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a heavy flexible cord or chain, left to adjust itself into a hanging catenary, and inverted, would support itself upon props perpendicular to the tangents at a and B.

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when a vanishes becomes = x, or nR = GH = a.

118. PROP. To point out the construction, and investigate the chief properties of the plat-band, or "flat arch," as it is sometimes called.

Let RR' be the proposed width, and кk the proposed thickness of a plat-band. As. sume a point P in the inferior prolongation of Kk the middle of the structure; and, sup. posing aa', ab, bc, cd,

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&c. the proposed thicknesses at bottom, of the truncated wedges of which the plat-band is to be constituted, let straight lines pa'a', paa, pbb, PCC, &c. be drawn, they will respectively show the directions in which the mutually abut. ting faces of the several wedges are to be cut, so that the whole shall be an equilibrated structure.

=

Now, 1st, If ak ak, be taken to represent half the weight of the central wedge, then rk perpendicular to it will represent the horizontal thrust throughout the plat-band, and consequently, the thrust, shoot, or drift, acting at R

or R'.

2dly, Therefore, by assuming P neårer or farther from RR', the thrust may be diminished or increased at pleasure.

3dly, No one of the wedges has a greater tendency to fall downwards than another; for those tendencies are throughout as their weights, each being represented by the successive lines ab, bc, cd, &c. on both sides the key-stone. The former are as the differences of the tangents ka, kb, kc, &c. to the radius pk; and the latter are as the areas of the trapezoids abвA, bсcв, &c. which are as ab + AB to be + BC, or as ab to be; the common height of all the trapezoids being equal to kи.

4thly, The pressure on each joint of the plat-band is VOL. II.

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proportional to the surface of that joint. For, pressure on as to pressure on bв, as Pa to rb, that is, as as to bв; and so throughout. The pressures being exerted perpendicu. larly to the respective surfaces, are evidently measured by lines in the directions of those surfaces (art. 32.) when we have assumed a horizontal line for the measure of gravity, and a vertical line to measure the horizontal thrust.

5thly, Hence also it follows that in this construction the pressure upon each square inch of joint, is a constant quantity throughout; being the same upon every square inch of the face in direction an, as upon every square inch of face in direction bâ, in direction cc, &c. to the extreme abutments RT, R'T'.

These properties will not be found co-existent in any other equilibrated structure.

119. Scholium. Yet this construction has a limitation which it is highly important to observe. To ensure stability, the distance of the centre of gravity of the semi-vault from the vertical PK, must exceed KV, the distance from the same vertical to the intersection of RV (a perpendicular to the abutment TR) with the top TT of the plat-band. Unless this condition be fulfilled, perpendiculars cannot be let fall from the centre of gravity upon both TR and кk; or, in other words, the semi-vault cannot be sustained by means of the two surfaces TR, and κk alone.

Let Rk kuh, kk = k, and rs =t, being the tangent of the ulterior angle of slope to the radius RS = k. Then the distance of the centre of gravity of the semi-vault кkRT from the middle, кk, of the key-stone will be

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Or, taking the limit of tottering equilibrium, we have

2. -3(h-k2)t+6hk2=0:

from which when two of the three letters are known the third may be found.

Suppose, for example, that a plat-band were constructed upon an equilateral triangle, or such that angle RPR' = 60°. Then Tst tan. 300 to rad. kk. Or, if кk = taken = 1, then ttan. 30° = = }√3.

Hence 3292900 — (h3-1)/3 + 6h = 0.

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k, be

From this equation h, in the case of the limit is found = {√ 37 + √ 3 = 3·7596.

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Consequently, in the proposed case, RR = 2h must be less than 7.5192, or than 74 times the thickness, кk, of the key-stone.

Of the Equilibrium of Vaults, regarding the Tenacity of

Cements.

