of the resistance of the wall necessary to sustain the pressure the wedge-shaped mass of earth AXY, being dependent for its amount upon the value of (so that different sections, such as XY, being taken, each different mass cut off by such section will require a different resistance of the wall to support it), may admit of a maximum value in respect to that variable. And if the wall be made strong enough to supply a resistance sufficient to support that wedge-shaped mass of earth whose inclination corresponds to the maximum value of P, and which thus requires the greatest resistance to support it; then will the earth evidently be prevented by it from slipping at any inclination whatever, for it will evidently not slip at that angle, the resistance necessary to support it at that angle being supplied; and it will not slip at any other angle, because more than the resistance necessary to prevent it slipping at any other angle is supplied. If, then, the wall supplies a resistance equal to the maximum value of P in respect to the variable, it will not be overthrown by the pressure of the earth on AX. Moreover, if it supply any less resistance, it will be overthrown; there not being a sufficient resistance supplied by it to prevent the earth from slipping at that inclination which corresponds to the maximum value of P. To determine the actual pressure of the earth on AX, we have then only to determine the maximum value of P in respect to . This maximum value is that which satisfies the conditions But differentiating equation (422) in respect to , we obtain by reduction dP di -=tu 28in. 2(1+)-sin. 2 cos. sin. (+0) (423)+ Let the numerator and denominator of the fraction in the The existence of this maximum will subsequently be shown: it is, how ever, sufficiently evident, that, as the angle is greater, the wedge-shaped mass to be supported is heavier; for which cause, if it operated alone, P would become greater as increased. But as increases, the plane XY becomes less inclined; for which cause, if it operated alone, P would become less as in. creased. These two causes thus operating to counteract one another, deter. mine a certain inclination in respect to which their neutralising influence is the least, and P therefore the greatest. Church's Diff. and Int. Cal., Art. 41. it follows, by substitution, that for every value of by which the first condition of a maximum is satisfied, the second dif ferential co-efficient becomes dᏢ Now it is evident from equation (423) that the condition =0 is satisfied by that value of which makes 2(1+0)= T-21, or And if this value be substituted for in equation (424), it which expression is essentially negative, so that the second condition is also satisfied by this value of . It is that, therefore, which corresponds to the maximum value of P; and substituting in equation (422), and reducing, we obtain for this maximum value of P the expression 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 pres DA pressure of a mass of so that the 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 gra vity is such as to cause each cubic foot of it to have a weight , represented in pounds by the formula Substituting this value for , 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 a 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. Hydrostatics, Art. 31. 321. The conditions necessary that a revetement wall may not 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 ", (equation 428) for , 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, 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 horizon. Let AB represent the surface of such a mass of earth, YX the plane along which the H rupture of the mass in Now the value of in this function is that which renders it a maximum (Art. 319). Expanding cot. (+), and dif ferentiating in respect to tan. 4, this value of is readily determined to be that which satisfies the equation cot.tan. sec. p 1/1+cot. B cot. Ф Substituting in equation (432), and reducing, ... (433). 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 ,, 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 pressure which, being applied to AX, would just be sufficient to 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 Z D Qatan. cot. (-)..... (436). |