Geotechnical Engineering

DHARM

322 GEOTECHNICAL ENGINEERING

The factor of safety against slippage may be written as:

F =^1 −

F

HG

I

KJ

=

F −

HG

I

KJ

= ′

γ

γ

φ

β

γγ

γ

φ

β

γ

γ

φ

β

wwtan

tan

tan

tan

.tan

sat tan ...(Eq. 9.7)

9.2.2 Infinite Slope in a Purely Cohesive Soil

Let us consider an infinite slope in purely cohesive soil as shown in Fig. 9.5.

zc

z

Purely

cohesive soil

Ledge

t

c

O s

b

D( , )stn

P(stnf, )f

Bs = c Strength envelope

A

Q

(a) Infinite slope in purely cohesive

soil-critical depth

(b) Relation between strength envelope

and angle of slope

Fig. 9.5 Infinite slope in a purely cohesive soil

For a particular depth z, the values of the normal and shear stresses at the base of the

element are given by Eqs. 9.2 and 9.3, i.e.,

σn = γ. z cos^2 β

and τ = γ. z sin β. cos β

If these are represented as co-ordinates on a σ – τ plot, point D is obtained. This should

lie on a line through origin O inclined at the angle of slope β, since

τ

σn

= tan β. If this point D

lies below the Coulomb strength envelope, s = c for the purely cohesive soil, the slope will be
stable.

The factor of safety against slippage will be AB

AD

, at a depth z from the surface.

∴ F = c/τ =

c

γββzsin cos

...(Eq. 9.8)

If the line OD is extended it will meet the horizontal strength envelope at a point, say P,
the foot of the perpendicular from P on to σ-axis being Q. The point P represents a stress
condition for a different depth, greater than z. At this point the shearing stress at the base of
the element equals the shearing strength of the soil; that is to say, failure is incipient at this
depth. In other words, the slope will be stable only up to a maximum depth zc, called the
critical depth, at which the shearing stress reaches the value of the shearing strength of the
soil, which is merely c in this case, as it is a purely cohesive soil. A ledge or some other material
with a sufficiently large strength exists below the soil of critical depth.
The critical depth zc can be evaluated by equating F to unity.