weight multiplied by a lever equal to half the base must exceed the sum of the elements of the wind's action. 15929. To determine the pressure of wind in pounds per square foot, equal to the stability of a square stalk, multiply the weight of the stalk in pounds, by twice its width in feet at the base, and divide the product by the square of its height in feet, and by the sum of its top and bottom breadths in feet. Let w=weight of stalk in pounds=90,000 p=pressure of wind in pounds per square foot equal to stability of stalk c=breadth of stalk at top in feet =2 feet Then 2,900 m * 23"=p=48 pounds per square foot. If the stalk be circular, then, to determine the pressure of wind, proceed as before, but replace the breadths by the diameter, and multiply the result by 2. Campin, Engineers' Pocket Remembrancer, 1863. Sect. X. BEAMS AND PILLARS. 1593. The woods used for the purposes of carpentry merit our attention from their importance for the purpose of constructing solid and durable edifices. They are often employed to carry great weights, and to resist great strains. Under these circumstances, their strength and dimensions should be proportioned to the strains they have to resist. For building purposes, oak and fir are the two sorts of timber in most common use. Stone has, doubtless, the advantage over wood: it resists the changes of moisture and dryness, and is less susceptible of alteration in the mass ; hence it ensures a stability which belongs not to timber. The fragility of timber is, however, less than that of stone, and its facility of transport is far greater. The greatest inconvenience attending the use of timber, is its great susceptibility of ignition. This has led to expedients for another material, and probably it will become greatly superseded by iron. 1594. Oak is one of the best woods that can be employed in carpentry. It has all the requisite properties; such as size, strength, and stiffness. Oaks are to be found capable of furnishing pieces 60 to 80 ft. long, and 2 ft. square. In common practice, beams rarely exceert 36 to 40 ft. in length, by 2 ft. square. 1595. In regard to its durability, oak is preferable to all other trees that furnish equal lengths and scantlings: it is heavier, better resists the action of the air upon it, as well as that of moisture and immersion in the earth. It is a saying relating to the oak, that it grows for a century, lasts perfect for a century, and takes a century to perish. When cut at a proper season, used dry, and protected from the weather, it lasts from 500 to 600 years. Oak, like other trees, varies in weight, durability, strength, and density, according to the soil in which it grows. The last is always in an inverse proportion to the slowness of its growth ; trees which grow slowest being invariably the hardest and the heaviest. 1596. From the experiments made upon oak and other sorts of wood, it is found that their strength is proportional to their density and weight; that of two pieces of the same species of wood, of the same dimensions, the heavier is generally the stronger. 1597. The weight of wood will vary in the same tree; usually the heaviest portions are the lower ones, from which upwards a diminution of weight is found to occur. In full. grown trees, however, this difference does not occur. The oak of France is heavier than that of England; the specific gravity of the former varying from 1000 to 1054, whilst the latter, in the experiments of Barlow, varies from 770 to 920. The weight, therefore, of an English cube foot of French oak is about 58 English pounds. Timber may be said to be well seasoned when it has lost about a sixth part of its weight. 1598. In carpentry, timber acts with an absolute and with a relative strength. For instance, that called the absolute strength is measured by the effort that must be exerted to break a piece of wood by pulling it in the direction of the fibres. The relative strength of a piece of wood depends upon its position. Thus a piece of wood placed horizontally on two points of support at its extremities, is easier broken, and with a less effort, than if it was inclined or upright. It is found that a smaller effort is necessary to break the piece as it increases in length, and that this effort does not decrease strictly in the inverse ratio of the length, when the thicknesses are equal. For instance, a piece 8 ft. long, and 6 in. square, placed horizontally, bears a little more than double of another, of the same depth and thickness, 16 ft. long, placed in the same way. In respect of the absolute force, the difference does not vary in the same way with respect to the length. The following are experiments by Rondelet, to ascertain the absolute force, the specimen of oak being of 861 specific gravity, and a cube foot, therefore, weighing 491. lbs Cohesive Force of Pieces drawn in the Direction of their Length. First experiment. A small rod of oak 0.0888 in. (= 1 French line) square, and 2.14 in, in length, broke with a weight of 115 lbs. avoirdupois Another specimen of the same wood, and of similar dimensions, broke with 105% Another specimen 11010 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 4391 lbs. avoirdupois Another specimen 418 Another specimen 4511 French, or 110 lbs. for each, French line = 0·0888 in. English). 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 1988of 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 Ibs. for 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. 