MIDDLE THIRD RULE
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
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
triangular profile fulfills this condition, when the dam is empty and there is
no water thrust on the upstream face.
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.
maximum compressive stress is at the heel of the am and i.e. equal to
( 1 + 6e/b)
e= b/2 - b/3 = b/6
the maximum stress at the heel is 2W/b/2
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.
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.
have to find out a relationship between the base width and height of dam (depth
of water) to satisfy the above condition.