Geotechnical Engineering

DHARM

324 GEOTECHNICAL ENGINEERING

For example, the depth z corresponding to the point T 2 is stable for the slope angle β > φ.
However, if β > φ, the slope can be stable only up to a limited depth, which is known as the
critical depth zc; the state of stress at this depth is represented by R, as already stated.

The equation of the strength envelope is given by:

s = c + σn tan φ

At failure, s = τf = c+σnftanφ

But σnf= γ. zc. cos^2 β

and τf = γ zc. sin β cos β, from Eqs. 9.2 and 9.3.

∴ γ zc. sin β cos β = c + γ zc. cos^2 β. tan φ

zc γ cos β (sin β – cos β tan φ) = c

∴ zc = (c/γ). 1/[cos^2 β (tan β – tan φ)] ...(Eq. 9.12)

Thus the critical depth is proportional to cohesion, for particular values of β and φ.

From Eq. 9.12,

c

γzc

= cos^2 β (tan β – tan φ) ...(Eq. 9.13)

The quantity c

γzc

is called the stability number Sn.

For any depth z less than zc, the factor of safety

F =

Shearing strength

Shearing stress

∴ F =

cz

z

+γ β φ

γββ

cos. tan

cos. sin

2

...(Eq. 9.14)

Since the factor of safety Fc with respect to cohesion,

Fc =

c

cm

, where cm = mobilised cohesion, at depth z,

Sn = c/γ zc = cm/γ z = c/Fc.γ z = cos^2 β (tan β – tan φ) ...(Eq. 9.15)

From Eqs. 9.13 and 9.15,

Fc =

z

z

c

This is the same as Eq. 9.11, as for a purely cohesive soil.

This is based on the assumption that the frictional resistance of the soil is fully devel-

oped. The actual factor of safety should be based on the simultaneous development of cohesion

and friction.

If there is a seepage parallel to the ground surface throughout the entire mass of soil, it

can be shown that:

c

γz =

cos^2 ββtan γ .tan

γ

′ φ−

F

HG

I

KJ

...(Eq. 9.16)

since effective stress alone is capable of mobilising shearing strength.