Hydraulic Structures: Fourth Edition

(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)

146 CONCRETE DAM ENGINEERING