Geotechnical Engineering

DHARM

184 GEOTECHNICAL ENGINEERING

The top flow line in each case is in close agreement with the parabola, except for a short
portion of its path in the end. It may be noted that the line of length a, which is a boundary of
the flow net, is neither a flow line not an equipotential. Since it is at atmospheric pressure, it
is a boundary along which the head at any point is equal to its elevation.
The following are the steps in the graphical determination of the top flowline for a
homogeneous dam resting on an impervious foundation:
(i) Sketch the base parabola with its focus at the downstream toe of the dam, as de-
scribed earlier.

(ii) For dams with flat slopes, this parabola will be correct for the central portion of the
top flow line. Necessary corrections at the entry on the upstream side and at the exist on the
downstream side are to be effected. The portion of the top flow line at entry is sketched visu-
ally to meet the boundary condition there i.e., the perpendicularity with the upstream face,
which is a boundary equipotential and the tangentiality with the base parabola.

(iii) The intercept (a + ∆a) is now known. The breakout point on the downstream dis-
charge face may be determined by measuring out ∆a from the top along the face, ∆a may be
may be obtained from Fig. 6.18.
(iv) The necessary correction at the downstream end may be made making use of one of
the boundary conditions at the exit, as shown in Fig. 6.17.

The seepage through all dams with flat slopes may be determined with good accuracy
from the simple equation (Eq. 6.16) which holds true for parabolic nets.

L. Casagrande’s Solution for a Triangular Dam

For triangular dams on impervious foundations with discharge faces at 90° or less to the hori-
zontal, L. Casagrande gives a simple and reasonably accurate solution for the top flow line
(Fig. 6.19):

d

AB

G Top flow line

E

a sina

a

a

H J

Equipotential lines

ht

F

Fig. 6.19 L. Casagrande’s method for the determination of top flow line
The top flow line starts at B instead of the theoretical starting point of the parabola, A;
the necessary correction at the entry is made as usual. The top flow line ends at E, the location
of which is desired, and is defined by the distance a.
Let z be the vertical co-ordinate measured from the tail water or foundation level. The
general equation for flow across any equipotential such as GH is given by:

q = k. iav (GH)