MIDDLE THIRD RULE
Every name implies, in gravity dams, the stability against thrust is provided by the weight of the dam, such that no develops along the base of the dam.
If all the forces acting on the dam, do not provide a resultant force which acts vertically at the mid point of the base. The loads get a uniform distribution of stress at the base. The load accentric resulting in a non-uniform resulting in a non-uniform at the base. In such cases, the stress distribution is by the equation.
Stress = W/b (1 +0R- 6e/b)
Where W is the vertical component of the resultant force.
B is the base width of the dam
e is the eccentricity
The stress at the base varies from W/b( 1 + 6e/b ) at one end to W/b( 1 - 6e/b ) at the other end of the base.
In order to have only compressive stress at the base, it is necessary that the resultant of all
forces falls within the Middle of the base, i.e., the eccentricity should not be more than b i.e., b . 3 6
The triangular profile fulfills this condition, when the dam is empty and there is no water thrust on the upstream face.
When the dam is empty, there is no water thrust acting on the dam. The only force is the weight of the Dam, acting vertically, passing through the upstream middle third point of the phase.
The stress distribution on the base is shown in the Figure.
The maximum compressive stress is at the heel of the am and i.e. equal to
W/b ( 1 + 6e/b)
Here e= b/2 - b/3 = b/6
Thus the maximum stress at the heel is 2W/b/2
The minimum stress is at the toe and is equal to W/b ( 1 + 6e/b), e being b/6 and W/b ( 1 - 6e/b) becomes Zero.
This shows there is neither compression, nor tension at the Toe.
When the dam is full, if the resultant cuts the base at the outer middle third point, the stress distribution gets reversed i.e., the maximum compression will be at the toe, and no stress at the heel as shown in the stress diagram.
Now, we have to find out a relationship between the base width and height of dam (depth of water) to satisfy the above condition.