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proportional to the surface of that joint. For, pressure on aA to pressure on bb, as Pa to Pb, that is, as aa to be ; and so throughout. The pressures being exerted perpendicularly 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 aa, as upon every square inch of face in direction bb, in direction co, &c. to the extreme abutments
RT, RT'. 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 trand Kk; or, in other words, the semi-vault cannot be sustained by means of the two surfaces TR, and kk alone.
Let Rk = k1 = h, Kk = k, and ts = 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 KkRT from the middle, Kk, of the key-stone will be
less than 7-5192, or than 74 times the thickness, Kk, 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. ore, to investigate the conditions of equilibrium V. for both these classes of dislocations. G;
121. Prop. Suppose f K that the arch off 'F' tend HT1 JX to fall vertically in one mass, by thrusting out the piers at the joints of frac. 4. ture, of, of'; it is requir. G to investigate the equa- : tions by which the equili- } | .
- ermin- I rl i * may be determia of EB'A'
Let 2A denote the whole weight of the arch lying be. tween of, and Ff", G 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 of, and 6' the place of its centre of gravity.
Now, Fv, Fv', being respectively perpendicular to Fs, Ff, the weight 2A may be understood to act from v, in the directions vP, vr', and pressing upon the two joints of, Ff". The horizontal thrust which it exerts on F, will be = a tan. Fv1 = A
ci - cot. FC1 = A . of and at the same time the vertical effort
will = A. 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, catteris paribus, the friction varies as the pressure, that is, here, as the weight, we shall have for the resisting force, f. A + f. p. Equating this with the
- ci - above expression, A. F we obtain for the first equation of equilibrium
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
the energy, A. in re. This is counteracted by the ver.
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, G', of the pier. Hence we have
A. - . FE = A. AE + p . An ;
FI ... . and, after a little redaction, there results for the second equation of equilibrium :
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,
Referring the weight, A, of the semi-arch from its centre of gravity to the direction of the vertical joint kK, its energy
is represented by A . EI and the resulting horizontal
thrust at A is, evidently, A. ft. * = A. ". The vertical
Fi ki 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
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 semiarch A at the point F, with the horizontal thrust before-mentioned, acting at the arm of lever EF. Thus we shall have
* - or = AUT - or ) . . . . (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 (i.) and (II.) 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
lost of materials, will be treated in a subsequent part of this votime,
124. THAT department of mechanics which relates to the circumstances and effects of bodies in motion (art. 5.) is of great extent, and of very comprehensive application. A selection of its most interesting topics will here be presented; but numerous other problems which, while they fall within its scope, require the aid of the fluxional analysis, will be solved in the collections in a subsequent part of this volume.
GENERAL LAWS OF MOTION, &c.
125. PRop. The quantity of matter, in all bodies, is in the compound ratio of their magnitudes and densities.
That is, b is as ma; where b denotes the body or quantity of matter, m its magnitude, and d its density.
For, by art. 10, in bodies of equal magnitude, the mass or quantity of matter is as the density. But, the densities remaining, the mass is as the magnitude ; that is, a double magnitude contains a double quantity of matter, a triple magnitude a triple quantity, and so on. Therefore the mass is in the compound ratio of the magnitude and density.
Corol. 1. In similar bodies, the masses are as the densities and cubes of the diameters, or of any like linear dimensions. —For the magnitudes of bodies are as the cubes of the diameters, &c.
Corol. 2. The masses are as the magnitudes and specific gravities.—For, by art. 10 and 17, the densities of bodies are as the specific gravities.
126. Scholium. Hence, if b denote any body, or the quantity of matter in it, m its magnitude, d its density, g its specific gravity, and a its diameter or other dimension; then, o: (pronounced or named as) being the mark for general proportion, from this proposition and its corollaries we have these general proportions:
b O. ma O. mg of a'd,