If the actual cross section of the dam is known, or if a cross section of the proposed dam be assumed, the stability against overturning may be determined graphically by either of the two following In Fig. 98, Q is the center of pressure of the water on the back of the wall. QB = IB. The point C is the center of gravity of the section — found as described in $ 935; and m is the middle of the base AB. H is the horizontal compo nent of the water pressure, and V the vertical component. W is the weight of a section of the dam a unit long. By moments, it is found that the resultant of V and W pierces the base AB in the point g; and by the triangle of forces it is found that the resultant of H and W + V pierces the base A B at r. As long as r lies within the base, the dam will not overturn.
The stability may also be determined by using the normal pressure F without re solving it into its components. Through Q, Fig. 99, draw a line, Qa, perpendicular to EB; through c, the center of gravity of the cross section, draw a vertical line ca. To any convenient scale lay off ab equal to the total pressure of the water against IB, and to the same scale make of equal to the weight of a unit section of the wall. Complete the parallelogram abet. The diagonal ae intersects the base of the wall at r; and as long as the center of pressure r lies between A and B, the wall will not overturn.
1. If the factor of safety against overturning be defined as the ratio of the resisting moment to the overturning moment, then in Fig. 98, for moments about A This value of the factor will not agree with that from equation 12, since in the former the vertical component of the pressure is in cluded in the overturning force while in the latter it is considered as a resisting force.
2. If the factor of safety be defined as the ratio of the force that would just overturn the dam to the force tending to overturn it, then from Fig. 99 the factor of safety is F' _ F = ab' ab; or This value of the factor agrees with that found by equation 12'; but does not agree with the value found by equation 12.
3. In the above methods of determining the factor of safety, no special account is taken of the fact that, owing to the unsym metrical cross section of the dam, the point in which the vertical through the center of gravity of the dam pierces the base, g, is on the right-hand side of m, the middle of the base; and consequently when there is no water pressure against the dam, there is a ten dency to overturn to the right instead of to the left, for any eccen tricity of pressure upon the foundation shows a tendency to over turn. Therefore the factor of safety found as above counts, as it were, from the initial condition of the dam. In the following
method it counts from what may be called the neutral condition of the dam.
If the factor of safety be defined as the ratio of the moment that would just overturn the dam about its toe, A, Fig. 98, to the actual moment tending to overturn it, then the value of the factor may be found as follows: Let H, = that portion of H which will cause the resultant to pierce the base at m, its middle point; and let H, = H — H,. Conceiving only H, as acting, there is no ten dency to overturn about A; and hence the resultant of the vertical forces may be considered as acting through m. If now H, be con ceived as coming into action, the resultant of (W + V) and H, must pierce AB at r, and H,. y = (W + V) rm. Then, according to the above definition, In the ordinary meaning of the term, equation 13 does not give the true factor of safety, although the result may under some con ditions approximate the true factor. This value will be called the approximate factor of safety. Equation 13 is strictly correct (1) when the resultant of the forces normal to the base pierces the base at its center, i.e., when the vertical cross-section is symmetrical and the external forces are horizontal or the vertical component of the external force is disregarded, since then it gives f = cc , as it should; and (2) when the resultant of all the forces passes through A, since then it gives f = 1, as it should.
Frequently equation 13 is more convenient than equation 12, 12', or 12", since the point r must always be determined to find the crushing stress and since the point m is very easily found, while a special construction is required for equations 12, 12', and 12".
The approximate value of the factor of safety, i.e., the value given by equation 13, is much used in discussions of the stability of dams, retaining walls, and arches. For example, a very com mon statement in considering the stability of such structures is: "If the center of pressure lies within the middle third of any sec tion, the factor of safety against overturning is at least 3." This statement assumes that equation 13 gives the true factor of safety against overturning, and that therefore if the center of pressure is within the middle third of any section, rm is equal to or less than ir 1; and hence, as Am = 1, equation 13 gives f = 3 or more. For dams and retaining walls, particularly the former, equation 13 frequently gives 3 for a factor of safety, when the true value is approximately 2; and hence the approximate formula should not be used for these structures. The approximate factor of safety is universally employed in discussions of the stability of arches in which the stresses are found by the thrust theory (the older and more common theory); but the formula is usually more accurate for arches than for dams and retaining walls, and besides the theory of the arch itself is not mathematically exact.