OUTLINES OF THE DESIGN.
Except for the effect of waves and ice (§ 957), the width on top could be zero. But in practice the top of the dam is generally used for a footway or a roadway, and hence a considerable width is required independent of any question of stability. Schuyler says that the top width need not be more than one tenth of the height, unless the top is used for a roadway.* For the top dimensions of a few of the highest masonry dams, see § 964-66.
In designing the vertical cross section of a gravity dam to resist still water, it is necessary to fulfill three con ditions: (1) to prevent sliding forward, equation 6, page 461, must be satisfied; (2) to resist overturning, equation 10, page 466, must be satisfied; and (3) to resist crushing, equation 22, page 473, must give safe maximum pressures when the reservoir is full and also when it is empty. and must also give a positive minimum pressure (i.e., must not give tension) when the reservoir is either full or empty. Limiting equation 22 to positive values is equivalent to keeping d less than 1, or equivalent to saying that the center of pressure shall always lie within the middle third of any horizontal joint. As the three equations of conditions really involve only three variables viz.: h, b„ and b',—the height of the dam and the batter of the two faces,—they can be satisfied exactly. However, as there is little or no danger of a dam's failing by sliding, provided it is safe against overturning and crushing, there are practically only two conditions to be fulfilled. Further, in a dam of the pure gravity type (the form here under consideration), there is no reason why the up-stream face may not be exactly vertical; and hence there are really only two variables—h and To design a dam it is necessary to satisfy the equation of con dition for successive comparatively thin horizontal layers, and use the dimensions of each elementary layer in finding the dimensions of the next lower one. The equations of condition may be satisfied either (1) by direct computation or (2) by trial.
1. The direct computation may be made either algebraically or graphically. To make a solution by the first method, state the condition for stability against overturning (equation 10, page 466) in terms of the length of the joint (1), the thickness on top (t), the height of the dam (h 1 ql, the depth of the water (hl,. the batter of the up-stream and down-stream faces (b' and b„ respectively), the weight of a cubic unit of water, and the weight of a cubic unit of masonry; and solve for 1. Then test the joint by equation 22, page 473. This method involves the solution of a quadratic of considerable complexity.
To solve the problem graphically, draw the section and determine the position of the center of pressure as in Fig. 98 or 99, page 466;
and then apply equation 22.
One or the other of the above processes is to be repeated suc cessively for each of the several layers beginning at the top.
2. To satisfy the conditions by trial, proceed as follows: The width at the top being known, assume dimensions for the first ele mentary horizontal layer, and test its stability by equations 10 (page 466) and 22 (page 473), the latter with and without the water pressure acting against the dam. If the first dimensions do not give results in accordance with the limiting conditions, other dimen sions must be assumed and the computations be repeated. A third approximation will rarely be needed.
The second method of satisfying the equations of condition is the one employed in the design of the New Croton Dam (§ 964); but later the equations for the first method of solution were worked out, and employed in checking the previously determined dimensions. - It is not necessary to attempt to satisfy these equations precisely, since there are a number of unknown and unknowable factors, as the weight of the stone, the quality of the mortar, the character of the foundation, the quality of the masonry, the static pressure under the mass, the amount of elastic yielding, the force of the waves and of the ice, etc., which have more to do with the ultimate stability of a dam than the mathematically exact profile. It is therefore sufficient to assume a trial profile, being guided in this by the cross sections of existing dams (§ 963-67) and by the principle stated in the next paragraph, and test it at a few points by applying the preceding equations; a few modifications to satisfy more nearly the mathematical conditions or to simplify the profile is as far as it is wise to carry the theoretical determination. Prof. Wm. Cain has shown * that the equations of conditions are nearly satisfied by a cross section composed of two trapezoids, the lower and larger of which is the lower part of a triangle having its base on the foundation of the dam and its apex at the surface of the water, and the upper trapezoid having for its top the predetermined width of the dam on top, and for its sides nearly vertical lines which intersect the sides of the lower trapezoid. The width of the dam at the bottom is obtained by applying the equations of condition as above. The relative batter of the up-stream and down-stream faces depends upon the relative factors of safety for crushing and overturning. The above section gives a factor of safety which increases from bottom to top—an important feature.