STABILITY AGAINST OVERTURNING. The horizontal pres sure of the water tends to tip the wall forward about the front of any joint, and is resisted by the moment of the weight of the wall. For the present, it will be assumed that the wall rests upon a rigid base, and therefore can fail only by overturning as a whole.
The conditions necessary for stability against overturning can be completely determined either by considering the moments of the several forces, or by the principle of resolution of forces. In the following discussion the conditions will be first determined by moments, and afterward by resolution of forces.
Overturning Moment. The pres sure of the water is perpendicular to the pressed surface. If the water presses against an inclined face, then the pressure makes the same angle with the horizontal that the surface does with the vertical. Since there is a little difficulty in finding the arm of this force, it is more convenient to. deal with the horizontal and vertical components of the pressure.
The overturning effect of the pressure of the water is equal to the moment of the horizontal component minus the moment of the vertical component.
The horizontal component can be found by equation 1, page 460. The arm of this force is equal to } h, and hence the moment tending to overturn the wall is equal to which, for convenience, represent by The amount of the vertical pressure against the up-stream face is given by equation 4, page 460. It acts vertically between I and B, Fig. 96, page 460, at a distance from B equal to IB; and its arm is 1— i h b'. Therefore, the moment of the vertical pressure on the inclined face is which, for convenience, represent by Of course, if the pressed face is vertical, M, will be equal to zero.
The net overturning moment is the sum of equations 7 and 8, or The moment of the pressure of the water, M,— M„ can be determined directly by considering the pressure of the water as acting perpendicular to IB at IB from B. The arm of this force is a line from A perpendicular to the line of action of the pressure. If the
cross section were known, it would be an easy matter to measure this arm on a diagram; but, in designing a dam, it is necessary to know the conditions requisite for stability before the cross section can be determined, and hence the above method of solution is the better.
The center of gravity can be found algebraically or graphically. There are several ways in each case, but the follow ing graphical solution is the simplest. In Fig. 97, draw the diagonals DB and AE, and lay off AJ = EL; then draw DJ, and mark the middle of it, Q. The center of gravity, 0, of the area ABED is at a distance from Q towards B equal to I QB. This method is applicable to any four sided figure.
The position of the center of gravity can also be found algebraically by the principle that the moment of the entire mass about any point, as A, is equal to the moment of the part ADK plus the moment of the portion DEFK plus the moment of the part EBF,—all about the same point, A.
The arm of the weight of the dam is Ag (_ ), and therefore the moment of the weight is which, for convenience, represent by M,.