217. Transverse Strength.-A piece of timber projecting from a wall, in which it is fixed, may be strained or broken by a weight suspended from the extremity, as in fig. 1, pl. XLII, or by a load uniformly distributed over it, as in the cantilivers of a roof. Figure 2 exhibits a piece of timber in the act of breaking; the bar moves round a pcint, A, at or near the middle of the depth; the fibres above, from A to CD, are supposed not to be broken, they are therefore in a state of tension, and the fibres below the point A, from A to B, are in a state of compression: both these forces equally counteract the efforts of the weight W; the force of extension being equal to that of compression. Figure 3 exhibits the manner in which a beam, supported at both its extremities, may be broken by the application of a force in the middle, or between its ends, as in the case of joists, binding-beams, and girders, which have not only to sustain their own weight, but also any accidental weights with which they may be loaded. This manner of exposing timber to fracture is the same as that represented at fig. 4, where the weights are substituted for the props and made to pull upwards, each weight being equal to half the weight suspended in the middle. Figure 5 represents a joist supported by two walls. We must here observe, that joists ought never to be firmly fixed in walls when they are inserted only nine or ten inches, as in common cases; for they would endanger the wall by causing it to bend or fracture, particularly when the wall is thin: however, when the wall is of sufficient thickness, and the timber inserted through the whole thickness, as in fig. 6, the effort to bend or fracture the wall will not be so great, and the timber, thus fixed, will be exceedingly stiff, the strength being increased by the mode of fixing. A joist, as in fig. 9, reaching over two areas of equal breadth, is much stronger than two joists of the same scantling, reaching over the two areas of the same breadth. Figure 7 exhibits another way in which timber may be broken, by being crushed, as in the case of columns, strong posts, principal rafters, &c. In figure 8, which represents a pair of rafters, supported on two opposite walls, a weight, W, suspended from the vertical angle A, compresses the rafters AB, AC, in the direction of their lengths. We have already shown the reader how to estimate the effect of such a weight in spreading out the walls in Art. 62. 218. Figure 9 shows the efforts of two equal weights to break two rafters; but this may be prevented by two struts, branching from the king-post to the points in the rafters under the places where the weights are applied. The effect of the weights is, therefore, to crush both rafters and struts. For a force applied upon the back of a rafter presses in the direction of the struts, which again press on the lower end of the king-post, and the king-post presses on the tops of the rafters, the lower ends of which press the extremities of the tie-beam, which is therefore brought into a state of tension. Therefore, the rafters are in a state of compression, and the king-post in a state of tension, while the struts are in a state of compression. 219. With regard to direct cohesion, the strength is as the number of ligneous fibres, and therefore as the area of fracture; but, in transverse strains, the case is very different, where, instead of a direct application of the force in the line of the fibres, it is made to act upon them in the direction of the fibres, by means of levers, and therefore the effect of the force depends on the proportions of the levers, as well as on the area of the section of fracture. GALILEO, to whom the physical sciences are so much indebted, was the first who undertook to investigate the subject upon pure mathematical principles. He considered solid bodies as being made up of numerous small fibres, applied parallel to each other: he assumed the force which resisted the action of power to separate them, to be directly as the area of a section per T pendicular to the length: that is, as the number of fibres of which the body is composed: he likewise supposed that bodies resisted lateral fracture by cohesion only, and that each particle was equally acted upon, and therefore the whole resistance to fracture of a rectangular beam, turning on a line in one of its sides perpendicular to the two edges, was the same as if the whole resistance had been comprised in the centre of gravity. bd2 Now, in a rectangular beam, the centre of gravity is distant from the axis half the depth of the beam. Therefore, let b be the breadth, and d the depth, of the beam; then the distance of the centre of gravity will be d, and the effort to resist fracture will be bdx d=d. Therefore, when a beam is solidly fixed in a wall, if I be the length, w the weight that will break it: From other investigations, which we shall not attempt to show here, Galileo endeavoured to prove that, whatever weight is required to break a beam fixed at