Civil Engineering Reference
In-Depth Information
line of intersection, as shown on Figure 2.18(b),
is wider than the envelope for plane failures.
The wedge daylight envelope is the locus of all
poles representing lines of intersection whose dip
directions lie in the plane of the slope face.
(a)
Weight
vector
Normal to
plane
Friction
cone,
= 35
°
2.6.4 Toppling failure
For a toppling failure to occur, the dip direction
of the discontinuities dipping into the face must
be within about 10
◦
of the dip direction of the face
so that a series of slabs are formed parallel to the
face. Also, the dip of the planes must be steep
enough for interlayer slip to occur. If the faces of
the layers have a friction angle
φ
j
, then slip will
only occur if the direction of the applied com-
pressive stress is at angle greater than
φ
j
with the
normal to the layers. The direction of the major
principal stress in the cut is parallel to the face
of the cut (dip angle
ψ
f
), so interlayer slip and
toppling failure will occur on planes with dip
ψ
p
when the following conditions are met (Goodman
and Bray, 1976):
f
=30
°
N
f
=80
°
(b)
f
=60
°
10
°
10
°
(
90
◦
−
ψ
f
)
+
φ
j
<ψ
p
(2.3)
These conditions on the dip and dip direction
of planes that can develop toppling failures are
defined on Figure 2.18(b). The envelope defining
the orientation of these planes lies at the opposite
side of the stereonet from the sliding envelopes.
Friction
cone
Legend
Envelopes of potential instability:
Wedges;
Plane failures;
Toppling failures;
Envelopes for
°;
Envelopes for
f
= 60
°.
f
= 80
2.6.5 Friction cone
Having determined from the daylight envelopes
whether a block in the slope is kinematically per-
missible, it is also possible to examine stability
conditions on the same stereonet. This analysis
is carried out assuming that the shear strength of
the sliding surface comprises only friction and the
cohesion is zero. Consider a block at rest on an
inclined plane with a friction angle of
φ
between
the block and the plane (Figure 2.19(a)). For an
at-rest condition, the force vector normal to the
plane must lie within the friction cone. When the
only force acting on the block is gravity, the pole
to the plane is in the same direction as the normal
Figure 2.19
Combined kinematics and simple stability
analysis using friction cone concept: (a) friction cone
in relation to block at rest on an inclined plane (i.e.
φ>ψ
p
); and (b) stereographic projection of friction
cone superimposed on “daylighting” envelopes.
force, so the block will be stable when the pole
lies within the friction circle.
The envelopes on Figure 2.19(b) show the
possible positions of poles that may form unstable
