Geotechnical Engineering

DHARM

458 GEOTECHNICAL ENGINEERING

This is twice the angle made by the plane on which the stress conditions are represented

by the point D on the Mohr’s circle. Hence, the angle made by the failure plane with the

horizontal is given by^12 (90°+ φ) or (45° + φ/2). Similarly, from the geometry of the Mohr’s

circle for the passive condition,

sin φ =

EC

OC

hv

hv

2

2

13

13

13

13

2

2

= −

+

= −

+

= −

+

()/

()/

σσ

σσ

σσ

σσ

σσ

σσ

,

since σh is the major principal stress and σv is the minor principal stress for the passive case.

This leads to

σ

σ

φ

φ

v

h

=

−

+

1

1

sin

sin

or σ

σ

φ

φ

h

v

= +

−

1

1

sin

sin

σ

σ

h

v

is the coefficient of lateral earth pressure and is denoted by Kp for the passive case.

∴ Kp =

1

1

45

2

+ 2

−

=°+F

HG

I

KJ

sin

sin

φ tan

φ

φ

...(Eq. 13.9)

The angle made by the failure plane with the vertical is (45° + φ/2), i.e., with the plane of
which the major principal stress acts.

Thus, the angle made by the failure plane with the horizontal is (45° – φ/2) for the
passive case.

The effective angle of friction, φ′, is to be used for φ, if the analysis is based on effective
stresses, as in the case of submerged or partially submerged backfills. These two states are the
limiting states of plastic equilibrium; all the intermediate states are those of elastic equilib-
rium, which include ‘at rest’ condition.

13.6.2Active Earth Pressure of Cohesionless Soil

Let us consider a retaining wall a vertical back, retaining a mass of cohesionless soil, the

surface of which is level with the top of the wall, as shown in Fig. 13.7 (a).

Kzog

z

H Cohesionless soil(unit weight : )g Pa

KHog

H/3

(a) Retaining wall with cohesionless

backfill (moving away from the fill)

(b) Active pressure

distribution with depth

Fig. 13.7 Active earth pressure of cohesionless soil—Rankine’s theory