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 pr portion 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. avoirdupois 5819.04 A third, 29-848 ft. long, of the same dimensions, bore before 2112.43 Whence it appears, that owing to the elasticity of the timber, the strength of the piece 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. Hence calling the primitive strength = a the number of times that the depth is contained in the length the length 1612. Suppose the primitive strength a=64.36 for each 1136 square line (=1 e 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 = 230.256 =43.2=b. ax d' d' we have 3 1613. The vertical depth being 5.330 or 63.960 lines. d2 will be 4089 88; substituting these values in the formula = 4067·99, instead of 4120-20, the mean result of two beams of the same scantlings in the experiments of Buffon. 1614. Mr. Gwilt has stated that the world generally, the architect and engineer especially, are indebted to Buffon, from whom certain tables have emanated, which were the result of laborious experiments and deserved much consideration. These several tables have been omitted in this edition of the Encyclopædia, as having been superseded by the more recent and scientific investigations in England of Robison, Young, Bevan, Rennie, Tredgold, Barlow, Hodgkinson, Fairbairn, Laslett, with others, from some of whose treatises passages have been adopted herein. The results of their more modern investigations have been 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. Of the Strength of Timbers in an Inclined Position, 1622. If we suppose the vertical piece AB (fig. 613a.) to become inclined to the base. as BD, the action of a vertical force at D will tend to cause the picce to bend, and thereby to lose much of its power of resistance to a force acting in the direction of its length. Suppose the radius AB or BD to represent the vertical force acting at D on the piece BD then, by the resolution of forces," it can be resolved into two forces, one acting in the direction of its length, and the other acting at right angles to its length. The former will be represented by the line Df, or the vertical force multiplied by the sine of the angle DBC; while the latter, at right angles to BD, will act at D, tending to bend the piece BD about its base B, and will be represented by the line Bf, or the vertical force multiplied by the cosine of the angle DBC. The piece is supposed to be fixed firmly at B, and may be considered as a beam fixed at one end and strained at the other by the force represented by Bf tending to break it about the end B. When the piece comes into the horizontal position, as BC, the vertical force acting at C will cease to produce any strain in the direction of its length, and the transverse strain will be represented by the line BC or AB acting at C, and straining the piece about its fixed end B. B Fig. 6.34. 1623. Example. Let AB be a piece of Riga fir 20 feet long, the scantling being 10 inches by 6 inches. First take it in the upright position AB, then (from par. 16316) the breaking weight W in tons of an oak pillar when the load acts vertically down AB is W = 4 d2+5 2146+200 = 11.24 tons. And the strength of a pillar of Riga fir being (par. 1631c.) five-sixths that of oak, we have W-9.37 tons for Riga fir. 1624. If we now place the piece in the horizontal position, as BC, the strain upon it from the load W, at C will be entirely transverse, and the breaking weight can be found from the formula (1629g.) for a beam supported at both ends and loaded at the middle. But as the load is at one end in the present case and the beam fixed at the other, we take one fourth of the weight given in the above-named formula, so that we have 3-21 bd2 1625. Now let the piece be placed in the inclined position, BD, making 60° with the horizontal. A load F, acting vertically at D, will be resolved into two others at right angles to each other, one acting longitudinally and equal to Fx sin 60°, and the other transversely at D and equal to Fx cos 60°, or F, which equals W, in the formula above for a beam fixed at one end and loaded at the other. Therefore the breaking weight applied vertically at D will be F=2W1 = 2.408 tons. = 1626. The breaking weight acting longitudinally is Fx sin 60° W = 9.37 tons, as found above. Therefore we have 1627. This value of F is, however, more than four times the value obtained for it when taking into consideration the transverse strain as found above, and we must take the smaller amount as the actual strength of the piece, which is therefore reduced in the proportion of 2-408 to 9.37, in consequence of its inclination from the vertical. 