Civil Engineering Reference
In-Depth Information
soil collapsing into the excavation is
P
a
and this is called the active force (see Sec. 24.1).
(In practice, vertical cracks may form in the ground near the top of the wall; I will
consider the influence of tension cracks later, but for the present I will assume that
they do not occur.) A mechanism can be constructed from a single straight slip surface
at an angle
and there must be slip surfaces between the soil and the wall as shown.
The forces acting on the triangular wedge are shown in Fig. 20.4(b). There is no
shear force between the soil and the smooth wall. The directions of all the forces are
known and the magnitudes of
P
a
and
N
are unknown; the magnitude of the shear
force
T
is given by
α
T
=
s
u
L
(20.7)
where
s
u
is the undrained strength and
L
is the length of the slip surface;
T
acts up
the surface as the wedge moves down into the excavation. With two unknowns the
problem is statically determinate and a solution can be found by resolution of the
forces; notice that if you resolve in the direction of the slip surface
N
does not appear
and
P
a
can be found directly. Alternatively, the solution can be found graphically by
constructing the closed polygon of forces in Fig. 20.4(c).
To obtain the limit equilibrium solution you must vary the angle
to find the max-
imum, or critical, value for
P
a
. If you do this you will find that the critical angle is
α
=
α
45
◦
and the limit equilibrium solution is
1
2
H
2
P
a
=
γ
−
2
s
u
H
(20.8)
Notice that if we put
P
a
=
0 we obtain
4
s
u
γ
H
c
=
(20.9)
which is a limit equilibrium solution for the undrained stability of an unsupported
trench.
This analysis can be extended quite simply to include the effects of foundation loads,
water in the excavation and shear stresses between the soil and a rough wall. The addi-
tional forces are shown in Fig. 20.5(a) and the corresponding polygon of forces is shown
Figure 20.5
Coulomb wedge analysis for a rough wall for undrained loading.
