Civil Engineering Reference
In-Depth Information
Figure 20.7
Coulomb wedge analysis for a rough wall for drained loading.
Again the analysis can be extended to include external loads, water in the excavation,
pore pressures and shear stresses between the soil and a rough wall. The additional
forces are shown in Fig. 20.7(a) and the corresponding polygon of forces is shown
in Fig. 20.7(b). For simplicity the water table is assumed to be the same in the soil
and in the excavation, so there is no seepage; later I will examine the case where the
excavation is dewatered and there is a steady state seepage flownet in the soil. The force
U
is the sum (or integral) of the pore pressures over the slip surface and is found by
summing Eq. (6.4) over the length
L
of the slip surface. The shear force
T
is given by
T
=
N
tan
φ
=
φ
(
N
−
U
) tan
(20.13)
Similarly, the shear force between the soil and the wall is given by
T
w
=
p
a
tan
δ
(20.14)
δ
δ
must be in
where
is the friction angle between the soil and the wall; obviously
≤
δ
≤
φ
depending on the roughness of the wall. Notice that the total
normal force on the vertical face of the soil is
P
a
+
the range 0
P
w
(i.e. the sum of the force from
the support prop and the force from the free water).
In Figs. 20.4 and 20.6 the major principal planes are horizontal because the shear
stress on the wall is zero and
h
. In Sec. 2.6 we found that zero extension lines
(i.e. lines of zero strain) were at angles
σ
>σ
z
45
◦
+
1
2
α
=
ψ
to the major principal plane and
τ
/
σ
ρ
were at angles
45
◦
+
1
2
ρ
planes where the stress ratio was
=
tan
α
=
to the
major principal plane. For undrained loading
ψ
=
0 and for drained loading, at the
ρ
=
φ
. Hence the critical surfaces in these limit equilibrium solutions
coincide with the critical planes and zero extension lines obtained from the Mohr circle
constructions discussed in Chapter 2. In Figs. 20.5 and 20.7 there are shear stresses
between the wall and the soil, so horizontal and vertical planes are not principal
planes and the critical surfaces are not necessarily at angles
critical state
45
◦
or 45
◦
+
1
2
φ
α
=
to
the horizontal.