1628. In Practical or Constructive CARPENTRY, Chap. III. Sect. IV., Tables of scantlings for timbers are given more immediately useful to the practical architect. But in consequence of the very large amount of information obtained since the first edition of this work, resulting from the investigations of scientific and practical experimentalists, the following condensed summary of the new ideas on the strength of BEAMS, GIRDERS, and PILLARS, both in timber and iron, are submitted for the consideration of the student. 1628a. The term beam is applied herein to large rectangular sections; that of girder to large irregular shapes: and those of bar and irons to small rectangular and irregular forms. 16286. Beams and girders are calculated for the following classes of buildings:I. Light workshops and factories, public halls, churches, and other buildings in which people only accumulate, with warehouses for light goods. For all these an allowance of 14 cwt., or 168 lbs., per square foot of floor surface will include the weight of the joisting, the flooring, and the load upon it. II. Storehouses for heavy goods, or factories in which heavy machinery and goods are placed. For these an allowance of 24 cwt., or 280 lbs, per square foot of floor surface will include the same weights. III. Ordinary dwelling houses. For these an allowance of 1 cwt., or 140 lbs., per square foot will include the same weights. IV The weight of a floor of timber has been calculated at 35 to 40 lbs. per square foot; 20 lbs. is usually allowed. A single joisted floor without counter floor, from 1260 to 2000 lbs. per square. A framed floor with counter flooring, from 2530 to 40C0 lbs. per square. Barrett's system at 78 lbs. A half brick arch floor, 70 lbs. A one brick arch floor, 120 lbs. Though Tredgeld allows 40 lbs per square foot for the weight of a ceiling, counter floor, and iron girders, with 120 lbs. per square foot more supposing the floor to be covered with people at any time = to 160 lbs., as the least stress, yet a warehouse floor, as required at the docks, is there calcu lated at about 17 lbs, including girders, which, with about 9 lbs, for plastering, allows 26 lbs. per sup. foot. V Partitions, or any other additional weights brought upon the floor, must also be taken into consideration. This is calculated at from 1480 to 2000 lbs. per square. VI. The weight of the load to be carried must always include that of the girder itself. STRAINS ON BEAMS AND GIRDERS. 1628c. These we shall consider under the heads I. TRANSVERSE STRAIN (1628g.), which consists partly of the action of Tension as well as of Compression, each of them being dependent upon the Cohesion of the material. Under II. TENSION (1630e.), will be considered the neutral axis (1630c.), deflection of beams (1630e.), with the modulus of elasticity (1630i), impact or collision (16300.), and the tensile strength (1630p.). Under III. COMPRESSION (1630w.) is considered Deflection of pillars, and Detrusion (1631n.). The subject IV. TORSION (1631x.) closes this section. 1629d. Timber is permanently injured if more than even of the breaking weight is placed on it. Buffon allowed, which is now the cu tom, for the safe load. Fairbairn states that for bridges and warehouses, cast iron girders should not be loaded with more thanor of the breaking weight in the middle. For ordinary purposes, for cast iron is allowed for the permanent load (Barlow). A little more than can be allowed for wrought iron beams, as that material, from its extensile capability, does not suddenly give way (Warr); but they should never be loaded with more than 4th (Fairbairn). Girders, especially those of cast iron, which are liable to be less strong than intended from irregularity in casting and cooling, should be proved before use to a little more than the extent of the safe load; this proof, however, should never exceed the half of the breaking weight, as the metal would be thoroughly weakened. Tredgold observes that a load of } of the breaking weight causes deflection to increase with time, and finally to produce a permanent set. The Board of Trade limits the working strain to 5 tons, or 11,200 lbs. per square inch, on any part of a wrought iron structure. 