﻿ Craig's Soil Mechanics 7th Edition - Mecânica dos Solos 2 - 44
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# Craig's Soil Mechanics 7th Edition

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```The special case of a vertical wall and a horizontal soil surface will now be
considered. For the undrained condition (�u ¼ 0), an expression for P can be obtained
by resolving forces vertically and horizontally. The total active thrust is given by the
maximum value of P, for which @P/@� ¼ 0. The resulting value is

Pa ¼ 1
2
�ðH2 � z20Þ � 2cuðH � z0Þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1þ cw

cu

� �s

For �u ¼ 0 the earth pressure coefficient Ka is unity and it is convenient to introduce
a second coefficient Kac, where

Kac ¼ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1þ cw

cu

� �s

For the fully drained condition in terms of tangent parameters c0 and �0, it can be
assumed that

Kac ¼ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ka 1þ cw

c0
�
h ir

In general, the active pressure at depth z can be expressed as

pa ¼ Ka�z� Kacc ð6:18Þ

where

Kac ¼ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ka 1þ cw

c

�
h ir
ð6:19Þ

Figure 6.14 Coulomb theory: active case with c > 0.

Coulomb’s theory of earth pressure 179

the shear strength parameters being those appropriate to the drainage conditions of
the problem. The depth of a dry tension crack (at which pa ¼ 0) is given by

z0 ¼
2c

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1þ cw

c

�
r
�
ﬃﬃﬃﬃﬃﬃ
Ka

p ð6:20Þ

The depth of a water-filled crack (z0w) is obtained from the condition pa ¼ �wz0w.
Hydrostatic pressure in tension cracks can be eliminated by means of a horizontal
filter.

Passive case

In the passive case, the reaction P acts at angle � above the normal to the wall surface
(or � below the normal if the wall were to settle more than the adjacent soil) and the
reaction R at angle � above the normal to the failure plane. In the triangle of forces,
the angle between W and P is 180� �
þ � and the angle between W and R is �þ �.
The total passive resistance, equal to the minimum value of P, is given by

Pp ¼ 1
2
Kp�H

2 ð6:21Þ

where

Kp ¼
sinð
þ �Þ

sin
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ½sinð
� �Þ�p �
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
sinð�þ �Þ sinð�þ �Þ

sinð
� �Þ
� 	s

0
BBBB@

1
CCCCA

2

ð6:22Þ

However, in the passive case it is not generally realistic to neglect the curvature of
the failure surface and use of Equation 6.22 overestimates passive resistance, ser-
iously so for the higher values of �, representing an error on the unsafe side. It is
recommended that passive pressure coefficients derived by Caquot and Kerisel [8]
should be used. Caquot and Kerisel derived both active and passive coefficients
by integrating the differential equations of equilibrium, the failure surfaces being
logarithmic spirals. Coefficients have also been obtained by Sokolovski [24] by
numerical integration.
In general, the passive pressure at depth z can be expressed as

pp ¼ Kp�zþ Kpcc ð6:23Þ

where

Kpc ¼ 2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Kp 1þ cw

c

�
h ir
ð6:24Þ

180 Lateral earth pressure

Kerisel and Absi [13] published tables of active and passive coefficients for a wide
range of values of �, �,
and �, the active coefficients being very close to those
calculated from Equation 6.17. For
¼ 90� and � ¼ 0�, values of coefficients
(denoted Kah and Kph) for horizontal components of pressure (i.e. Ka cos � and Kp
cos �, respectively) are plotted in Figure 6.15.

30

20

10

5

4

3

2

1

0.5

0.4

0.3

0.2

0.1
10° 15° 20° 25° 30° 35° 40° 45°

0

0

0.67φ′

0.33φ′

0.67 φ′
φ′

δ

δ

φ′

Active

Passive

Kph

Kah

φ′

Figure 6.15 Coefficients for horizontal components of active and passive pressure.

