The weights of the layers, wad, aabb, etc., are now computed and plotted to a convenient scale, in the vertical line The distance from 0 to each of the several points, A, B, C, etc., represents the total weight of masonry Wa, 11'b, etc., above the corresponding sec tion a-a, etc., of the clam.
The line of action of the total weight of masonry above each horizontal section must now be found. This may be done by taking moments about a vertical line, or it may be done graphically as follows: The center of gravity of each layer into which the section of the dam has been divided is determined and marked (as shown by the points enclosed by circles). From the point lay off on the line the distances from the centers of gravity of the layers to any vertical line, as o4'. (The scale used in laying off these distances, 01— ai, etc., is here made larger than that used for the section of the clam.) Assume a pole, P, and draw strings to the weight line 0-K, then from a point on the vertical through draw the equilib rium polygon as shown, finding the positions of the resultant lines of action, Wk, TVi, 1171, etc. The distances of these lines from the vertical through are the same as the distances of the respective centers of gravity from the line o-k' on the section of the dam. Plotting these lines of action and drawing them to intersection with the corresponding horizontal sections upon which they act, we find the line of pressure for the dam with no water pressure against it.
When water pressure is against the dam to its full height, the horizontal against any portion h feet in depth below the surface is H=31.25 These pressures may be computed for each of the horizontal sections, and each resultant pressure acts on a horizontal line at one-third the height from the section to the surface of the water, as shown, Ha, Ilb, etc.
On the horizontal line 0-K', make the distances 0-A', 0-B', etc., equal the values of the water pressures Ha, Ilb, etc., to the same scale as used for the weights of masonry. The lines A-A', B-B', etc., now represent, in direction and amount, the resultant pressures, Ra, Rb, Rc, etc., upon the various horizontal sections aa, bb, cc, etc.,
of the dam. Lines drawn parallel to these directions through the intersections of the corresponding H and TV lines of action give the points of application of these resultants upon the various sections, and locate the lines of pressure with water against the darn to the top.
137. Design of Profile.—In designing a profile, for a dam we com mence at the top with the assumed thickness and find by trial the required base thickness for each horizontal layer, making each thick ness such that the line of pressure remains everywhere within the middle third of the section. This may be done by the use of the formulas given in Section 135 or by the graphical method of Section 136.
The crushing stress upon the masonry must also be kept within safe limits.
Let b=the width of the section; x= the distance from the outer edge to the point of appli cation of R; y=the distance from the inner edge to the point of appli cation of TV.
The maximum crushing stress at the outer edge of the section is given by the formula, The maximum crushing stress at the inner edge is The allowable crushing stress depends upon the quality of masonry used, and the conditions under which the darn is to be constructed. In high clams, where the front face of the clam has considerable batter, the pressure allowed at the outer face is often made less than that at the inner face. The maximum pressure at the inner edge of a section occurs when the clam carries no water pressure, and the resultant pressure on the base is vertical. The maximum pressure at the outer edge occurs when full water pressure is on the darn, the batter of the outer face is greater than that of the inner face, and the resultant pressure is inclined, only its vertical component being con sidered in determining the stress. For these reasons Rankine's recommendation that the allowable unit crushing stress at the outer edge be made less than that at the inner edge has been followed by some designers. Pressures of from S to 15 tons per square foot have been allowed in a number of large dams of massive rubble or cyclopean concrete.