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# Craig's Soil Mechanics 7th Edition

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```to be zero. The
rigorous treatment of this type of problem, with both stresses and displacements being
considered, would involve a knowledge of appropriate equations defining the stress–
strain relationship for the soil and the solution of the equations of equilibrium and
compatibility for the given boundary conditions. It is possible to determine displace-
ments by means of the finite element method using suitable computer software,
provided realistic values of the relevant deformation parameters are available. How-
ever, it is the failure condition of the retained soil mass which is of primary interest and
in this context, provided a consideration of displacements is not required, it is possible
to use the concept of plastic collapse. Earth pressure problems can thus be considered
as problems in plasticity.
It is assumed that the stress–strain behaviour of the soil can be represented by the

rigid–perfectly plastic idealization, shown in Figure 6.1, in which both yielding and
shear failure occur at the same state of stress: unrestricted plastic flow takes place at
this stress level. A soil mass is said to be in a state of plastic equilibrium if the shear
stress at every point within the mass reaches the value represented by point Y0.
Plastic collapse occurs after the state of plastic equilibrium has been reached in part of

a soil mass, resulting in the formation of an unstable mechanism: that part of the soil

Figure 6.1 Idealized stress–strain relationship.

mass slips relative to the rest of the mass. The applied load system, including body forces,
for this condition is referred to as the collapse load. Determination of the collapse load
using plasticity theory is complex and would require that the equilibrium equations, the
yield criterion and the flow rule were satisfied within the plastic zone. The compatibility
condition would not be involved unless specific deformation conditions were imposed.
However, plasticity theory also provides the means of avoiding complex analyses. The
limit theorems of plasticity can be used to calculate lower and upper bounds to the true
collapse load. In certain cases, the theorems produce the same result which would then be
the exact value of the collapse load. The limit theorems can be stated as follows.

Lower bound theorem

If a state of stress can be found, which at no point exceeds the failure criterion for the
soil and is in equilibrium with a system of external loads (which includes the self-
weight of the soil), then collapse cannot occur; the external load system thus constitutes
a lower bound to the true collapse load (because a more efficient stress distribution
may exist, which would be in equilibrium with higher external loads).

Upper bound theorem

If a mechanism of plastic collapse is postulated and if, in an increment of displace-
ment, the work done by a system of external loads is equal to the dissipation of energy
by the internal stresses, then collapse must occur; the external load system thus
constitutes an upper bound to the true collapse load (because a more efficient mechan-
ism may exist resulting in collapse under lower external loads).

In the lower bound approach, the conditions of equilibrium and yield are satisfied
without consideration of the mode of deformation. The Mohr–Coulomb failure criterion
is also taken to be the yield criterion. In the upper bound approach, a mechanism of plastic
collapse is formed by choosing a slip surface and the work done by the external forces is
equated to the loss of energy by the stresses acting along the slip surface, without
consideration of equilibrium. The chosen collapse mechanism is not necessarily the true
mechanism but it must be kinematically admissible, i.e. the motion of the sliding soil mass
must be compatible with its continuity andwith any boundary restrictions. It can be shown
that for undrained conditions the slip surface, in section, should consist of a straight line or
a circular arc (or a combination of the two); for drained conditions the slip surface should
consist of a straight line or a logarithmic spiral (or a combination of the two). Examples of
lower and upper bound plasticity solutions have been given by Atkinson [1] and Parry [16].
Lateral pressure calculations are normally based on the classical theories of Rankine

or Coulomb, described in Sections 6.2 and 6.3, and these theories can be related to the
concepts of plasticity.

6.2 RANKINE’S THEORY OF EARTH PRESSURE

Rankine’s theory (1857) considers the state of stress in a soil mass when the condition of
plastic equilibrium has been reached, i.e. when shear failure is on the point of occurring

162 Lateral earth pressure

throughout the mass. The theory satisfies the conditions of a lower bound plasticity
solution. The Mohr circle representing the state of stress at failure in a two-dimensional
element is shown in Figure 6.2, the relevant shear strength parameters being denoted by
c and �. Shear failure occurs along a plane at an angle of 45� þ �/2 to the major principal
plane. If the soil mass as a whole is stressed such that the principal stresses at every point
are in the same directions then, theoretically, there will be a network of failure planes
(known as a slip line field) equally inclined to the principal planes, as shown in Figure 6.2.
It should be appreciated that the state of plastic equilibrium can be developed only if
sufficient deformation of the soil mass can take place.
Consider now a semi-infinite mass of soil with a horizontal surface and having a

vertical boundary formed by a smooth wall surface extending to semi-infinite depth, as
represented in Figure 6.3(a). The soil is assumed to be homogeneous and isotropic.
A soil element at any depth z is subjected to a vertical stress �z and a horizontal stress
�x and, since there can be no lateral transfer of weight if the surface is horizontal, no
shear stresses exist on horizontal and vertical planes. The vertical and horizontal
stresses, therefore, are principal stresses.

Figure 6.2 State of plastic equilibrium.

Rankine’s theory of earth pressure 163

If there is a movement of the wall away from the soil, the value of �x decreases as the
soil dilates or expands outwards, the decrease in �x being an unknown function of the
lateral strain in the soil. If the expansion is large enough, the value of �x decreases to a
minimum value such that a state of plastic equilibrium develops. Since this state is
developed by a decrease in the horizontal stress �x, this must be the minor principal
stress (�3). The vertical stress �z is then the major principal stress (�1).
The stress �1 (¼�z) is the overburden pressure at depth z and is a fixed value for

any depth. The value of �3 (¼�x) is determined when a Mohr circle through the
point representing �1 touches the failure envelope for the soil. The relationship
between �1 and �3 when the soil reaches a state of plastic equilibrium can be derived
from this Mohr circle. Rankine’s original derivation assumed a value of zero for the
shear strength parameter c but a general derivation with c greater than zero is given
below to cover the cases in which undrained parameter cu or tangent parameter c

0

is used.
Referring to Figure 6.2,

sin� ¼
1
2
ð�1 � �3Þ

1
2
ð�1 þ �3 þ 2c cot�Þ

; �3ð1þ sin�Þ ¼ �1ð1� sin�Þ � 2c cos�

Figure 6.3 Active and passive Rankine states.

164 Lateral earth pressure

; �3 ¼ �1 1� sin�
1þ sin�
� �

� 2c
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð1� sin2 �Þ

q
1þ sin�

0
@

1
A

; �3 ¼ �1 1� sin�
1þ sin�
� �

� 2c
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1� sin�
1þ sin�
� �s

ð6:1Þ

Alternatively, tan2 (45� � �/2) can be substituted for (1� sin�)/(1þ sin�).
As stated, �1 is the overburden pressure at depth z, i.e.

�1 ¼ �z

The horizontal stress for the above condition is defined as the active pressure (pa) being
due directly to the self-weight```