120. When the operation of cements is taken into the consideration, the conditions to ensure equilibrium are more easily investigated than when the gravitating tendency of the superincumbent matter is alone regarded. If the cohesive energy of the cement were insuperable, the arch might then be considered as one mass, which would be every where secure, whatever its form might be, provided the piers or abutments were sufficiently strong to resist the horizontal thrust. And, although this property cannot safely be imputed to any cement (strong as many cements are known to be), yet, in a structure, whose component parts are united with a very powerful cement, the matter above an arch will not yield, as when the whole is formed of simple wedges, or as when it would give way in vertical columns, but by the separation of the entire mass into three, or at most, into four pieces that is, either into the two piers, and the whole mass between them, or into the two piers, and the including mass splitting into two at its crown. It may be advisable, there. fore, to investigate the conditions of equilibrium for both these classes of dislocations.

121. PROP. Suppose that the arch Fff'F' tend to fall vertically in one mass, by thrusting out the piers at the joints of frac ture, rf, f'; it is requir to investigate the equations by which the equili brium may be determined.

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Let 2a denote the whole weight of the arch lying be. tween Ff, and Ff', & the centre of gravity of one half of that arch, the centre of gravity of the whole lying on cv; let P be the weight of one of the piers, reckoned as high as rf, and 6' the place of its centre of gravity.

Now, FV, F'v', being respectively perpendicular to Ff, F'f', the weight 2A may be understood to act from v, in the directions VF, VF, and pressing upon the two joints Ff, Ff'. The hori zontal thrust which it exerts on F, will be a tan. FVI = À

CI

cot. Fcl A. ; and at the same time the vertical effort

will = A.

FI

Now, the first of these forces tends to thrust out the solid AF horizontally, an effort which is resisted by friction; and since it is known that, cæteris paribus, the friction varies as the pressure, that is, here, as the weight, we shall have for the resisting force, ƒ. a+ƒ. P. Equating this with the we obtain for the first equation of

above expression, a .

CI

FI

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Moreover, the horizontal thrust that tends to overturn the pier AF about the angle A, must be regarded as acting at the arm of lever FE, and, therefore, as exerting altogether

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the energy, A. FE. This is counteracted by the ver

FI

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tical stress A, operating at the horizontal distance AE, and by the weight P, acting at the distance AD; DG' being the vertical line passing through the centre of gravity, c', of the pier. Hence we have

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and, after a little redaction, there results for the second equation of equilibrium :

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122. PROP. Suppose that each of the two halves kr, kr', of the arch, tend to turn about the vertex k, removing the points F, and F': it is required to investigate the conditions of equilibrium in that case.

EQUILIBRIUM OF ARCHES.

Referring the weight, A, of the semi-arch from its centre of gravity to the direction of the vertical joint kê, its energy FH ; and the resulting horizontal

is represented by A. EI

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thrust at a is, evidently, a . kx

FI

=A.

FH The vertical

kI

stress is = P + A; and therefore the friction is represented by f. P+f. A. Equating this with the above value of the horizontal thrust, that the pier AF may not move horizontally, we have

FH

f.P=A( kı

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(1.)

Then, considering the arch and piers as a polygon capable of moving about the angles A, F, k, F', A', we must, in order to equilibrium, balance the joint action of P and the semi. arch A at the point F, with the horizontal thrust before-menThus we shall have tioned, acting at the arm of lever EF.

P. AD + A . AEA. tion, there results

FH EF: from which, after due reduc

ki

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AE

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EF

EF

(II.)

123. Corol. Hence it will be easy to examine the stability of any arch whose parts are cemented as in the hypotheses of these two propositions. Assume different points such as F, in the arch, for which let the numerical values of the equations (1.) and (11.) be computed. To ensure stability, the first members of those equations, which represent the resistance to motion, must exceed the second members; the weakest points will be those in which the excess of the first above the second member is the least.

If the dimensions of the arch were given, and the thickness of the pier required, the same equations would serve for its determination*.

The principles adopted in the two last propositions are due to De la Hire, and Coulomb, respectively. For a more comprehensive view of this interesting subject, the student may consult Hutton's Tracts, vol. i., the Appendix to Bossut's Mechanics, and Berard's Treatise on the Statics of Vaults and Domes. The pressure of earth, and the strength of materials, will be treated in a subsequent part of this volume.

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