Farther, the angle whose tangent is-f+ √1+f2 is half Let, therefore, BF be the slope which loose earth would, of itself, naturally assume then, the line BE which determines the triangle of earth that exerts the greatest horizontal stress against the vertical wall bisects the angle CBF. 134. SCHOLIUM. Sandy and loose earth takes a natural declivity of 60° from the vertical; stronger earth will take a declivity of 53°. Therefore, for a terrace of loose earth we have i = 30°; for another of strong and close earth i = 2610. Hence, for the former kind, where tan. 30° = √3, the value of the stress is fa's, and that of the momentum of the stress a3s. For the latter kind, where tan. 26° = nearly, the stress = a's, its momentum = a3s. 135. The horizontal stress and momentum being thus known, it is easy to proportion to them the resistance of the wall ABCD. Let b = AB, while BC = a, and let s be the spec. grav. of the wall. For brick, s = 2000, for strong earth, s = 1428. Then the momentum of the resistance referred to the point AB, being ab's; we shall have ab's=a's (for strong earth) Thus, if a = 39-37 feet, s and s as above, we shall find b=9-326 feet. EXAM. 2. Supposing the earth of the same kind as in the above example, s to s, as 4 to 5, and the height of the wall and bank each 12 feet; required the thickness of the wall, being rectangular. Ans. 2-986 feet. Note. The preceding investigation proceeds upon the principles assumed by Coulomb and Prony. They who wish to go thoroughly into this subject, and have not opportunity to make experiments, may advantageously consult Traité Expérimental, Analytique et Pratique de la Poussée des Terres, &c. par M. Mayniel. ON THE FLEXIBILITY, STRENGTH, AND A piece of solid matter may be exposed to, at least, four distinct kinds of strains: viz. 1st. It may be pulled, or torn, asunder, as in the case of ropes, stretchers, king-posts, tie-beams, &c. 2dly. It may be crushed, as in the case of pillars, posts, and truss-beams. 3dly. It may be broken across, as in the case of a joint or rafter. 4thly. It may be wrenched, or twisted, as in the case of the axle of a wheel, the nail of a press, &c. The complete investigation of these particulars, only in their principal varieties, would require a volume. The student who wishes to go into the inquiry with scientific precision, may consult M. Girard's Treatise on the Resistance of Solids, an interesting essay on the Flexibility of Wood, by M. Dupin, in Journal de l'Ecole Polytechnique, tome 10, Tredgold's Principles of Carpentry, and Mr. Barlow's va. luable Essay on the Strength and Stress of Timber. Having attended many of the experiments recorded in the lattermentioned work, I can with confidence recommend its principal results as accurate and useful; and shall, therefore, refer to the work itself for the experiments and investigations from which the following formulæ and rules are deduced. Let I denote the length, a the breadth, d the depth of a rectangular beam, all in inches, w the weight with which it is loaded in the middle (being supported at both ends), d the deflection occasioned by that weight, and E the measure of wl3 add the elasticity: then it is found that = E is a constant quantity, for the same timber; or, which amounts to the wl3 same, that = δ. Ead3 This formula is equally applicable to beams fixed at one end, and loaded at the other, and those which are supported at both ends and loaded in the middle; only the value of E in the one case will be to that in the other, as 32 to 1. For the ultimate deflection of beams before their rupture, 12 the theorem is = u, where A is the last deflection. dA If the resistance of a rod an inch square be s, then ad's will be the resistance of a beam the same length, whose breadth is a and depth d: also, if the angle of deflection be A, and the breaking weight be w; then 1. When the beam is fixed at one end, and loaded at the other. lw cos. lw cos. adas, or ada s, a constant quantity. 2. When the beam is supported at each end, and loaded in the middle. lwsec2 Alwsec = ad's, or s, constant. 4ad2 3. When the beam is fixed at each end, and loaded in the middle. lwsec2 lwsec2 A = ad's, or s, constant. 6ad2 sec A = ad's, or 4. When the beam in either of the two last cases is loaded at any other point than the centre. We shall have, in the former case, by denoting the two unequal lengths by m and n, mnw ι and in the second, 2mnw 31 sec2 A ad's, or still the same constant quantity. And the first formula will also apply to a beam fixed at any given angle of inclination; observing only, that the angle A, in this case, will represent the angle of the beam's inclination, increased or diminished by the angle of its deflection, according as its first position is ascending or descending; or rather, it will denote the angle of the beam's inclination at the moment of fracture. In all these cases, when it is only intended to apply the results to the common application of timber to architectural and other purposes, the angle of deflection may be omitted, and the equations then become simply, 3. Sy 4. = s, 6ad lad2 The absolute value of direct cohesion on a square inch is c = s'd (d-D); where D 2 is the depth of the natural axis, or of the line which separates the compressed from the stretched portion of the wood. The subjoined portion of data for different kinds of wood, results from the union of these formule with experiments. Pitch Pine Red Pine.... New Eng. Fir. .... Do. Spec. 2. 657 605 7359700 1341 553 757 5967400 1102 .. 1368 10000 1116 9947 1131 10707 .... 753 588 5314570 1108 693 531 Norway Spar 1172 .... 7655 7352 577 648 5832000 1474 1492 12180 Other tables and observations on the cohesive strength of metals, &c. are given in a subsequent part of this volume. Solution of Practical Problems, from the preceding Data. PROB. I. To find the Strength of Direct Cohesion of a Piece of Timber of any given Dimensions. Rule.-Multiply the area of the transverse section, in inches, by the value of c, in the preceding table of data, and the product will be the strength required. Note. If the specific gravity be not the same as the mean tabular specific gravity; say, as the latter is to the former, so is the above product to the correct result. EXAM. 1. What weight will it require to tear asunder a piece of teak 3 inches square, the specific gravity being 745? Ans. 139950lbs. EXAM. 2. What weight will break vertically a cylinder of ash, 2 inches in diameter, and specific gravity 700? Ans. 50166lbs. PROB. II. To compute the Deflection of Beams fixed at one 3. Divide the latter product by the former, for the deflec. tion sought. EXAM. 1. An ash batten, 3 inches square, is fixed in a wall, and projects from it 4 feet. If a weight of 200lbs. be hung on its extremity, how much will it be deflected? Ans. 11⁄2 inches. EXAM. 2. What would the same beam be deflected if a prop or shore, proceeding from the wall, met it at half its length ? Here, without repeating the operation, as we know that the deflections are as the cubes of the lengths; and as by means of the shore the length is reduced to one half the former, viz. to 2 feet, we have 43:23:: 14 inches (former deflec.): 1×231 43 3 4 824 = of an inch, answer. Ехлм. 3. A batten of New England fir, 6 feet long and 4 inches deep, by 21⁄2 inches in breadth, is fixed at one end, and loaded, uniformly throughout its length, with 200lbs., how much will its extremity be deflected ? Note. The same rule will apply, when the weight is dis. tributed throughout the length, by multiplying the second product by 12 instead of 32. PROB. III. To compute the Deflection of Beams, supported at each End, and loaded in the Middle with any given Weight. Rule. 1. Multiply the tabular value of E by the breadth and cube of the depth, both in inches. VOL. II. 51 1 1 |