Civil Engineering Reference
In-Depth Information
with the slope respectively; the angle subtended at the centre of the circle is 2
i
l
. The
Mohr circle just touches the undrained failure envelope and so the state of stress
in the slope does not exceed the undrained failure criterion. From the geometry of
Fig. 21.10(b), making use of Eq. 21.12, a lower bound for the critical slope angle is
given by
tan
i
l
=
τ
s
u
s
σ
s
=
(21.13)
γ
H
cos
2
i
l
and hence
1
2
sin
−
1
2
s
u
i
l
=
(21.14)
γ
H
Comparing Eqs. (21.11) and (21.14), the upper bound solution exactly equals the
lower bound solution and so both must equal the exact solution. Hence the critical
slope angle
i
c
for undrained loading of an infinite slope is given by
1
2
sin
−
1
2
s
u
i
c
=
(21.15)
γ
H
(b) Drained loading - no seepage
Figure 21.11(a) shows a mechanism of plastic collapse for an infinitely long slope
whose angle to the horizontal is an upper bound
i
u
. The mechanism is a single slip
surface at a depth
z
and there is a block of soil length
l
measured down the slope; as
before, the forces
F
1
and
F
2
that act on the vertical sides are equal and opposite. The
displacement diagram for an increment of displacement
w
is shown in Fig. 21.11(b),
where the direction of the increment of displacement makes an angle
δ
ψ
=
φ
c
to the
slip surface.
Figure 21.11
Mechanism of plastic collapse for an infinitely long slope in dry soil.
