For a square rod, whose base was 1·066 in. area (=1 square in. French), 12-792 in.

56-16 × 5
6

=6739 lbs.

6863

56'16
2

=4043 lbs.

- 3703

56.16
3

2696 lbs.

2881

=1348 lbs.

674 lbs.

=

337 lbs.

56-16
6
56.16
12
56.16
24

high, we have 144 x For a rod 25-584 (=24 French) in. high, 144 × For a rod 38-376 (=36 French) in. high, 144 × For a rod 51160 (48 French) in. high, 144 × For a rod 63.960 (=60 French) in. high, 144 × For a rod 76-752 (=72 French) in. high, 144 + The rule by Rondelet above given was that also adopted by MM. Perronet, Lamblardie, and Girard. In the analytical treatise of the last-named, some experiments are shown, which lead us to think it not very far from the truth. From the experiments, moreover, we learn, that the moment a post begins to bend, it loses strength, and that it is not prudent, in practice, to reduce its diameter or side to less than one tenth of its height.

1602. In calculating the resistance of a post after the rate of only 10.80 for every 1'066 superficial line English (=1 line super. French), which is much less than one quarter of the weight under which it would be crushed, we shall find that a square post whose sides are 1066 ft.(= 1 ft. French) containing 22104-576 English lines (=20736 French), would sustain a weight of 238729 lbs. or 106 tons. Yet as there may be a great many circumstances, in practice, which may double or triple the load, it is never safe to trust to a post the width of whose base is less than a tenth part of its height, to the extent of 5 lbs. per 1066 line; in one whose height is fifteen times the width of the base, 4 lbs. for the same proportion; and when twenty times, not more than 3 lbs.

Horizontal Pieces of Timber.

1603. In all the experiments on timber lying horizontally, as respects its length, and supported at the ends, it is found that, in pieces of equal depth, their strength diminishes in proportion to the bearing between the points of support. In pieces of equal length between the supports, the strength is as their width and the squares of their depths. We here con

tinue M. Rondelet's experiments.

1604. A rod of oak 2-132 in. (2 in. French) square, and 25.584 in. (24 in. French) long, broke under a weight of 2488 32 lbs., whilst another of the same dimensions, but 19.188 in. (18 in. French) bore 3353 40; whence it appears that the relative strength of these two rods was in the inverse ratio of their length. The proportion is 19.188: 25·584::2488-32: 3317.76, instead of 3353-40 lbs., the actual weight in the experiment.

1605. In another rod of the same wood, 2·132 in. wide and 3·198 deep, and 25·584 in. bearing, it broke with a weight of 5532 lbs. In the preceding first-mentioned experiment it was found that a rod of 2·132 in. square, with a bearing 25-584 in. bore 2488.32 lbs. Supposing the strength of the rods to be exactly as the squares of their heights, we should have 4.54 (2·132o): 10·23 (3·1982): 2488 32: 5598 7 lbs. ; which the second rod should have borne, instead of 5532 lbs. There are numberless considerations which account for the discrepancy, but it is one too small to make us dissatisfied with the theory.

1606. In a third experiment on the same sort of wood, the dimension of 3.198 in. being laid flatwise, and the 2.132 in. depthwise, the bearing or distance between the supports being the same as before, it broke with a weight of 3578 lbs. whence it follows that the strength of pieces of wood of the same depth is proportional to their width. Thus, comparing the piece 2.132 in. square, which bore 2488 lbs., we ought to have 2·132; 3.198 ::2488-32 3624-48, instead of 3573 lbs.

1607. From a great number of experiments and calculations made for the purpose of finding the proportion of the absolute strength of oak, to that which it has when lying horizontally between two points of support, the most simple method is to multiply the area of the piece in section by half the absolute strength, and to divide the product by the number of times its depth is contained in the length between the points of support.

1608. Thus, in the experiments made by Belidor on rods of oak 3 French (=3·198 English) ft. long, and i French (=1.066 in. English) in. square, the mean weight under which they broke was 200-96 lbs. averdupois. Now, as the absolute strength of oak is from 98 to 110 lbs. for every in. (=1 French line), the mean strength will be 104 and 52 lbs. for its half, and the rule will become (144 lines, being =1 French in.) 207 30 lbs., instead of 200·96 lbs.

144 x 52

36 (F. in.)

888

576(=144x4) x 52 18

1609. Three other rods, 2 French in. square (2·132 Eng.), and of the same length between the supports, broke with a mean weight of 17118 lbs. By the rule =1658 88 lbs. averdupois. Without further mention of the experiments of Belidor, we

The

may observe, that those of Parent and others give results which confirm the rule. experiments, however, of Buffon, having been made on a larger scale, show that the strength of pieces of timber of the same size, lying horizontally, does not diminish exactly in the proportion of their length, as the theory whereon the rule is founded would indicate. It becomes, therefore, proper to modify it in some respects.

1610. Buffon's experiments show that a beam as long again as another of the same dimensions will not bear half the weight that the shorter one does.

A beam, 7-462 ft. long, and 5.330 in. square, broke with a weight of

Another, 14-924 ft. long, of the same dimensions, broke with a weight of

A third, 29.848 ft. long, of the same dimensions, bore before breaking

Thus

12495 06 lbs. averdupois.

