Cohesive Force of Pieces drawn in the Direction of their Length. A small rod of oak 0·0888 in. (= 1 French line) square, and 2·14 115 lbs. averdupois. 105 110 The mean weight, therefore, was, in round numbers, 110 lbs. A rod of the same wood as the former, 0·177 inch (=2 French lines) square, and 2.14 inches long, broke with a weight of Another specimen Another specimen = 35 160 459 lbs. averdupois. = 4 square lines 0-0888 in. English). The mean weight, therefore, was 436 lbs. for an area 3 in. ( French, or 110 lbs. for each, French line 1599. Without a recital of all the experiments, we will only add, that after increasing the thickness and length of the rods in the several trials, the absolute strength of oak was found to be 110 lbs. for every 1888 of an inch area (=1 French line superficial). The Strength of Wood in an upright Position. 1600. If timber were not flexible, a piece of wood placed upright as a post, should bear the weights last found, whatever its height; but experience shows that when a post is higher than six or seven times the width of its base, it bends under a similar weight before crushing or compressing, and that a piece of the height of 100 diameters of its base is incapable of bearing the smallest weight. The proportion in which the strength decreases as the height increases, is difficult to determine, on account of the different results of the experiments. Rondelet, however, found, after a great number, that when a piece of oak was too short to bend, the force necessary to crush or compress it was about 49-72 lbs. for every 10000 1888 of a square inch of its base, and that for fir the weight was about 56.16 lbs. Cubes of each of these woods, on trial, lost height by compression, without disunion of the fibres; those of oak more than a third, and those of fir one half. 10000 1601. A piece of fir or oak diminishes in strength the moment it begins to bend, so that the mean strength of oak, which is 47.52 lbs. for a cube of an inch, is reduced to 2.16 lbs. for a piece of the same wood, whose height is 72 times the width of its base. From many experiments, Rondelet deduced the following progression : — For a cube, whose height is 1, the strength =1 12, 24, 36, 48, 60, 72, Thus, for a cube of oak, whose base is 1066 in. area (=1 square in. French) placed upright, that is, with its fibres in a vertical direction, its mean strength is expressed by 144* × 47·52-6842 lbs. From a mean of these experiments, the result was (by experiment) in lbs. averdupois For a rod of the same oak, whose section was of the same area by 12.792 in. high 6853 47.52 x 5 5735 3144 =2281 lbs. =1140 lbs. For a rod 76-752 (=72 French) in. high, the strength is 144 × The French inch, consisting of 144 lines. 8089 For a square rod, whose base was 1·066 in. area (=1 square in. French), 12-792 in. =6739 lbs. 56.16 × 5 56'16 =4043 lbs. 6863 - 3703 56.16 =2696 lbs. 2881 56.16 =1348 lbs. = 674 lbs. = 337 lbs. 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 51.160 (=43 French) in. high, 144 × For a rod 63.960 (=60 French) in. high, 144 x 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. 56.16 1602. In calculating the resistance of a post after the rate of only 10-80 for every 1066 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 continue 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: 55987 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 3573 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 (=1066 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.) 144 × 52 =207.30 lbs., instead of 200.96 lbs. 36 (F. in.) 889 1609. Three other rods, 2 French in. square (2·132 Eng.), and of the same length be576(=144×4) × 52 tween the supports, broke with a mean weight of 1711.8 lbs. By the rule =1658.88 lbs. averdupois. Without further mention of the experiments of Belidor, we 18 may observe, that those of Parent and others give results which confirm the rule. The 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 Thus 12495'06 lbs. averdupois. 5819.04 A third, 29-848 ft. long, of the same dimensions, bore before breaking 2112.48 By the rule, the results should have been, for the 7-462 ft. beam 12495.60 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 length = b α The general formula will be, × dd b add dd = 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 b we have =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, 6499, 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 43-2 4089 88 =4120.20, as in the experiment. 3 64 99 x 4089-88 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. TABLE I. Experiments on Pieces of Timber 4-264 inches square, supposing the absolute Strength 60.1344. Mean Relative Breaking Effort Strength Weight accord-accord- calcuing to ing to lated on Experi- Calcula- relative ment. tion. Strength. Absolute Relative Strength. From Experiment. Weight in Pounds averdupois. Experiments on Pieces of Timber 5-330 inches square, supposing the absolute Experiments on Pieces of Timber 6.396 inches square, supposing the absolute 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 |