Geotechnical Engineering

DHARM

496 GEOTECHNICAL ENGINEERING

construction is used. It simply means that, in the force triangle, W should be taken as the

weight of the trial sliding wedge plus qs. cos β.

Equation 13.58 may also be written as follows:

γ 1 = γ

β

αβ

+

+

L

N

M

O

Q

P

2 q

Hs

.cos

sin( )

...(Eq. 13.59)

γ 1 = γ

β

ααβ

+

+

L

N

M

O

Q

P

2 q

H

.cos

sin .sin( )

...(Eq. 13.60)

If the intensity of surcharge is specified as q per unit sloping area, Eq. 13.58 gets modi-
fied as

γ 1 = γ+ ′

F

HG

I

KJ

2 q
Hs ...(Eq. 13.61)
It may be shown that the location of the failure plane is not changed, as also the direc-
tions of the forces on the sliding wedge. When surcharge is added, all forces increase in the
same ratio. The ratio of the additional thrust due to surcharge to that without surcharge is
sometimes called the ‘surcharge ratio’.
Since the expression for the modified unit weight consists of two terms—one of the unit
weight of soil and the other relating to the surcharge term, the thrust may be looked upon as
being composed of that without surcharge and the contribution due to the surcharge. The
weight of the soil wedge and the thrust due to it are proportional to H^2 , while the weight of
surcharge is proportional to the surface dimension of the wedge, or to H; hence, the contribu-
tion of the surcharge to the lateral pressure is proportional to H. Thus, the lateral pressure
due to surcharge is constant over the height of the wall. The distribution of the lateral pres-
sure with depth is, therefore, as shown in Fig. 13.36 (b). The pressures σ 1 and σ 2 may be
obtained if the thrusts with and without surcharge are determined.
If the ground surface is surcharged with different intensities q 1 , q 2 etc., as shown in Fig.
13.37, the Culmann-curve may have several maximum P-values. The maximum of the several
maximum values, the so-called “maximum maximorum”, is then taken as the active thrust: Pa
= Pa max max. This value also determines the position of the most dangerous rupture surface,
BC.

y B

(+)fd

f

Pmax max

Pa max

y-line

A

q 1

q 2

a

Fig. 13.37 Culmann’s method for surcharges of different intensities