GRAVITY DAM ANALYSIS 145
where∑Vis the resultant vertical load above the plane considered, exclus-
ive of uplift, ∑M* is the summation of moments determined with respect
to the centroidof the plane, yis the distance from the neutral axis of the
plane to the point where zis being determined and Iis the second
moment of area of the plane with respect to its centroid.
Applied to a regular two-dimensional plane section of unit width
parallel to the dam axis, and with thickness Tnormal to the axis, equation
(3.31) may be rewritten as
z∑
T
V
12 ∑
T
V
3ey
(3.32)and, at yT/2,
z∑
T
V
^1
6
T
e
(3.33a)i.e. for the reservoir full load state, at the upstream face,
zu∑
T
V
^1
6
T
e
(3.33b)and, at the downstream face,
zd∑
T
V
^1
6
T
e
(3.33c)whereeis the eccentricity of the resultant load, R, which must intersect
the plane downstream of its centroid for the reservoir full condition. (The
signs in equations (3.33b) and (3.33c) interchange for the reservoir empty
condition of loading.)
The eccentricity is determined by evaluating the moments, M*, given
by
e∑M*/∑Vwhere∑Vexcludes uplift.
It is evident from equation (3.33b) that, for eT/6, upstream face
stress,zu, will be negative, i.e. tensile. This is not permissible in view of
the limited and unpredictable tensile strain capacity of concrete (the
classic ‘middle-third’ rule). Total vertical stresses at either face are
obtained by the addition of external hydrostatic pressures.