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 1888 of an inch, is reduced to 2•ló 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 12, 888 every T0000 24, 6 36, 48, 60, 72, upright, that is, with its fibres in a vertical direction, its mean strength is ex- 6853 For a rod of the same oak, whose section was of the same area by 12.792 in. high 47.52 x 5 5735 for a rod 25.584 (=24 French) in. high, the strength is 144 x "C" =3421 lbs. • 3144 For a rod 38•376 (=36 French) in. high, the strength is 144 x 47.52 = 2281 lbs. 2936 47.52 For a rod 51.160 ( = 48 French) in. high, the strength is 144 * =1140 lbs. 47.52 For a rod 63.960 (=60 French) in. high, the strength is 144 x = 570 lbs. 47 52 For a rod 76-752 (=72 French) in. high, the strength is 144 x = 285 lbs. For a cube of fir, whose sides are 1.066 in. area (=1 square in. French), placed as before, with the fibres in a vertical direction, we have 144 x 56.16=8037 lbs. 8089 47.52 3 6 12 • The French inch, consisting of 144 lines. 56-16 x 5 2 56.16 3 6 24 For a square rod, whose base was 1.066 in. area ( =) square in. French), 12.792 in. - 6863 56.16 For a rod 25.584 (=24 French) in. high, 144 x =4043 lbs. • 3703 For a rod 38.376 (=36 French) in. high, 144 x = 2696 lbs. . 2881 56-16 56.16 = 674 lbs. 56.16 For a rod 76.752 (=72 French) in. high, 144 + = 337 lbs. 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, morto over, 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 1.066 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 1 *066 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 (2.1324): 10:23 (3:198?):: 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 fatwise, 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 8 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. avoirdupois. Now, as the absolute strength of oak is from 98 to 110 lbs. for every to 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. 36 TF. in.) 1 609. Three other rods, 2 French in. square (2.132 Eng.), and of the same length between the supports, broke with a mean weignt of 1711.8 lbs. By the rule 576(=114x1)x 52 =1658-88 lbs. avoirdupois. Without further mention of the experiments of Belidor, we 144 x 52 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 preportion 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. Thus A beam, 7.462 ft. long, and 5.330 in. square, broke with a weight of 12495 06 lbs. avoirdupois Another, 14.924 ft. long, of the same dimensions, broke with a weight of 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 for that of 14.924 6247.80 for that of 29.848 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 catenarian 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. the number of times that the depth is contained in the length the length ada da 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 5.330 = 43.2=b. 1613. The vertical depth being 5.330 or 63.960 lines, da will be 4089.88; substituting these values in the formula axd? _??? 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, 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 64-99X 4089 88 =4120.20, as in the experiment. The scientific world generally, the architect and engineer especially, are indebted to the person from wliom the tables which follow have emanated. They are the result of laborious experiments, no doubt, and as such, deserve much consideration. But they must not lead us to ignore the labours in the cause of such men in England as Robison, Young, Bevan, Rennie, Tredgold, Barlow, Hodgkinson, and later, Fairbairn, with others, from some of whose treatises we have adopted passages; nor to speak disparagingly of the results of those who have endeavoured to benefit both the architect and engineer, by bringing the aid of mathematical investigations, to found upon their experiments, safe and general rules for practice. 1 b 230-256 64:36 x 4089-88 492 4089 88 3 3 43-2 4089 88 3 TABLES OF EXPERIMENTS, TABLE I Esperiments on Pieces of Timber 4.264 inches square, supposing the absolute Strength 60.1344. TADLE II. Experiments on Pieces of Timber 5.330 inches square, supposing the absolute Strength 64.36. s 101.52 2.665 7.462 16 $ 12,7177 64.37 58.32 95.58 12,496 58.31 12,496 2.665 | 12,177 S [112:32 2.842 8.528 19 s 10,692 63.58 56.67 110-16 10,626 57.94 10,750 3109 | 10,449 127.44 3.198 9,072 9.594 213 125.28 3.464 62.20 54:42 8,991 8,635 56:58 9,429 124.20 3:731 8,856 142:56 3:375 7,803 10.660 24 140-40 3.731 60.40 51.76 7,614 7,765 5572 8,357 138.78 4.264 7,668 168.48 5.886 12.792 283 6,534 63.50 51.54 166:32 6,644 54.99 6,748 6:132 6,588 5,832 5,819 52.26 4,600 8794 5,616 $ 225.72 8.616 17.056 383 4,779 65.12 51.30 221.40 4,810 50:53 4,738 8.705 4,617 250*56 8.528 19:188 43 4,050 64.99 49:44 249:48 8.705 3,942 4,120 48.80 4,066 284.04 9.416 21.320 48 65.60 48.32 3,537 279:72 10.660 3,624 47.08 3,530 3,429 29.452 523 309:48 11.992 68.34 49-33 3,213 3,364 45.35 3,092 s 334.80 11.726 25.584 57 2,3767 60.76 40 02 331.56 14:491 2,502 43.62 2,726 2,295 393:12 19.188 29.848 67} 63.42 39.24 s 1,944 2,112 40:16 388.80 23.452 2,151 1,890 TABLE III. Strength 56.88. $ 20,790 70462 14 60:44 54.50 20,635 51.84 9196 13662 2.132 | 20,142 160.92 2.487 8.528 16 57.75 52.28 Š 16,956 157.68 2576 16,804 | 51:12 15,562 | 16,578 179.28 2.664 9.594 18 s 14,526 56.09 4961 14,292 50.40 14,547 177.66 3.020 | 13,878 203.04 3.198 s 12,393 10'660 20 54.23 47.05 200.88 3:430 ( 11,907 12,197 49.68 12,877 Ec |