section – Fig. 9.10) increases the seepage length, with a corresponding
reduction in the exit gradient. It will also provide a smooth erosion-
resistant transition for the accelerating flows approaching the weir. The
upstream end of the approach slab is firmly secured to the upstream sheet-
piling or to the vertical concrete cut-off. It is usually monolithic with the
weir section in order to provide additional resistance to sliding.
9.1.5 Pressure distribution under the foundation floor of a weir/barrage
Applying the theory of complex variables (of potential theory) involved with
the seepage flow under a flat floor (see Section 2.6) a Laplace differential
equation can be formulated which on integration with appropriate boundary
conditions suggests that the pressure head (P) at any point beneath the floor
is a fraction, , of the total head, Hs(see Fig. 9.4). Thus a solution to Laplace
equation (Khosla et al., 1954; Leliavsky, 1965) can be written as
P/Hs(1/π) cos^1 (2x/b) (9.6)
for the underside of the floor where bis the total floor length and xis the
distance from the centre of the floor to the point where the uplift pressure
head is P.
Equation (9.6), based on the potential theory of the seepage flow,
suggests entirely different distribution to that of Bligh’s creep theory (i.e.
linear distribution – see Fig. 9.5).
In reality the weir foundations are composite in construction consist-
ing of floor slabs (horizontal or sloping), pilings or cut-off structures and a
direct solution of the Laplace equation is not feasible. Khosla et al.(1954),
in dealing with this problem, introduced a method of independent vari-
ables splitting the composite weir/barrage section into a number of simple
forms of known analytical solutions and by applying some corrections in
transferring the results to the composite section.
The simple standard forms of a composite section are:
i) a straight horizontal floor of negligible thickness with a sheet pile at
either end;
ii) a straight horizontal floor of finite thickness (depressed floor) with
no cut-off piles;
iii) a straight horizontal floor of negligible thickness with an intermedi-
ate sheet pile.
Khoslaet al.produced pressure charts (see Figs 9.6 and 9.7) from which
the pressures at key points (junction points of the floor and the pile and
the bottom corners of the depressed floor) of these elementary forms can
372 DIVERSION WORKS
