Craig's Soil Mechanics 7th Edition
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Craig's Soil Mechanics 7th Edition

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108 kN=m

Figure 6.21 Example 6.6. (Reproduced from K. Terzaghi (1943) Theoretical Soil Mechanics, John
Wiley & Sons Inc., by permission.)

194 Lateral earth pressure

The horizontal thrust on the wall is given by

Pa cos � ¼ 105 kN=m

Other failure surfaces would have to be chosen in order that the maximum value of
total active thrust can be determined.
For the inclined drain shown in Figure 6.21(c), the flow lines and equipotentials

above the drain are vertical and horizontal, respectively. Thus at every point on the
failure plane the pore water pressure is zero. This form of drain is preferable to the
vertical drain. In this case

Pa ¼ 1
2
Ka�satH

2

For �0 ¼ 38� and � ¼ 15�, Kah (¼Ka cos �) ¼ 0:21 (from Figure 6.15). The horizontal
thrust is

Pa cos � ¼ 1
2
� 0:21� 20� 62 ¼ 76 kN=m

For the case of no drainage system behind the wall, the pore water is static, and
therefore the horizontal thrust

¼ 1
2
Ka�

0H2 cos � þ 1
2
�wH

2

¼ 1
2
� 0:21� 10:2� 62

� �
þ 1

2
� 9:8� 62

� �

¼ 39þ 176 ¼ 215 kN=m

6.7 EMBEDDED WALLS

Cantilever walls

Walls of this type are mainly of steel sheet piling and are used only when the retained
height of soil is relatively low. In sands and gravels these walls may be used as
permanent structures but in general they are used only for temporary support. The
stability of the wall is due entirely to passive resistance mobilized in front of the wall.
The principal limit state is instability of the retained soil mass causing rotation or
translation of the wall. Limit states (4) to (7) listed for gravity walls (Section 6.6)
should also be considered. The mode of failure is by rotation about a point O near the
lower end of the wall as shown in Figure 6.22(a). Consequently, passive resistance acts
in front of the wall above O and behind the wall below O, as shown in Figure 6.22(b),
thus providing a fixing moment. However this pressure distribution is an idealization
as there is unlikely to be a complete change in passive resistance from the front to the
back of the wall at point O. To allow for over-excavation it is recommended that the

Embedded walls 195

soil level in front of the wall should be reduced by 10% of the retained height, subject
to a maximum of 0.5m. A minimum surcharge pressure of 10 kN/m2 should be
assumed to act on the soil surface behind the wall.
Design is generally based on the simplification shown in Figure 6.22(c), it being

assumed that the net passive resistance below point O is represented by a concentrated
force R acting at a point C, slightly below O, at depth d below the lower soil surface.
The traditional method of analysis involves determining the depth d by equating
moments about C, a factor of safety F being applied to the restoring moment, i.e.
the available passive resistance in front of the wall is divided by F. The value of d is
then increased arbitrarily by 20% to allow for the simplification involved in the
method, i.e. the required depth of embedment is 1.2d. However, it is advisable to
evaluate R by equating horizontal forces and to check that net passive resistance
available over the additional 20% embedded depth is equal to or greater than R.
The translation limit state is satisfied if the horizontal resisting force is greater than or
equal to the disturbing force. Cantilever walls can also be analysed by applying partial
factors.

Anchored or propped walls

Generally, structures of this type are either of steel sheet piling or reinforced concrete
diaphragm walls, the construction of which is described in Section 6.9. Additional
support to embedded walls is provided by a row of tie-backs or props near the top of
the wall, as illustrated in Figure 6.23(a). Tie-backs are normally high-tensile steel
cables or rods, anchored in the soil some distance behind the wall. Walls of this type
are used extensively in the support of deep excavations and in waterfront construction.
In the case of sheet pile walls there are two basic modes of construction. Excavated
walls are constructed by driving a row of sheet piling, followed by excavation or
dredging to the required depth in front of the wall. Backfilled walls are constructed
by partial driving, followed by backfilling to the required height behind the piling. In
the case of diaphragm walls, excavation takes place in front of the wall after it has been
cast in situ. Stability is due to the passive resistance developed in front of the wall
together with the supporting forces in the ties or props.

Figure 6.22 Cantilever sheet pile wall.

196 Lateral earth pressure

Free earth support analysis

It is assumed that the depth of embedment below excavation level is insufficient to
produce fixity at the lower end of the wall. Thus the wall is free to rotate at its lower
end, the bending moment diagram being of the form shown in Figure 6.23(b). The
limit states to be considered are instability of the retained soil mass causing rotation or
translation of the wall, the vertical equilibrium of the wall and states (4) to (7) listed in
Section 6.6. To satisfy the rotation limit state, the restoring moment about the anchor
or prop must be greater than or equal to the overturning moment. The horizontal
forces on the wall are then equated to zero, yielding the minimum value of anchor or
prop force required to satisfy the translation limit state. Finally, if appropriate, the
vertical forces on the wall are calculated, it being a requirement that the downward
force (e.g. the component of the force in an inclined tie back) should not exceed the
(upward) frictional resistance available between the wall and the soil on the passive
side minus the (downward) frictional force on the active side.
Four methods of introducing safety factors into the calculations have evolved, as

described below. These methods also apply to the analysis of embedded cantilever
walls.

1 The depth of embedment at which the wall is on the point of collapse is calculated
by equating moments to zero, using the fully mobilized values of active and
passive pressures. This depth is then multiplied by a factor (Fd), known as the
embedment factor. This method is not recommended because of the empirical way
in which the factor Fd is introduced but if it were to be used the design should also
be checked by one of the other methods.

2 The factor of safety (Fp) is expressed as the ratio of the restoring moment to the
overturning moment, i.e. the former must exceed the latter by a specified margin.
Fully mobilized values of active and passive pressures are used in calculating the
moments. Gross soil pressures are used in the calculations, i.e. the active and
passive pressures are not combined in any way. Being the source of the restoring
moment, gross passive resistance only is factored.

3 The factor of safety is defined in terms of shear strength. The shear strength
parameters are divided by a factor (Fs) before the active and passive pressures are

Figure 6.23 Anchored sheet pile wall: free earth support method.

Embedded walls 197

calculated, the depth of embedment then being determined by equating the
overturning and restoring moments. The factor is thus applied to the parameters
of greatest uncertainty. The water pressures in an effective stress analysis, of
course, should not be factored. This method satisfies the requirements of EC7,
Fs being the equivalent of the partial material factor �m, and of BS 8002, Fs
becoming the mobilization factor M.

4 The factor of safety (Fr) is applied to moments, as in method 2, but is defined as the
ratio of the moment of net passive resistance to the moment of net active thrust.
The concept of net passive resistance is illustrated in Figure 6.23(c), the active thrust
over the embedded depth being subtracted from the passive resistance. This method
was proposed by Burland et al. [7] because of the lack of consistency