Coulomb’s theory of earth pressure 181

Compaction-induced pressure

In the case of backfilled walls, lateral pressure is also influenced by the compaction
process, an effect that is not considered in the earth pressure theories. During back-
filling, the weight of the compaction plant produces additional lateral pressure on the
wall. Pressures significantly in excess of the active value can result near the top of the
wall, especially if it is restrained by propping during compaction. As each layer is
compacted, the soil adjacent to the wall is pushed downwards against frictional
resistance on the wall surface. When the compaction plant is removed the potential
rebound of the soil is restricted by wall friction, thus inhibiting reduction of the
additional lateral pressure. Also, the lateral strains induced by compaction have a
significant plastic component which is irrecoverable. Thus, there is a residual lateral
pressure on the wall. A simple analytical method of estimating the residual lateral
pressure has been proposed by Ingold [10].
Compaction of backfill behind a retaining wall is normally effected by rolling. The

compaction plant can be represented approximately by a line load equal to the weight
of the roller. If a vibratory roller is employed, the centrifugal force due to the vibrating
mechanism should be added to the static weight. The vertical stress immediately below
a line load Q per unit length is derived from Equation 5.14:

�z ¼ 2Q

z

Then the lateral pressure on the wall at depth z is given by

pc ¼ Kað�zþ �zÞ

When the stress �z is removed, the lateral stress may not revert to the original value
(Ka�z). At shallow depth the residual lateral pressure could be high enough, relative to
the vertical stress �z, to cause passive failure in the soil. Therefore, assuming there is no
reduction in lateral stress on removal of the compaction plant, the maximum
(or critical) depth (zc) to which failure could occur is given by

pc ¼ Kp�zc

Thus

Kað�zc þ �zÞ ¼ 1
Ka
�zc

If it is assumed that �zc is negligible compared to �z, then

zc ¼ K
2
a�z
�

¼ K
2
a

�

2Q

zc

182 Lateral earth pressure

Therefore

zc ¼ Ka
ﬃﬃﬃﬃﬃﬃﬃ
2Q

�

s

The maximum value of lateral pressure ( pmax) occurs at the critical depth, therefore
(again neglecting �zc):

pmax ¼ 2QKa

zc

¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2Q�

r ð6:25Þ
The fill is normally placed and compacted in layers. Assuming that the pressure pmax

is reached, and remains, in each successive layer, a vertical line can be drawn as a
pressure envelope below the critical depth. Thus the distribution shown in Figure 6.16
represents a conservative basis for design. However, at a depth za the active pressure
will exceed the value pmax. The depth za, being the limiting depth of the vertical
envelope, is obtained from the equation

Ka�za ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2Q�

r

Thus

za ¼ 1
Ka

ﬃﬃﬃﬃﬃﬃﬃ
2Q

�

s
ð6:26Þ

0 pmax pc 0 pmax pc

zc zc

za
z

z

pp

pa

σz

Figure 6.16 Compaction-induced pressure.

Coulomb’s theory of earth pressure 183

6.4 APPLICATION OF EARTH PRESSURE THEORY

TO RETAINING WALLS

In the Rankine theory the state of stress in a semi-infinite soil mass is considered, the
entire mass being subjected to lateral expansion or compression. However, the move-
ment of a retaining wall of finite dimensions cannot develop the active or passive state
in the soil mass as a whole. The active state, for example, would be developed only
within a wedge of soil between the wall and a failure plane passing through the lower
end of the wall and at an angle of 45� þ �/2 to the horizontal, as shown in Figure
6.17(a); the remainder of the soil mass would not reach a state of plastic equilibrium. A
specific (minimum) value of lateral strain would be necessary for the development of
the active state within the above wedge. A uniform strain within the wedge would be
produced by a rotational movement (A0B) of the wall, away from the soil, about its
lower end and a deformation of this type, of sufficient magnitude, constitutes the
minimum deformation requirement for the development of the active state. Any
deformation configuration enveloping A0B, for example, a uniform translational
movement A0B0, would also result in the development of the```