(b) Horizontal shear stresses
Numerically equal and complementary horizontal (zy) and (yz) shear
stresses are generated at any point as a result of the variation in vertical
normal stress over a horizontal plane.
It is normally sufficient to establish the boundary, i.e. upstream and
downstream,values. If the angles between the face slopes and the verti-
cal are respectively uupstream and ddownstream, and if an external
hydrostatic pressure, pw, is assumed to operate at the upstream face, then
u(pw zu)tanu (3.34a)
and
dzdtand. (3.34b)
Between the boundary values given by equations (3.34a) and (3.34b) the
variation in shear stress is dependent upon the rate of change in vertical
normal stress. A graphical solution may be used if it is considered neces-
sary to determine the parabolic distribution generally assumed to apply.
(c) Horizontal normal stresses
The horizontal stresses on vertical planes, y, can be determined by
consideration of the equilibrium of the horizontal shear forces operating
above and below a hypothetical element within the dam. The difference in
shear forces is balanced by the normal stresses on vertical planes. Bound-
ary values for yat either face are given by the following: for the upstream
face,
yupw(zu pw)tan^2 u; (3.35a)
for the downstream face,
ydzdtan^2 d. (3.35b)
(d) Principal stresses
Principal stresses 1 and 3 may be determined from knowledge of zand
yand construction of a Mohr’s circle diagram to represent the stress con-
ditions at a point, or by application of the equations given below: for the
major principal stress,
1
z
2
y
max, (3.36a)