Geotechnical Engineering

DHARM

SEEPAGE AND FLOW NETS 181

B

C

D F

AB

S

KJ

Top flow

line

P(x, z)

Kozenys

base

parabola

ht

z

Q

H

G

C M

x

0/3(EB)

E

D

d

Directrix

F

(c) Earth dam with flow net consisting of confocal parabolas

(d) Common case of earth dam and the top flow line (A. Casagrande, 1940)
Fig. 6.15 Flow net consisting of confocal parabolas (After Taylor, 1948)
(ii) With A as centre and AF as radius, draw an arc to cut the water surface (extended)
in J. The vertical through J is the directrix. Let this meet the bottom surface of the dam in M.
(iii) The vertex C of the parabola is located midway between F and M.
(iv) For locating the intermediate points on the parabola the principle that it must be
equidistant from the focus and the directrix will be used. For example, at any distance x from
F, draw a vertical and measure QM. With F as center and QM as radius, draw an arc to cut the
vertical through Q in P, which is the required point on the parabola.

(v) Join all such points to get the base parabola. The portion of the top flow line from B
is sketched in such that it starts perpendicular to BD, which is the boundary equipotential
and meets the remaining part of the parabola tangentially without any kink. The base pa-
rabola meets the filter perpendicularly at the vertex C.

The following analytical approach also may be used:

With the origin of co-ordinates at the focus [Fig. 6.15(d)], PF = QM

xz^22 + = x + S ...(Eq. 6.13)

∴ x =

()zS

S

22

2

−

...(Eq. 6.14)

This is the equation to the parabola.