Geotechnical Engineering

DHARM

LATERAL EARTH PRESSURE AND STABILITY OF RETAINING WALLS 457

Rankine (1857) was the first to investigate the stress conditions associated with the
states of plastic equilibrium in a semi-infinite mass of homogeneous, elastic and isotropic soil
mass under the influence of gravity or self-weight alone. The concept as postulated by Rankine
in respect of a cohesionless soil mass is shown in Fig. 13.6.
Let us consider an element of unit area at a depth z below the horizontal ground sur-
face. Let the unit weight of the cohesionless soil be γ. The vertical stress acting on the horizon-
tal face of the element σv = γ.z. Since any vertical plane is symmetrical with respect to the soil
mass, the vertical as well as horizonatal planes will be free of shear stresses. Consequently,
the normal stresses acting on these planes will be principal stresses. The horizontal principal
stress, σh, or the lateral earth pressure at rest in this case, is given by K 0. σv, or K 0. γ.z. The
element is in a state of elastic equilibrium under these stress conditions.
Horizontal movement or deformation of the soil mass can change the situation. For
example, if the soil mass gets stretched horizontally, the lateral stress or horizontal principal
stress gets reduced and reaches a limiting minimum value. Any further stretching will induce
plastic flow or failure of the soil mass. This limiting condition is one of plastic equilibrium at
which failure is imminent and is referred to as the ‘active’ state. Subsequent failure, if it
occurs, is active failure. It is said to be active because the weight of the soil it self assists in
producing the horizontal expansion or stretching.
On the other hand, if the soil mass gets compressed horizontally, the lateral pressure or
horizontal principal stress increases and reaches a limiting maximum value; any further com-
pression will induce plastic flow or failure of the soil mass. This limiting condition also is one of
plastic equilibrium at which failure is imminent, and is referred to as the ‘passive’ state. Sub-
sequent failure, if it occurs, is passive failure. It is said to be passive because the weight of soil
resists the horizontal compression.
The conditions of stress in these two cases are illustrated in Fig. 13.6 (a) and (b) respec-
tively and known as the ‘Active Rankine State’ and the ‘Passive Rankine State’ respectively.

The orientation or pattern of the failure planes as well as the lateral pressures in these
two states may be obtained from the corresponding Mohr’s circles of stress representing the
stress conditions for these two states as shown in Fig. 13.6 (c).

From the geometry of the Mohr’s circle, for active condition,

sin φ =

DC

OC

vh

vh

1

1

13

13

13

13

2

2

=

−

+

=

−

+

=

−

+

()/

()/

()

()

()

()

σσ

σσ

σσ

σσ

σσ

σσ

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

This leads to

σ

σ

φ

φ

h

v

= −

+

1

1

sin

sin

σ

σ

h

v

is known as the coefficient of lateral earth pressure and is denoted by Ka for the active

case.

∴ Ka =

1

1

45

2

− 2

+

=°−F

HG

I

KJ

sin

sin

φ tan

φ

φ

...(Eq. 13.8)

(by trignometry.)

AC 1 D = 90° + φ, from ∆OC 1 D.