Lg cosec. tan.&= =cot. μ1Lq sec. (394). If the inclination of the roof be made to vary, the span remaining the same, P will attain a minimum value when It is therefore at this inclination of the roof of a given span, whose trusses are of the simple form shown in the figure, that the least pressure will be produced upon the feet of the rafters. If represent the limiting angle of resistance between the feet of the rafters and the surface of the tie, the feet of the rafters would not slip even if there were no mortice or notch, provided that z were not greater than ₫ (Art. 141.), orcot not greater than tan. 7, or 303. The thrust upon the feet of the rafters of a roof in which the tie-beam is suspended from the ridge by a king-post. B It will be shown in a subsequent portion of this work (see equation 558) that, in this case, the strain upon the king-post BD is equal to ths of the weight of the tie-beam with its load. Representing, therefore, the weight of each foot in the length of the tie beam by μ, and proceeding exactly as in the last article, we shall obtain for the pressure P upon the feet of the rafters, and its inclination to the vertical, the expressions * If the surfaces of contact be oak, and thin slips of oak plank be fixed under the feet of the rafters, so that the surfaces of contact may present parallel fibres of the wood to one another (by which arrangement the friction will be greatly increased), tan. 448 (see p. 152.); whence it follows that the rafters will not slip, provided that their inclination exceed cot.-1.96, or 46° 10'. P=IL{(2μg sec. 1+ 2)2 + (μg sec. + 2)2 cot.2}* .. (397). 304. The stability of a wall sustaining the thrust of a roof, having no tie-beam. B a Let it be observed, that in the equation to the line of resistance of a wall (equation 377); the terms P sin. & and P cos. a represent the horizontal and vertical pressures on each foot of the length of the summit of the wall; and that the former of these pressures is represented in the case of a roof (Art. 302.) by L cosec., and the latter by L sec. ; whence, substituting these values in equation (377), we obtain for the equation to the line of resistance in a wall sustaining the pressure of a roof, without a tie-beam in which expression a represents the thickness of the wall, k the distance of the feet of the rafters from the centre of the summit of the wall, L the span of the roof, u the weight of a cubic foot of the wall, and μ, the weight of each square με foot of the roofing. The thickness a of the wall, so that, being of a given height h, it may sustain the thrust of a roof of given dimensions with any given degree of stability, may be determined precisely, as in Art. 293., by substituting h for x in the above equation, and a-m for y, and solving the resulting quadratic equation in respect to a. If, on the other hand, it be required to determine what must be the inclination of the rafters of the roof, so that . เ being of a given span L it may be supported with a given degree of stability by walls of a given height h and thickness a; then the same substitutions being made as before, the resulting equation must be solved in respect to instead of a. The value of a admits of a minimum in respect to the variable. The value of, which determines such a minimum value of a, is that inclination of the rafters which is consistent with the greatest economy in the material of the wall, its stability being given. 305. The stability of a wall supported by buttresses, and sustaining the pressure of a roof without a tie-beam. The conditions of the stability of such a wall, when supported by buttresses of uniform thickness, will evidently be determined, if in equation (388) we substitute for P cos. ≈ and P sin. a their values μL sec. and L cosec.; we shall thus obtain 1 μ¡ Lí‡h ̧ cosec. ¡—7 sec.‚)={μ(a ̧2h ̧+2α ̧à ̧h ̧+}a ̧3h ̧)−m{μ} L sec. 1+μ{a}h+'agh2} · · (400). From which equation the thickness a2 of the buttresses necessary to give any required stability m to the wall may be determined. If the thickness of the buttresses be different at different heights, and they be surmounted by pinnacles, the conditions of the stability are similarly determined by substituting for P sin. a and P cos. a the same values in equations (390) and (392). To determine the conditions of the stability of a Gothic building, whose nave, having a roof without a tie-beam, is supported by the rafters of its two aisles, or by flying buttresses, which rest upon the summits of the walls of its aisles, a similar substitution must be made in equation (383). If the walls of the aisles be supported by buttresses, equation (383) must be replaced by a similar relation obtained by the methods laid down in Arts. 299. and 301.; the same substitution for P sin. a and P cos. a must then be made. 306. The conditions of the stability of a wall supporting a H shed roof. Let AB represent one of the rafters of such a roof, one extremity A resting against the face of the wall of a building contiguous to the shed, and the other B upon the summit of the wall of the shed. E R F It is evident that when the wall BH is upon the point of being overthrown, the extremity A will be upon the point of slipping on the face of the wall DC; so that in this state of the stability of the wall BH, the direction of the resistance R of the wall DC on the extremity A of the rafter will be inclined to the perpendicular AE to its surface at an angle equal to the limiting angle of resistance. Moreover, this direction of the resistance R which corresponds to the state bordering upon motion is common to every other state; for by the principle of least resistance (see Theory of the Arch) of all the pressures which might be supplied by the resistance of the wall so as to support the extremity of the rafter, its actual resistance is the least. Now this least resistance is evidently that whose direction is most nearly vertical; for the pressure upon the rafter is wholly a vertical pressure. But the surface of the wall supplies no resistance whose direction is inclined farther from the horizontal line AE than AR; AR is therefore the direction of the resistance. Resolving R vertically and horizontally, it becomes R sin. and R cos. 4. Representing the span BF by L, the inclination ABF by 1, the distance of the rafters by q, and the weight of each square foot of roofing by μ, (Art. 10.), R sin. μι +P cos. a=p, Lq sec. and R cos. -P sin. a=0; also the perpendiculars let fall from A on P and upon the vertical through the centre of AB, are represented by L cos. (a+1) sec. and L; therefore (Art. 7). P cos. (a+1)=L, 9. Eliminating between these equations, we obtain R= If the rafter, instead of resting at A against the face of the wall, be received into an aperture, as shown in the figure, so that the resistance of the wall may be applied upon its inferior surface instead of at its extremity: then drawing AE perpendicular to the surface of the rafter, the direction AR of the resistance is evidently inclined to that line at the given limiting angle . Its inclination to the Π horizon is therefore represented by -1+0. 2 ing this angle for in equations (401) and (402), Substitut Lu, q sec. P= Substituting in equations (377) and (379) for P sin. a, P cos. α, 307. THE PLATE BANDE OR STRAIGHT ARCH. N F TC ME Let MN represent any joint of the plate bande ABCD, whose points of support are A and B; PA the direction of the resistance at A, WQ a vertical through the centre of gravity of AMND, TR the direction of the resultant pressure upon MN; the directions of TR, WQ, and PA intersect, therefore, in the same point O. Let OAD=α, AM=x, MR=y, AD=H, AB=2L, weight (404). |