1628e. Of all the circumstances tending to invalidate theoretical calculations, the sun is about the worst. Mr. Clark writes, about the Britannia tubular bridge: "Although the tubes offer so effectual a resistance to deflection by heavy weights and gales of wind, they are nevertheless extremely sensitive to changes of temperature, so much so that half an hour's sunshine has a much greater effect than is produced by the heaviest trains or the most violent storm. They are, in fact, in a state of perpetual motion, and after three months' close observation, during which their motions were recorded by a self-registering instrument, they were observed never to be at rest for an hour. The same may almost he said of the large bridges over the dock passages. The sun heats the top flange, whilst the wind, a'ter sweeping along the water, impinges on the bottom flange, keeping it cool and causing it to contract, whilst the top flange is being expanded by the sun, so putting a camber on the bridge much exceeding the deflection caused by the heaviest working load. At the Mersey Docks the top flanges of the bridges are painted white, to assist in meeting the difficulty." TRANSVERSE STRAIN. length 1628g. The strength of beams in general is directly as the breadth, directly as the breadth x depth2 square of the depth, and inversely as the length; thus But a certair supposed quantity must, however, be added to express the specific strength of any material a quantity only obtained by experiments on that material. This supposed quantity is re presented by S. We then obtain breadth x depth2 x S = breaking weight. Therefore, it length experiments, a simple transposition of the quantities evolves the value of S; thu length x breaking weight S, which S then becomes a constant. As regards the usual form breadth x depth2 of a cast-iron girder, using C as a constant for a signification in a girder, similar to that o S in a beam, the formula area of section x depth2x0 breaking weight. The values of and C are only applicable to a beam or girder of a similar sectional form to that fror length = which the value was derived, since this constant expresses tue specific strength of that form of section. 1628h. Another formula for estimating the strength of beams rests on the knowledge of the resistance (or r) offered by any material to fracture by a tensile or crushing force, and the depth of the neutral axis (or n) of this area in the beam; the latter, of course, rx breadth x depth? cannot be calculated, except from experiment. The rule is = breaking weight. See RESISTANCE, in Glossary. nx length 16281. TABLE OF THE TRANSVERSE STRENGTH OF TIMBER: 1 Inch Square, 1 Foot Long. 1628k. The results of Barlow, Nelson, Moore, Denison, and some others, are collected In the above table, which gives a mean of the whole (Warr); Barlow's values are also noted separately, being those usually supplied in the Handbooks; and obtained by Barlow's formula, =S, from experiments on a projecting beam or arm; or from the formula 1x BW 1 x BW a d' 4 a da =S, when a beam supported at the ends is under trial. A measurable set is produced by a straining force very much less than that to which the material will be likely to be exposed in practice. Without having this principle in mind, the differences between the actual breaking weight and the permanent set weight of some writers will be misunderstood. The practical man, however, will use one third or some other proportion of these values, as noticed in par. 1628d. (See another Table, par. 1630s.). 16281. TABLE OF THE TRANSVERSE STRENGTH OF METALS: 1 Inch Square, 1 Foot Long. 1628m. Fairbairn's experiments on cast irons obtained from the principal iron-works, and made into bars 1 inch square and 5 feet long, proved that the longer beams are weaker than the shorter in a greater proportion than their respective lengths; that the strength does not increase quite so rapidly as the square of the depth; that the deflection of a beam is proportional to the force or load; and that a set occurs with a small portion of the breaking weight. In 59 experiments, the strongest; Ponkey In 59, experiments, the weakest; Plaskynas Spec. Grav. Break. Wt. Ult, deflect. 1.74, hard. 6.916 357 lbs. 1.36, soft. For ton, No. 2, hot blast Mean value 440 lbs., affording for the specific strength, S=1980 lbs., or='884 tons. the rule including n, a comparison of two specimens gave n=2.63. 1628n. Morries Stirling has considerably strengthened cast iron by adding a portion of malleable cast iron. Four experiments, by Hodgkinson, gave the following results :No. 2 quality (20 per cent. scrap), bars 9 ft. long, 2 ins. square No. 3 S=2248 (15 per cent. scrap). 16280. Hodgkinson also found the average breaking weight in pounds of a bar of cast iron, 1 inch square and 4 feet 6 inches long between the supports, to be as follows:Average of 21 samples of hot blast iron Average of 21 samples of cold blast iron The superior transverse strength of cold blast iron equals nearly 2 per cent. R. Stephenson experimented, in 1846 and 1847, on bars of different kinds of cast iron, 1 inch square and 3 feet bearing. The results are given in the Civil Engineer, 1850, pp. 194-9. SHAPES OF BEAMS AND GIRDERS. 1628p. "Calculation affords the following shapes for iron beams, as being enabled to de the most work with the least expenditure of substance. Beams supported at one end: L |