 1 |40"·0| 1"·0 | 1"·0 | 152 | 152 | 0·0 |
| 2 |40"·0| 0"·5 | 1"·0 | 77 | 76 | -1·0 |
| 3 |40"·0| 1"·0 | 0"·5 | 38 | 38 | 0·0 |
| 4 |40"·0| 0"·5 | 0"·5 | 19 | 19 | 0·0 |
| 5 |30"·0| 1"·0 | 0"·5 | 59 | 51 | -8·0 |
| 6 |30"·0| 0"·5 | 0"·5 | 25 | 25 | 0·0 |
| 7 |20"·0| 1"·0 | 0"·5 | 74 | 76 | +2·0 |
| 8 |20"·0| 0"·5 | 0"·5 | 36 | 38 | +2·0 |
| 9 |10"·0| 1"·0 | 0"·5 | 154 | 152 | -2·0 |
| 10 |10"·0| 0"·5 | 0"·5 | 68 | 76 | +8·0 |
+-------+-----+--------+------+------------+----------+------------+
381. In the first column is a series of figures for convenience of
reference. The next three columns are occupied with the dimensions of
the beams. By span is meant the distance between the points of support;
the real length is of course greater; the depth is that dimension of
the beam which is vertical. The fifth column gives the mean of two
observations of the breaking load. Thus for example, in experiment
No. 5 the two beams used were each 36" × 1" × 0"·5, they were placed
on points of support 30" distant, so the span recorded is 30": one of
the beams was broken by a load of 58 lbs., and the second by a load
of 60 lbs.; the mean between the two, 59 lbs., is recorded as the
mean breaking load. In this manner the column of breaking loads has
been found. The meaning of the two last columns of the table will be
explained presently.
382. We shall endeavour to elicit from these observations the laws
which connect the breaking load with the span, breadth, and depth of
the beam.
383. Let us first examine the effect of the span; for this purpose we
bring together the observations upon beams of the same section, but
of different spans. Sections of 0"·5 × 0"·5 will be convenient for
this purpose; Nos. 4, 6, 8, and 10 are experiments upon beams of this
section. Let us first compare 4 and 8. Here we have two beams of the
same section, and the span of one (40") is double that of the other
(20"). When we examine the breaking weights we find that they are 19
lbs. and 36 lbs.; the former of these numbers is rather more than half
of the latter. In fact, had the breaking load of 40" been ¾ lb. less,
18·25 lbs., and had that of 20" been ½ lb. more, 36·5 lbs., one of the
breaking loads would have been exactly half the other.
384. We must not look for perfect numerical accuracy in these
experiments; we must only expect to meet with approximation, because
the laws for which we are in search are in reality only approximate
laws. Wood itself is variable in quality, even when cut from the same
piece: parts near the circumference are different in strength from
those nearer the centre; in a young tree they are generally weaker, and
in an old tree generally stronger. Minute differences in the grain,
greater or less perfectness in the seasoning, these are also among the
circumstances which prevent one piece of timber from being identical
with another. We shall, however, generally find that the effect of
these differences is small, but occasionally this is not the case, and
in trying many experiments upon the breaking of timber, discrepancies
occasionally appear for which it is difficult to account.
385. But you will find, I think, that, making reasonable allowances
for such difficulties as do occur, the laws on the whole represent the
experiments very closely.
386. We shall, then, assume that the breaking weight of a bar of 40"
is half that of a bar of 20" of the same section, and we ask, Is this
generally true? is it true that the breaking weight is inversely
proportional to the span? In order to test this hypothesis, we can
calculate the breaking weight of a bar of 30" (No. 6), and then compare
the result with the observed value; if the supposition be true, the
breaking weight should be given by the proportion—
30" : 40" :: 19 : Answer.
The answer is 25·3 lbs.; on reference to the table we find 25 lbs. to
be the observed value, hence our hypothesis is verified for this bar.