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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. a and P sin. a their values L sec. and L consec. ; we shall thus obtain

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μ‚L (‡h, cosec. 1—1 sec. 1) =‡o (a ̧,'h ̧+2a,a‚h+—a,”h‚)—w

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From which equation the thickness a, 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 substi tuting 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 shed roof.

Let AB represent one of the rafters of such a roof, one ex

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tremity 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.

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 direc tion 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 resist

ance.

Resolving R vertically and horizontally, it becomes R sin. and R cos. p. 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, (Art. 10.), R sin. +P cos. a=μ, 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).
PL cos. (a+)sec.

P cos. (a+1)=L, 7. tions, we obtain

L. Lu, 7 sec. 4, and hence

Eliminating between these equa

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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 suface 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 hori

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zon is therefore represented by 1+.

2

Substituting this angle for in equations (401) and (402),

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Substituting in equations (377) and (379) for Psin. a, P cos. α, their values determined above, all the conditions of the stability of a wall supporting such a roof will be determined.

307. THE PLATE BANDE OR STRAIGHT ARCH.

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A Q ME

B

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 0. Let OAD=a, AM=x, MR=y, AD=H, AB=2L, weight of cubic foot of material of arch,. Draw Rm a perpendicular upon PA produced; then by the principle of the equality of moments,

Rm. P=MQ. (weight of DM).

But Rmx cos. ay sin. a, MQ=r, weight of DM = H; also resolving P vertically,

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Whence we obtain, by substitution in the preceding equation, and reduction,

L(x-y tan. a)=x2. . . . . (406),

which is the equation to the line of resistance, showing it to be a parabola. If, in this equation, L be substituted for x, and the corresponding value of y be represented by Y, there will be obtained the equation Y tan. a=L, whence it appears that a is less as Y is greater; but by equation (405), Pis less as a is less. P, therefore, is less as Y is greater;. but Y can never exceed H, since the line of resistance cannot intersect the extrados. The least value of P, consistent with the stability of the plate bande, is therefore that by which Y is made equal to H, and the line of resistance made to touch the upper surface of the plate bande in F.

Now this least value of P is, by the principle of least resistance (see Theory of the Arch), the actual value of the resistance at A,

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Eliminating a between equations (405) and (407),

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Multiplying equations (405) and (407) together,

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Now P sin. a represents the horizontal thrust on the point of support A. From this equation it appears, therefore, that the horizontal thrust upon the abutments of a straight arch is wholly independent of the depth H of the arch, and that it varies as the square of the length L of the arch; so that the stability of the abutments of such an arch is not at all diminished, but, on the contrary, increased, by increasing the depth of the arch. This increase of the stability of the abutment being the necessary result of an increase of the vertical pressure on the points of support, accompanied by no increase of the horizontal thrust upon them.

308. The loaded plate bande.

It is evident that the effect of a loading, distributed

uniformly over the extrados of the plate bande, upon its stability, is in every respect the same as would be produced if the load were removed, and the weight of the material of the bande increased so as to leave the entire weight of the structure unchanged. Let μ, represent the weight of each cubic foot when thus increased, the weight of each

cubic foot of the load, and H, the height of the load; then μ2HL=μ‚HL+~2H ̧L,

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The conditions of the stability of the loaded plate bande are determined by the substitution of this value of , for μ, in the preceding article.

309. Conditions necessary that the voussoirs of a plate bande may not slip upon one another.

It is evident that the inclination of every other resultant pressure to the perpendicular to the surface of its corresponding joint, is less than the inclination of the resultant

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pressure or resistance P, to the perpendicular to the joint AD. If, therefore, the inclination he not greater than this limiting angle of resistance, then will every other corresponding inclination be less than it, and no voussoir will therefore slip upon the surface of its adjacent voussoir. Now the tangent of the inclination P to the perpendicular to AD is represented by cot. a

or by

2H

L

(equation 407); the required condition is therefore

determined by the inequality,

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