which expression represents the actual pressure of the earth on a surface AX of the wall, whose width is one foot and its depth x.

REVETEMENT WALLS.

320. If, instead of a revetement wall sustaining the pressure of a mass of earth, the weight of each cubic foot of which is represented by 1, it had sustained the pressure of a fluid, the weight of each cubic foot of which was re

so that the pressure of a mass of

earth upon a revetement wall (equation 427), when its surface is horizontal (and when its horizontal surface extends, as shown in the figure, to the very surface of the wall), is identical with that of an imaginary fluid whose specific gravity is such as to cause each cubic foot of it to have a weight 2, represented in pounds by the formula

Substituting this value for μ, in equations (416) and (419), we determine therefore, at once, the lines of resistance in revetement walls of uniform and variable thickness, under the conditions supposed, to be respectively

where σ represents the ratio of the specific gravity of the material of the wall to that of the earth. The conditions of the equilibrium of the revetement wall may be determined from the equation to its line of resistance, as explained in the case of the ordinary wall.

may not

321. The conditions necessary that a revetement wall be overthrown by the slipping of the stones of any course upon those of the subjacent course.

These are evidently determined from the inequality (420) by substituting μ2 (equation 428) for μ, in that inequality; we thus obtain, representing the limiting angle of resistance of the stones composing the wall by 9, to distinguish it from that of the earth,

P

1

(x-e)3

42/2ax + x2 tan. a

(431);

where represents the ratio of the specific gravity of the material of the wall to that of the earth.

As before, it may be shown from this expression that the tendency of the courses to slip upon one another is greater in the lower courses than the higher.

322. The pressure of earth whose surface is inclined to the

P

horizon.

G G

H

Let AB represent the surface of such a mass of earth, YX the plane along which the rupture of the mass in contact with the surface AX of a revetement wall tends to take place, AX=x, B AXY=1, XAB=3.

Then if W be taken to represent the weight of the mass AXY, it may be shown, as in Art. 319., equation (421), that PW cot.(+9).

x sin.

Now W=AX. AY. sin. ß, AY=sin. (+B);

x2 sin. sin. ẞ
1

fore Wsin. (+6)

x2

= cot. +cot. B

; there

Now the value of in this function is that which renders it a maximum (Art. 319.). Expanding cot. (+9), and differentiating in respect to tan. 1, this value of is readily determined to be that which satisfies the equation

cot.tan. + sec. √1+cot. ß cot. Substituting in equation (432), and reducing,

.....

(433).

(434).

From which equation it is apparent, that the pressure of the earth is, in this case, identical with that of a fluid, of such a density that the weight 3, of each cubic foot of it, is represented by the formula

The conditions of the equilibrium of a revetement wall sustaining the pressure of such a mass of earth are therefore determined by the same conditions as those of the river wall (Arts. 313. and 316.).

323. THE RESISTANCE OF EARTH.

Let the wall BDEF be supported by the resistance of a mass of earth upon its surface AD, a pressure P, applied to its opposite face, tending to overthrow it. Let the surface AH of the earth be horizontal; and let Q represent the pres

sure which, being applied to AX, would just be sufficient to

P

2

E

cause the mass of earth in contact with that portion of the wall to yield; the prism AXY slipping backwards upon the surface XY. Adopting the same notation as in Art. 319., and proceeding in the same manner, but observing that RS is to be measured here on the opposite side of TS (Art. 241.), since

the mass of earth is supposed to be upon the point of slipping upwards instead of downwards, we shall obtain

Now it is evident that XY is that plane along which rupture may be made to take place by the least value of Q: in the above expression is therefore that angle which gives to that expression its minimum value. Hence, observing that equation (436) differs from equation (422) only in the sign of , and that the second differential (equation 426) is rendered essentially positive by changing the sign of p, it is apparent (equation 427) that the value of Q necessary to overcome the pressure of the earth upon AX is represented by

324. It is evident that a fluid would oppose the same resistance to the overthrow of the wall as the resistance of the earth does, provided that the weight, of each cubic foot of the fluid were such that

so that the point in AX at which the pressure Q may be conceived to be applied, is situated at 3ds the distance AX.

325. The stability of a wall of uniform thickness which a given pressure P tends to overthrow, and which is sustained by the resistance of earth.

Let y be the point in which any section XZ of the wall

would be intersected by the resultant of the pressures upon the wall above that section, if the whole resistance Q, which the earth in contact with AX is capable of supplying, were called into action. Let BX =x, Xy=y, BA=e, BE=a, Bp=k, weight of cubic feet of material of wall=μ, inclination of P to vertical =0. Taking the moments about the point y of the pressures applied to BXZE, we have, by the principle of the equality of moments, observing that XQ=3(x-e), and that the perpendicular from Y upon P is represented by x sin. 0-(k-y) cos. 0,

Px sin. -(ky) cos. }=(x-e)Q+(a−y) pax;

or substituting for Q its value (equation 437), and solving in respect to y,

y—¿a‚(x—e)3+ ¿μa2x—P(x sin. ◊ — k cos. ◊)

y=

P cos. +μax

(439).

Now it is evident that the wall will not be overthrown upon any section XZ, so long as the greatest resistance Q, which the superincumbent earth is capable of supplying, is sufficient to cause the resultant pressure upon EX to intersect that section, or so long as y in the above equation has a positive value; moreover, that the stability of the wall is determined by the minimum value of y in respect to x in that equation, and the greatest height to which the wall can be built, so as to stand, by that value of x which makes y=0.