(b) In order to determine the factor of safety, it is necessary to find the amount of thrust. From formula 1 of Art. 26, Then, the factor of safety is equal to the resistance divided by the thrust, or 5,692.5 ÷ 4,500 1.27. Ans. = EXAMPLE 2.-Let the wall in example 1 be built of granite weighing 170 pounds per cubic foot. Required: (a) its resistance to sliding; (b) the factor of safety. SOLUTION. (a) Substituting in formula 2, R = .75 X 8+3 X 12 X 170 = 8,415 lb. Ans. (b) The factor of safety is equal to the resistance divided by the thrust, which was determined above. Then, the factor of safety is equal to 8.415 ÷ 4,500 = 1.87. Ans. 29. Resistance to Overturning.-The resistance to overturning, called the moment of stability of the wall, is the static moment of the wall with reference to the point about which rotation tends to take place, that is, about the toe C, Fig. 12, of the dam; it is equal to the weight of the wall multiplied by the horizontal distance between the toe and a vertical line through the center of gravity of the wall. order that there may be equilibrium, the moment of stability of the dam per foot of length must be equal to the moment M of the horizontal thrust, as found in Art. 26. In FIG. 12 The almost invariable section or profile of a masonry dam, except for very high dams, is a trapezoid, as shown in Fig. 12. The section may be divided into a rectangle DE, whose area is b, H, and a triangle EFC, whose area is (b,b,)H. The static moment of the rectangle about the toe C is The static moment of the triangle about the toe C is, since 26 the center of gravity is horizontally distant from C, 1⁄2 — × 3 For the static moment M2 of the whole area, we have, then, 2 The moment of stability M, of the dam, per foot of length, is w M., denoting by w the weight of the material per cubic foot. Replacing M. by its value, we have, In order that the dam may be secure, M, must be at least equal to M; hence, the least moment of stability that is consistent with equilibrium is given by the formula (see EXAMPLE.-What is the moment of stability of the trapezoidal wall of example 1, Art. 28? 30. Design of the Profile.-In designing masonry dams, the height H is determined by the amount of water to be stored, taken in connection with the nature of the place in which the reservoir is built; the weight w is given by the material of which the dam is built; the top width b, and the factor of safety are assumed. From these data, the width of base b, can be calculated. Consider first the resistance to sliding. If a factor of safety is used, formula 1 of Art. 26, for the horizontal thrust, becomes T 31.25 j H'. = The resistance to sliding, from formula 2 of Art. 28, EXAMPLE 1.-The height of a trapezoidal wall is 30 feet; width at top, 6 feet; weight per cubic foot, 140 pounds; and factor of safety, 2.5. To determine the required bottom width of the wall to resist sliding. SOLUTION. Here, j = 2.5, H = 30, w = 140, and b2 tuting these values in the last equation, = 6; substi 31. To determine the breadth of base of a trapezoidal wall to resist overturning, the moment of stability given by formula 1 of Art. 29 must be equal to the moment of thrust given by formula 2 of Art. 26. Introducing the factor of safety j, formula 2 of Art. 26 becomes Making the moment of thrust equal to the moment of stability, EXAMPLE. The dimensions of a trapezoidal wall and the factor of safety being the same as those given in the example of the preceding article, determine the width of base to resist overturning. SOLUTION.-Substituting the given values in the formula, EXAMPLES FOR PRACTICE 1. The depth of water pressing against a dam is 75 feet. (a) What is the horizontal thrust per foot of length of dam? (b) What is the overturning moment, in foot-pounds, about the outer toe of the dam, per foot of length of dam? Ans. 2. A trapezoidal wall 25 feet high is 4 feet bottom, and weighs 130 pounds per cubic foot. to sliding? {8 f(a) 175,780 lb. (b) 4,395,900 ft.-lb. wide on top, 12 feet at What is its resistance Ans. 19,500 lb. 3. What is the moment of stability of the wall in example 2? Ans. 199,330 ft.-lb. 4. The height of a trapezoidal wall is 50 feet; width at top, 8 feet; weight per cubic foot, 170 pounds; and factor of safety, 2.5. What is the required bottom width to resist: (a) sliding? (b) overturning? Ans. {30.6 ft. (a) 53.3 ft. 32. Remarks on Stability.-In examining the results given by the formulas of Arts. 30 and 31, it is evident that a much greater width of base is required to insure security against sliding than is required to guard against overturning. It must be kept in mind, however, that, as already stated, only the mere friction of stone on stone is taken into account when calculating the resistance to sliding; that is, only such friction as might occur if two level surfaces of stone were brought in contact. When it is remembered that a well-bound piece of masonry is by no means in this condition, but is knit together in a more or less homogeneous mass, it will be seen that the tendency to move forwards is counteracted, not by mere friction alone, but also by the resistance to shearing of the stonework. This is a very strong combination, and makes the total resistance so great that experience proves that, when a dam is safe against overturning, it is safe against being moved forwards bodily on its base. If, however, the whole dam were placed on a smooth surface, such as a timber grillage, particularly if the planks were laid in the same direction as the pressure, or if it rested on a yielding clay, with only a small depth of foundation, very serious doubts might exist as to whether it would remain immovable, and careful examinations and calculations would be necessary. In such cases, the coefficient of friction may fall considerably below .75. As, however, all masonry dams should stand on a rock foundation, into which the footing course is well embedded, no danger of their moving bodily forwards need be apprehended if the stability is satisfactory as regards overturning. 33. Average Dimensions.-Calculations made with various practical values for w and b, show that a bottom width equal to from H to H will always give a satisfactory factor of safety, and in nearly all cases the smaller of these two values, that is, b1 secure profile. = 2 H will give a perfectly HIGH MASONRY DAMS 34. General Considerations.-The trapezoidal profile hitherto considered is the one almost universally adopted for masonry dams up to 50 or 60 feet in height. Beyond this limit, it would no longer be economical nor, in very high dams, practicable. So far, only resistance to sliding and ΠΑΙ B FIG. 13 FIG. 14 FIG. 15 overturning has been considered, but in very high dams. another element of destruction must be taken into account; namely, the crushing of the material under its own weight. In the case of symmetrical figures, the amount of pressure per square unit of base is obtained by dividing the whole |