Geotechnical Engineering

DHARM

182 GEOTECHNICAL ENGINEERING

The value of S may be determined, either graphically or analytically. The graphical
approach consists in measuring FM after the determination of the directrix. Analytically S
may be got by substituting the coordinates of A(d, ht) in Eq. 6.13:
S = dh d^22 +−t ...(Eq. 6.15)
For different values of x, z may be calculated and the parabola drawn. The corrections
at the entry may then be incorporated.
A simple expression may be got for the rate of seepage. In Fig. 6.15 (d), the head at the
point G equals S, that along FC is zero; hence the head lost between equipotentials GH and
FC is S. Equation 6.1 may now be applied to the part of the flow net GCFH; nf and nd are each
equal to 3 for this net.
∴ q = k. H. 3/3 = k. S ...(Eq. 6.16)
Alternatively, the expression for q may be got analytically as follows:
q = k. i. A

= k.

d

d

z

x

. z for unit length of the dam.

But z = (2x S + S^2 )1/2 from equation 6.13.

∴ d

d

S

xS S

S

xS S

z

x

=

+

=

+

1

2

2

()() 22212 //^212

Substituting,

q = k. S

() 2 xS+S^212 /

. (2x S + S^2 )1/2 = k. S,

as obtained earlier (Eq. 6.16).

6.6.2 Top Flow Line for a Homogeneous Earth Dam Resting on an Impervious

Foundation

In the case of a homogeneous earth dam resting on an impervious foundation with no drainage

filter directly underneath the dam, the top flow line ends at some point on the downstream

face of the dam; the focus of the base parabola in this case happens to be the downstream toe

of the dam itself as shown in Fig. 6.16.

G

E a

Da

a

Discharge face

FC

Top flow line

Break-out point

AB

D

Fig. 6.16 Homogeneous earth dam with no drainage filter

The slope of the ‘discharge face’, EF, with the base of the dam is designed α, measured

clockwise. This can have values of 90° or more also depending upon the provision of rock-toe or

a drainage face. If the points at which the base parabola and the actual top flow line meet the