Adopting the theory of Coulomb*, M. Garidel has arrived at an equationt which becomes identical with equation (472) in respect to that particular case of the more general conditions embraced by that equation, in which =1 and 9=0.

By an ingenious method of approximation, for the details of which the reader is referred to his work, M. Garidel has determined the values of the angle of rupture Y, and the P

quantity, in respect to a series of different values of a and

B.

6. The results are contained in the tables which will be found at the end of this volume.

P 2,2

The value of being known from the tables, and the

values of Y and Ye from eanations (464), (465), the line of resistance is determined by the substitution of these values in equation (454). The line of resistance determines the point of intersection of the resultant pressure with the summit of pier; the vertical and horizontal components of this resultant pressure are moreover known, the former being the weight of the semi-arch, and the other the horizontal thrust on the key. All the elements necessary to the determination of the stability of the piers (Arts. 289 and 312) are therefore known.

It will be observed that the amount of the horizontal thrust for each foot of the width of the soffit is determined by multiplying the value of

shown by the tables, by the

square of the radius of the intrados in feet, and by the weight of a cubic foot of the material

*See Mr. Hann's Theory of Bridges, Art. 16.; also p. 24. of the Memoir on the Arch by the author of this work, contained in the same volume.

Tables des Poussées des Voutes, p. 44. Paris, 1837. Bachelier.

NOTE 1. PART IV.

The length of an elementary arc ds of the intrados AS subtending the angle de is expressed by rde; an elementary volume of the arch will therefore be expressed by rdedr; the perpendicular distance of the centre of gravity of this volume from the vertical line CE is r sin. ; the moment of this volume, with regard to CE, is therefore rd@dr×r sin.0=r'dr sin. Ode; then from (Art. 81.) equation (20) there obtains

NOTE 2. PART IV.-General integrals of equations 464, 465.

The general integral, (equation 464)

S1+3—cos. (—) sec. }} cos. Odo

=(1+8) cos. Odb

Ssec. 1 (cos. O cos. ¿+sin. O sin. () cos. Odê=S(1+3) cos. Udo —

9-1

sec. L

..ƒ { 1+3—cos. (0—1) sec. 1 } cos. Odê=(1+,3) sin. 6—

(sin. 20 cos. -sin. cos. 20).

The general integral, S{(1+8)—sec. ¿ cos. (0—¿)} sin. O cos. Od8, (equation

L sin. cos. d. 0=

cos. 20d cos. 0+/tan. sin. 20d sin. 0=

0+/tan.

-Scos. "Od cos. 0+

- cos. '64 sin. '8

—sec. ¿ cos. (0—1 ) } sin. & cos. 6d. +de=−1(1+,3) cos. "8+

In equation (427), (Art. 319), by making =0, we obtain P1 x2; since

π

tan. =1, and this answers to the case of the horizontal pressure of a perfect fluid like water. From this expression there obtains dP-u,xdx, to express the elementary pressure at any depth z below the surface. This depth in (Art. 341), equation (473), is TV=AD+AB=AD+AC¬BC=ßR+R-R cos., ..dP=uR(1+3—cos. 6) R1 (1+ß-cos. 6)=uR {1+3-cos. } sin. Odo

.. P=X=μR3 {1+3—cos. 0 } sin. Odo.
= x = n {

PART V.

THE STRENGTH OF MATERIALS.

ELASTICITY.

345. From numerous experiments which have been made upon the elongation, flexure, and torsion of solid bodies under the action of given pressures, it appears that the displacement of their particles is subject to the following laws.

1st. That when this displacement does not extend beyond a certain distance, each particle tends to return to the place which it before occupied in the mass, with a force exactly proportional to the distance through which it has been displaced.

2dly. That if this displacement be carried beyond a certain distance, the particle remains passively in the new position which it has been made to take up, or passes finally into some other position different from that from which it was originally moved.

The effect of the first of these laws, when exhibited in the joint tendency of the particles which compose any finite mass to return to any position in respect to the rest of the mass, or in respect to one another, from which they have been displaced, is called elasticity. There is every reason to believe that it exists in all bodies within the limits, more or less extensive, which are imposed by the second law stated above.

The force with which each separate particle of a body tends to return to the position, from which it has been displaced varying as the displacement, it follows that the force with which any aggregation of such particles, constituting a finite portion of the body, when extended or compressed within the limits of elasticity, tends to recover its form, that is the force necessary to keep it extended or

458

compressed, is proportional to the amount of the extension or compression; so that each equal increment of the extending or compressing force produces an equal increment of its extension or compression. This law, which constitutes perfect elasticity, and which obtains in respect to fluid and gaseous bodies as well as solids, appears first to have been established by the direct experiments of S. Gravesande on the elongation of thin wires.*

It is, however, by its influence on the conditions of deflexion and torsion that it is most easily recognized as characterizing the elasticity of matter, under all its solid forms, within certain limits of the displacement of its particles or elements, called its elastic limits.

ELONGATION.

346. To determine the elongation or compression of a bar of a given section under a given strain.

Let K be taken to represent the section of the bar in square inches, L its length in feet, its elongation or compression in feet under a strain of P pounds, and E the strain or thrust in pounds which would be required to extend a bar of the same material to double its length, or to compress

*For a description of the apparatus of S. Gravesande, see Illustrations of Mechanics, by the Author of this work, 2d edition, p. 30. In one of his experiments, Mr. Barlow subjected a bar of wrought iron, one square inch in section, to strains increasing successively from four to nine tons, and found the elongations corresponding to the successive additional strains, each of one ton, to be, in millionths of the whole length of the bar, 120, 110, 120, 120, 120. In a second experiment, made with a bar two square inches in section, under strains increasing from 10 tons to 30 tons, he found the additional elongations, produced by successive additional strains, each of two tons, to be, in millionths of the whole length, 110, 110, 110, 110, 100, 100, 100, 100, 95, 90. From an extensive series of similar results, obtained from iron of different qualities, he deduced the conclusion that a bar of iron of mean quality might be assume to elongate by 100 millionth parts, or the 10,000th part, of its whole length, under every additional ton strain per square inch of its section. (Report to Directors of London and Birmingham Railway. Fellowes, 1835.)

The French engineers of the Pont des Invalides assigned 82 millionth parts to this elongation, their experiments having probably been made upon iron of inferior quality. M. Vicat has assigned 91 millionth parts to the elongation of cables of iron wire (No. 18.) under the same circumstances, MM. Minard and Desormes, 1,176 millionth parts to the elongation of bars of oak. (Illust. Mech., p. 393.)

The experiments of Prof. Robison on torsion show the existence of this property in substances where it might little be expected; in pipe-clay, for instance.