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arch, which, since gh is given, varies as the sine of Hgk, or hhk: wherefore, the force impelling the voussoir is as the square of the sine of hik.
3dly, The wedge impelled in a direction perpendicular to the curve endeavours to split the arch, and therefore to move one segment about the fulcrume, the other about the fulcrum f. Hence the force of the voussoir acting on the levers Hf, He, being as either of the perpendiculars fr, eq, is as the sine of the angle for or hik.
We have supposed the centre of curvature of the arches at the points A, a, h, H, to be at c : but this is merely to prevent the figure from being too complex, and makes no alteration in the nature of the demonstration.
Corol. Hence, if the height of the wall incumbent on any point H of the intrados is inversely as the cube of the sine of hhk into radius of curvature at that point, or directly as cube of the secant of the angle formed by him and the horizon, and inversely as the radius of curvature, all the voussoirs will endeavour to split the arch with equal forces, and will be in perfect equilibrium with each other.
The general expression, therefore, for the thickness GH over any point of an arch, is
where r = rad. of curvature at the vertex
The radii of curvature for the different curves are determinable by the method of fluxions, or by other means: they are here supposed known.
1. Suppose, for example, it were required to find the requisite thickness over any point of a circular arc, to ensure equilibration, the thickness a = DK, at the crown of the arch being given.
Here, rad. of curv. at H IC = rad. of curv. at D t that is R = r. and sec. Rht = sec. arc. DH.
ar Conseq. GH = sec.” DH X -- = sec.” DH . a. q r
Hence we have a convenient logarithmic expression for computation; viz. Log. DK -- 3 log. sec. DH = log. GH.
In this example, the curve of equilibration, GKs, runs up to an infinite height over B, the springing of a semicircular arch. But over a portion of 30° or 35° on each side the vertex, as DH, the curve ke of the extrados accords very well with what would be required for a roadway.
Ex. 2. Determine the requisite thickness for equilibration over any point of a parabola. T
3. _ (p +4+)* 2Vp = *-ij-, p. ii p (p + 4r) So that the extrados is a parabola equal to the intrados, and every where vertically equidistant from it.
Ex. 3. To determine the requisite thickness over any point of a cycloidal arch.
Here, putting DK = a, DR=r, Dc-d; we have, from the known properties of the cycloid, the tangent Ht parallel to the corresponding chord sD, , or angle THR = Z dsR, so = V/dr, sR = V/da — a ";
By computing the value of GH for several corresponding values of DR, and co, and thence constructing the extrados by points, it will, as in the figure, appear analogous to that for the circle, but rather flatter till it approach the extremities of the arch, where the curve runs off to infinity, as in the case for the circle. o * ,
ExAM. 4. To determine the requisite thickness over any point of an elliptical arch. Here, taking r, y, and a, as before, take AC = t, DC = c, HQ = or, being perpendicular to the tangent HT. Then, by the property of the ellipse,
- iro Radius of curvature at H = R = #. p being the parame
2:2 ter to CD = "c
_ 4oc" o'c". d d _ to ... R = H----, an r (rad. curv. at D) – F.
= ——a = (c-a')" CR” as before, a convenient expression for logarithmic operation. Here, again, computing values of GH for several assumed values of co, the curve of the extrados may thence be constructed, and, like that for the cycloid, it will be sound rather flatter than that for the circle, but still analogous to it. ExAM. 5. For the Catenary. (See the fig. to Exam. 2.) Here, put DR = r, GR = y, DG = z, t = tension at the vertex D when the chain hangs from A and B. Then, by
Rad. curv. at G = —- = n, and therefore at D where z
Corol. If a = t, or the thickness at the crown equal to a line whose weight expresses the tension,
then GH = a + z = kD + DR. Corol. 2. If a * t, the exterior curve will proceed
downwards Corol. 3. If DK, the thickness at the crown, be very small
compared with t, then will the thickness over H be nearly
: both ways from K.
118. Prop. To point out the construction, and investigate the chief properties of the plat-band, or “flat arch,” as it is sometimes called.
the proposed thickness S of a plat-band. As- Rof do B | sume a point P in the inferior prolongation NM of kk the middle of the structure ; and, supposing aa, ab, be, cd, &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 abutting faces of the several wedges are to be cut, so that the whole shall be an equilibrated structure. woNow, 1st, If ak = ak, be taken to represent half the weight of the central wedge, then pk 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 nearer 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, be, 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 trape
zoids abn A, bech, &c. which are as ab + AB to be + BC, or
as ab to be ; the common height of all the trapezoids being
equal to kic. o 4thly, The pressure on each joint of the plat-band is Vol. II. 25