Civil Engineering Reference
In-Depth Information
24.8 Calculation of earth pressures - undrained loading
Active and passive pressures for undrained loading can be calculated using either the
upper and lower bound methods or the limit equilibrium method. The procedures are
similar to those described in the previous section for drained loading.
A limit equilibrium solution for the active pressures on a smooth wall was obtained
in Sec. 20.3 from the limit equilibrium method using the Coulomb wedge analysis (see
Fig. 20.4). The solution was
1
2
H
2
P
a
=
γ
−
2
s
u
H
(24.6)
where
s
u
is the undrained strength. Assuming that the stresses increase linearly with
depth,
σ
=
σ
−
2
s
u
(24.7)
a
v
where
v
is the total vertical stress. It is relatively simple to show that the passive
pressure for undrained loading is given by
σ
σ
p
=
σ
v
+
2
s
u
(24.8)
These expressions for active and passive earth pressures for undrained loading can be
written as
σ
=
σ
−
K
au
s
u
(24.9)
a
v
σ
=
σ
+
K
pu
s
u
(24.10)
p
v
where
K
au
and
K
pu
are earth pressure coefficients for undrained loading.
The solutions with
K
au
2 are for a smooth vertical wall with a level ground
surface. Tables and charts are available giving values for
K
au
and
K
pu
for other cases
including rough walls where the shear stress between the soil and the wall is
s
w
.
From Eq. (24.7) the active earth pressure for undrained loading appears to become
negative (i.e. in tension) when
=
K
pu
=
σ
v
<
2
s
u
(24.11)
This is impossible as the soil is not glued to the wall and a tension crack opens up as
shown in Fig. 24.10(a). This is the same kind of tension crack as found near the top
of slopes (see Sec. 20.8) and the critical depth
H
c
of a water-filled crack is
2
s
u
γ
−
γ
H
c
=
(24.12)
w
If the crack is not filled with water put
0 into Eq. (24.12). Notice that the position
of the active force
P
a
has been lowered and if the crack is filled with water it is free
water (not pore water) and applies a total stress to the wall. If there is a surface stress
q
as shown in Fig. 21.10(b), the tension crack will close entirely when
q
γ
w
=
>
2
s
u
.
