The Hoover dam has a semi-circular shape, as shown below. The dam is 220 [m] tal

Question

The Hoover dam has a semi-circular shape, as shown below. The dam is 220 [m] tall and the maximum water level behind the dam is 180 [m]. Consider the following coordinate system: x is tangent to the arclength of the dam, y is normal to the dam surface , and z is upwards from the downstream river. If the arc-length of the top of the dam is 379 [m], and assuming the dam is of constant cross-sectional area and has the same radial value into and out of the page, determine the following:

a) The magnitude of force acting in the horizontal direction

b) The magnitude of force acting in the vertical direction

c) The total magnitude of the resultant force

d) The line of action of the resultant force in terms of x, y and z coordinates.

Explanation / Answer

First we need to calculate area of dam over which water force will act.

Arc Length = 379 m

Perimeter of semi-circle = 3.14 X R

R = 120.70 m

Water height = 180 m

Surface area = 180 X 379 = 68220 m2, Unit wt of water = 10 kN/m3

Total water force in horizontal direction - Pressure X Area - (10 X 180 X 0.5) X (180 X 379) = 61398000

However this force is acting normal to the circular arc.Hence half of it will cancel with each other.

Force in horizontal direction will be - 30699000 kN

Force acting in vertical direction (on dam) will be zero.

The total magnitude of resultant force is 30699000 kN

Resultant force will be in y direction. Force in horizontal direction will cancel each other.

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