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# How is math used to calculate the pressure behind a dam?

There are many engineering considerations and calculations needed when building a dam, the most important being the water pressure behind the structure. Engineers know that as the elevation of water behind a dam increases, the height and density of the water causes higher pressures at the bottom of the dam. Thinking in mathematical terms, the horizontal force acting on the dam is the integral of the water pressure over the area of the dam that is in contact with the water. The force exerted by the water pushes horizontally on the dam face, and this is resisted by the force of static friction between the dam and the bedrock foundation on which it rests. The water also tries to rotate the dam about a line running along the base of the dam; the torque resulting from the weight of the dam acts in the opposite sense.

Hoover Dam, an impressive structure on the Colorado River, could only have been made possible by engineers designing it in such a way that the walls were thick enough to withstand massive water pressure.

For example, take the water pressure on a dam, such as the Hoover Dam on the Colorado River between Arizona and Nevada. Before building that dam, engineers needed to know the pressure not only along the entire structure, but also—and especially—at its base. In general, the pressure exerted by water equals the density times the depth, in which the density of water is 62.4 pounds per cubic foot. For Hoover Dam, this gives a pressure of 37,440 pounds per square foot, or 18.72 tons per square foot; the pressure calculated at half the height of the dam is 9.36 tons per square foot. This is why the base width of the dam is 1,660 feet (201.2 meters)—it is thicker to compensate for the increase in pressure at the bottom of the dam—while the width of the crest of the dam is only 45 feet (13.7 meters).

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