5819.04

2112.48

By the rule, the results should have been, for the 7·462 ft. beam 12495.60

6247.80
3123.90

Whence it appears, that owing to the elasticity of the timber, the strength of the pieces, instead of forming a decreasing geometrical progression, whose exponent is the same, forms one in which it is variable. The forces in question may be represented by the ordinates of a species of catenarean curve.

1611. In respect, then, of the diminution of the strength of wood, it is not only proportioned to the length and size, but is, moreover, modified in proportion to its absolute or primitive force and its flexibility; so that timber exactly of the same quality would give results following the same law, so as to form ordinates of a curve, exhibiting neither inflection nor undulation in its outline: thus in pieces whose scantlings and lengths form a regular progression, the defects can only be caused by a difference in their primitive strength; and as this strength varies in pieces taken from the same tree, it becomes impossible to establish a rule whose results shall always agree with experiment; but by taking a mean primitive strength, we may obtain results sufficiently accurate for practice. For this purpose, the rule that nearest agrees with experiment is

1st. To subtract from the primitive strength one third of the quantity which expresses the number of times that the depth is contained in the length of the piece of timber.

2d. To multiply the remainder thus obtained by the square of the length.

3d. To divide the product by the number expressing the relation of the depth to the length.

Hence calling the primitive strength

= @

the number of times that the depth is contained in the length
the depth of the piece

= b

= d

the length

x dd add dd b

b 3

1612. Suppose the primitive strength a=64.36 for each 1.136 square line (=1 line French), we shall find for a beam 5·330 in. square, by 19-188 ft. long, or 230-256 inches, that the proportion of the depth to the length 1613. The vertical depth being 5-330 or 63.960 lines, dd will be 4089-88; substituting these values in the formula axdd

=4067·99, instead of 4120-20, the mean result of two beams of the same scantlings in the experiments of Buffon. But as the mean primitive strength of the beams is, according to the second of the following tables, 64'99, instead of 64·36, which has been taken for the mean strength of all the pieces given in that table, we ought to have found less. Thus taking 64.99, we have

4089 88

3

4120-20, as in the experiment.

64-99×4089-88

43-2

The scientific world generally, the architect and engineer especially, are indebted to the person from whom the tables which follow have emanated. They are worth more than all which hitherto has been done in this country; and our surprise is great that in most of the various treatises on timber and carpentry, some whereof have resulted from no mean hands, more importance has been given to theoretical instruction than to that which might have been deduced from experiments. The treatises, indeed, on mechanical carpentry almost seem to have been written more with the view of perplexing than of assisting the student.

TABLES OF EXPERIMENTS.

Experiments on Pieces of Timber 4.264 inches square, supposing the absolute

Strength 60.1344.

Experiments on Pieces of Timber 5·330 inches square, supposing the absolute

Experiments on Pieces of Timber 6.396 inches square, supposing the absolute

12,393 12,197 49-68 12,877

Experiments on Pieces of Timber 7-462 inches square, supposing the absolute

Strength 57.85.

Experiments on Pieces of Timber 8.528 inches square, supposing the absolute

Strength 55.08.

1614. The five preceding tables give a view of the results of experiments by Buffon upon beams 4.264, 5·330, 6·396, 7·462, and 8.528 inches square, of different lengths, as compared with those found by the modified rule above given (1660.).

1615. The first column shows the length of the pieces in English feet. The second, the proportion of their depth to their length. The third, the weight of each piece in pounds averdupois. The fourth, the weight borne before breaking. The fifth, the absolute or primitive strength, that is, independent of the length. The sixth, that strength reduced in the ratio of the proportion of the depth to the length of the pieces given in the

second column. The seventh column gives the weight borne

before breaking, independent of their own weight. The eighth, the mean effort with which the pieces broke, including half their weight, the other half acting on the points of support. The ninth shows the reduced strength

of the pieces in respect of the proportions of the depth to the length, supposing the primitive strength equal for all the pieces in the same table. The tenth column gives the result of the calculation according to the rule above given.

1616. In order to give an idea of the method of representing the strength of wood of the same scantling, but of different lengths, by the ordinates of a curve, we annex fig. 612. to explain by it the result of the experiments of Buffon, given in the second table. The ordinates of the polygon N, O, P, Q, &c. represent the results of the experiments made upon beams 5.330 in. square, of different lengths, whose primitive strength varied in each piece.

1617. The ordinates of the regular curve, m, l, i, h, g, f, e, d, c, b, Z, show the results of the calculation according to the rule, taking the same primitive strength for each piece.

1618. After what has been said in a preceding page, it is easy to conceive that the primitive unequal strengths would form an irregular polygon, whereof each point would answer to a different curve; whilst, supposing the same primitive strength to belong to each piece, there should be an agreement between the strengths and scantlings which constitute a regular curve.

1619. Thus it is to be observed that the points O and P of the regular polygon only vary from the regular curve, m, l, k, i, &c., because the ordinate LO is the product of a primitive strength diminished by the mean primitive strength which produced the ordinate of the curve KP. Hence the point P is above the properly correspondent point k.

1620. For the same reason, the point c is above its corresponding point X, because the relative ordinate Ce is the product of a primitive strength greater than the mean which produced the