Geotechnical Engineering

DHARM

STABILITY OF EARTH SLOPES 327

The depth of tension crack is given by:
hc = 2c/γ ...(Eq. 9.18)
(The concept and derivation of this is given in Ch. 13).
The effect of the tension crack is to shorten the arc along which shearing resistance gets
mobilised to AB′ and to reduce the angle θ to θ′.

In computing the factor of safety F against slippage, θ′ is to be used instead of θ, and the
full weight W of the soil within the sliding surface AB to compensate for any water pressure
that may be exerted, if the crack gets filled with rain water.

The Swedish Method of Slices for a Cohesive-frictional Soil

If the soil is not purely cohesive but is a cohesive-frictional one or if it is partially satu-
rated, the undrained strength envelope shows both c and φ values. The total stress analysis
can be adapted to this case by dividing the area within the slip circle into a number of vertical
slices, as shown in Fig. 9.10.

q

O

r

R 1 R 2

W N

T

(a) Slip circle divided into slices (b) Forces on a typical slice

Fig. 9.10 Swedish method of slices for a cohesive-frictional soil

The forces on a typical slice are given in Fig. 9.10 (b). The reactions R 1 and R 2 at the

sides of the slice are assumed equal. The weight W of the slice is set-off at the base of the slice

to a convenient scale. The known directions of its normal component N and the tangential

component T are drawn to complete the vector triangle. The values of N and T are scaled-off.

Taking moment about the centre of rotation,

Sliding moment = r Σ T (reckoned positive if clockwise)

Restoring moment = r (crθ + Σ N tan φ) (reckoned positive if counterclockwise)

∴ Factor of Safety F =

(tan)cr N

T

θφ+Σ
Σ ...(Eq. 9.19)
The effect of a tension crack can be allowed for as in the previous case. But the depth of
the crack in this case is given by:

hc = (2c/γ). tan (45° + φ/2) ...(Eq. 9.20)
The tangential components of a few slices at the base may cause restoring moments. In
that case these may be considered negative, thus reducing the denominator in Eq. 9.19. Alter-
natively these may be added to the numerator.

The values of W, ΣN, and ΣT may be found conveniently in a number of ways.