point where the line of resistance intersects the base of P.xM+w.xC=Q.xN+μah.xK. Now xM=xs sin. xsM=(HK-Ht) sin. x = {h−(Hp+st) Ph sin. a-(k+a)cos. a}-{pa2h+m(P cos. a+uah+w) (381). This expression may be placed under the form Q=(Pcos. a+pah+w)sec. ß. P{b cos.a-h sin. a+(k+ža) cos. a}+μah (}a+b)+wb. (b+m) cos.ßt If the numerator of the fraction in the second member of this equation be a positive quantity (as in all practical cases it will probably be found to be) the value of Q manifestly diminishes with that of m. Now the least value of m, consistent with the stability of the wall, is zero, since the line of resistance no where intersects the extrados; the least value of Q (the shore being supposed necessary to the support of the wall) corresponds, therefore, to the value zero of m; moreover this least value of the thrust upon the shore consistent with * The weight 2w of the shore may be conceived to be divided into two equal parts and collected at its extremities. †The expression (b+m) cos. ß may be placed under the form b cot.ẞ sin.ẞ+m cos. ẞ=c sin. ß+m cos. ß, where c represents the height CE of the point against which the prop rests. the stability of the wall is manifestly that which it sustains. when the wall simply rests upon it, the shore not being driven so as to increase the thrust sustained by it beyond that just necessary to support the wall.* This least thrust is represented by the formula The thrust which must be given to the prop in order that there may be given to the wall any required stability, determined by the arbitrary constant m, is determined by equation (381). The stability will diminish as the value of m is increased beyond a, and the wall will be overthrown inwards when it exceeds a. 296. The stability of a wall sustained by more than one shore in the same plane. Let EF, ef be shores in the same plane, sustaining the wall ABCD, and both necessary to D its stability; so that if EF were removed the wall would turn over upon f, and if ef were removed, upon some point between F and C. If the thrust of the shore EF be only that just necessary to sustain the tendency of the wall to overturn upon f, it is evident that the line of resistance must pass through that point; but if the thrust exceed that just necessary to the equilibrium, or if the shore be driven, then the line of resistance will intersect fg in some point x. Let fx=m; then representing the thrust upon EF by Q, the distances ƒD and fi by h and b, and the angle EFC by ẞ, the value of Q is evidently determined by equation (380). * This case presents an application of the principle of least resistance. (See Theory of the Arch.) If z be taken in like manner to represent the point where the line of resistance intersects the base of the wall, and Cz=m1, CE=b1; Сe=b2, Cƒe=ß1, CD=h1, the thrust upon the prop ef by Q, and its weight by 2w,; then the sum of the moments about the point z of Q and Q1, and the weight μah, of the wall, equals the sum of the moments of P, w, and w1; or Q1(b2+m1) cos. ß1 + Q(b1 +m ̧) cos. ß+μah ̧(¦a—m ̧) 2 = P {h, sin. a − (k + ļa−m ̧) cos. a} + (w + w1)m, . . : . (382). Substituting the value of Q in this equation, from equation (381), and solving in respect to Q,, the thrust upon the prop ef will be determined, so that the stability of the wall, upon its section fg and upon its base CB, may be m and m1 respectively. If m1=m, the portions of the wall above and below fg are equally stable. If m1=m=0, the thrust upon each shore is only that which is just necessary to support the wall, or which is produced by its actual tendency to overturn. In this case we have (Psin.a-pa2)(h ̧b—hb1)+P(b ̧ —b)(k+ļa) cos. a bb2 cos. Bi Q1 = the value of Q being determined by equation (380). 297. The stability of a structure having parallel walls, one of which is supported by means of struts resting on the summit of the other. F P Let AB and CD be taken to represent the walls, and EF one of the struts; the thrust Q upon the strut may be determined precisely as in Art. 295. So that the line of resistance may intersect the base of the wall AB at a given distance m from the extrados (see note, p. 418.) Let m, represent the distance Dx from the extrados at which the line of resistance intersects the base of the wall CD; then taking the moments of the pressures applied to the wall CD about the point x, as in Art. 295., and observing that besides the pressure Q the weight w of one half the strut is applied at E, we have Q{h, sin. ẞ+(k,+}a,—m,) cos. ẞ}=μ‚«‚h1(}a‚—m])+(k,+}a,−m,)w'; in which equation h, and a, are taken to represent the height and thickness of the wall CD, k, the distance of the point E on which the strut rests from the axis of the wall, ẞ the inclination of the strut to the vertical, and μ, the weight of a cubic foot of the material of the wall. Substituting for Q its value from equation (381), and reducing, {h sin. a-(k+ja) cos. a} −}ua3h+m(Pcos. «+uah+w)_μ{a}h}{}a,−m})+(k ̧+ja1−m})w . By this equation is determined that relation between the dimensions of the two walls and the amount of the insistent pressure P, by which any required stability may be assigned to each wall of the structure. If m=0, the pressure upon the strut will be that only which is produced by the tendency of AB to overturn; and the value of m, determined from the above equation will give the stability of the external wall on this supposition. If m=0 and m, 0, both walls will be upon the point of overturning, and the above equation will express that relation between the dimensions of the wall and the amount of the insistent pressure, which corresponds to the state of the instability of the structure. The conditions of the stability, when the wall AB is supported by two struts resting upon the summit of the wall CD, may be determined by a method similar to the above (see Art. 296.) The general conditions of the stability of the structure discussed in this article evidently include those of a GOTHIC BUILDING having a central nave, whose walls are supported, under the thrust of its roof, by the rafters of the roof of its side aisles. By a reference to the principles of the preceding article, the discussion may readily be made to include the case in which a further support is given to the walls of the (383). nave by flying buttresses, which spring from the summits of the walls of the aisles. The influence of the buttresses which support the walls of the aisles upon the conditions of the stability of the structure forms the subject of a subsequent article. 298. The stability of a wall sustaining the floors of a dwelling. The joists of the floors of a dwelling-house rest at their extremities upon, and are sometimes notched into, pieces of timber called wall-plates, which are imbedded in the masonry of the wall. They serve thus to bind the opposite sides of the house together; and it is upon the support which the thin walls of modern houses receive from these joists, that their stability is sometimes made to depend.* Representing by w the weight of that portion of the flooring which rests upon the portion ABCD of the wall, and the distance BE by c, taking K B x, as before, to represent the point where the line of resistance intersects the base of the wall, and measuring the moments from this point, we have xN.Q+xK.pah+xB. w=xM.P; whence, taking the same notation as in the preceding articles, and substituting, cQ+(a−m)μah+(a−m)w= {h sin. a−(k+ļa−m)cos. a} P; ... Qc={h sin a-(k+ja) cos. a}P-pa2h-wa+m(Pcos.a+pah+w).. (384); from which expression it appears that Q is less as m is less. When, therefore, the strain upon the joists is that only which * A house thus constructed evidently becomes unsafe when its wallplates or the extremities of its joists begin to decay. |