one end, double that weight is necessary to break a beam of equal breadth and depth, with twice the length, when supported at both ends. 220. But Marriotte, a Member of the French Academy, discovered the inaccuracy of Galileo's theory, and was fortunate enough to arrive at the true one. The discovery of Marriotte attracted the attention of the philosopher Leibnitz, who, through some strange oversight, concluded that every fibre acted by tension only, and instead of acting with an equal force, each exerted a power of resistance proportional to the distance of extension, and that the beam turned on a line in one of its sides as a fulcrum. The high name and authority of Leibnitz caused his inaccurate views to be followed, and, till within a few years, the labours of Marriotte were not appreciated according to their value. A complete investigation of the resistance of materials, according to the direction and situation of the forces applied, would require a volume. The reader who wishes for more information on this subject, cannot do better than consult the third Edition of Mr. Barlow's valuable Essay on the Strength and Stress of Timber;" and Mr. Tredgold's "Practical Treatise on the Strength of Iron and other Metals." We propose here to give the rules for plain rectangular beams, and in as simple a mode as possible. 221. The strength of beams of the same kind, and fixed in the same mani.er, in resisting a transverse force, is simply as their breadth, as the square of their depth, and inversely as their length. Thus, if a beam be twice as broad as another, it will also be twice as strong; but if it be twice as deep, it will be four times as strong: for the increase of depth not only doubles the number of the resisting particles, but also gives each of them a double power, by increasing the length of the levers on which they act. The increase of the length of a beam must also obviously weaken it, by giving a mechanical advantage to the power which tends to break it: and some experiments appear to show, that the strength is diminished in a proportion somewhat greater than that in which the length is increased. The strength of a beam, supported at both ends, is twice as great as that of a single beam of half the length, which is fixed at one end; and the strength of the whole beam is again nearly doubled, if both the ends be firmly fixed; and the stiffness follows the same proportions, as far as the fixing and manner of supporting is concerned. These proportions, combined with the following series of experiments, will be sufficient to enable the carpenter to compute the strength of the beams which are used in buildings, more complicated forms we do not attempt to give, as they are chiefly executed in iron. 222. Results of Experiments on the Strength of various Specimens of wood, from the Minutes of Evidence on the Timber Trade, taken before a Committee of the House of Commons, p. 22. The trials were made upon pieces carefully selected as to quality and grain; the pieces were two feet in length and one inch square, and all of them from split portions of timber. The order of Strength as ascertained by their being broken by the application of weight. 223. An account of the specific gravity, strength, and deflection of the several kinds of Foreign Fir, as found by Mr. Peter Barlow, by experiments made under his inspection, from timber supplied from his Majesty's Yard, at Woolwich, and delivered in evidence before the Committee on the Timber Trade, by Sir Robert Seppings. The pieces tried were eight feet long, two inches square, supported by props seven feet apart, and had the weight placed in the middle of each piece. 224. The following series of experiments were made by Mr. George Buchanan, Civil Engineer, of Edinburgh, before the students of the School of Arts of that city. They were made with a peculiar apparatus he had constructed for the purpose, on the principle of the hydrostatic press. The experiments on timber were made on bars of Memel fir, supported at the ends, and the force applied at the middle of the length From the evidence afforded by his experiments, Mr. Buchanan is satisfied that it is unsafe to load a beam with more than half the weight that would break it. Hence, when we say it should be loaded with one-fourth of the breaking weight only, the allowance is only double that required for absolute safety. 225. We have next to detail an interesting series of experiments made by Mr. George Rennie, on bars of cast-iron, in which all the bars had the same area of section, but differently formed; consequently, these experiments show, at once, the advantage to be derived by adopting different forms for the sections of beams. The bars were supported at the ends, and loaded in the middle, all the bars were cast from the cupola. The last column in this, as well as in the preceding tables, we have calculated, and added for the purpose of rendering the experiments available in calculation, the mode of doing which we have shortly to describe. |