300. The stability of a pier or buttress sur• mounted by a pinnacle. Let W represent the weight of the pinnacle, and e the distance of a vertical through its centre of gravity from the edge C of the pier: then assuming to be the point where the line of resistance intersects the base of the pier, and taking the same notation as before, equation (387) will evidently become P{h, sin. a-(l—m) cos. a} = {a,—m+}a,} h‚a,‚μ‚+ h Substituting for μ, its value, or, writing μ for "1, and reducing,. n 1 P(h, sin. a—l cos. a)=fv (a,'h,+2a,a,h,+2a,h,) α We-m {P cos. +W+(a,h,+1^,^) } + If a, represent the thickness of that pier by which the wall will just be sustained under the pressure, taking m=0, and solving in respect to a,, a,na, + h1 h2 TED B of moments, THE GOTHIC BUTTRESS. 301. In Gothic buildings the thickness of a buttress is not unfrequently made to vary at two or three different heights above its base. Such buttress is represented in the accompanying figure. The conditions by which any required sta bility may be assigned to that portion of it whose base is be may evidently be determined by equation (390). To determine the conditions of the stability of the whole buttress upon CD, let the heights of the points Q, a, and b above CD be represented by h,, h, and h,; let DE=a,, DF-a,, FC=a,, Cx=m,; then adopting, in other respects, the same notations as in Arts. 299 and 300. Since the distances from a of the verticals through the centres of gravity of those portions of the buttress whose bases are DE, DF, and FC respectively, are (a,+a,+a,m,), (a,+ḍa, -m,) and (a,m) we have, by the equality P{h, sin. a—(l—m,) cos. a} =(a,+a, +‡a,—m,) h‚a‚μ+ μ (a,+1a,—m,)h,a,—+({a,—m,) h,a,+W(e—m,) . n . . ..... (392). This equation establishes a relation between the dimensions of the buttress and its stability, by which any one of those dimensions which enter into it may be so determined. as to give to m, any required value, and to the structure any required degree of stability. (See APPENDIX.) It is evident that, with a view to the greatest economy of the material consistent with the given stability of the but tress, the stability of the portion which rests upon the base be should equal that of the whole buttress upon CE; the value of m, in the preceding equation should therefore equal that of m in equation (390) If m be eliminated between these two equations, it being observed that h, and 7, in equa tion (390) are represented by h1-h, and h2-h, in equation (392), a relation will be established between a,, a,, a, h, ha, h,, which relation is necessary to the greatest economy of 3 material; and therefore to the greatest stability of the struc ture with a given quantity of material. THE STABILITY OF WALLS SUSTAINING ROOFS. 302. Thrust upon the feet of the rafters of a roof, the tie beam not being suspended from the ridge. P If, be taken to represent the weight of each square foot of the roofing, 2L the span, the inclination BAC of the rafters to the horizon, q the distance between each two principal rafters, and a the inclination to the vertical of the resultant pressure P on the foot of each rafter; then will L sec. represent the length of each rafter, and Lg sec. the weight of roofing borne by each rafter. Let the weights thus borne by each of the rafters AB and BC be imagined to be collected in two equal weights at its extremities; the conditions of the equilibrium will remain unchanged, and there will be collected at B the weight supported by one rafter and represented by Lq sec., and at A and C weights, each of which is represented by Ly sec. . Now, if Q be taken to represent the thrust produced in the direction of the length of either of the rafters AB and BC, then (Art. 13.) Lq sec. = 2Q cos. ABC: but ABC-21; therefore cos. ABC= sin. ‹; therefore 2Q sin. =,Lq sec. ; に rafter are the required presResolving Q and Q cos. ¿, The pressures applied to the foot A of the thrust and the weight Lq sec. ; and the sure P is the resultant of these two pressures. vertically and horizontally, we obtain Q sin. or Lg sec. and Lg cosec.. The whole pressure applied vertically at A is therefore represented by Lg sec. 4, and the whole horizontal pressure by La cosec. ; whence it follows (Art. 11.) that tan. a= La cosec. =cot.. (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 mor tice or notch, provided that a were not greater than (Art. 141.), or cot. not greater than tan. 9, or not less than cot.-1 (2 tan. )* (396). ❤ 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. It will be shown in a subsequent portion of this work the last article, we shall feet of the rafters, and expressions (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, and proceeding exactly as in obtain for the pressure P upon the its inclination to the vertical, the PL (24,7 sec. 1+,)2+(u, sec. 1+5) cot. . . . . (397). * 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 par allel fibres of the wood to one another (by which arrangement the friction will be greatly increased), tan. =48 (see p. 133.); whence it follows that the rafters will not slip, provided that their inclination exceed cot.-196, or 46° 10'. 304. The stability of a wall sustaining the thrust of a roof, having no tie-beam. Let it be observed, that in the equation to the line of resistance of a wall (equation 377), the terms P sin. a 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. 4, 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, the weight of a cubic foot of the wall, and, the weight of each square foot of the rooting. 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 a 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 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 t, which determines such a minimum value of a, is that inclination of the rafters which is |