Civil Engineering Reference
In-Depth Information
7.3 Analysis of wedge failure
The factor of safety of the wedge defined in
Figure 7.3, assuming that sliding is resisted only
by friction and that the friction angle
φ
is the same
for both planes, is given by
the actual factor of safety of the wedge cannot be
determined from the stereonet, because it depends
on the details of the geometry of the wedge, the
shear strength of each plane and water pressure,
as described in the following sections.
The trend
α
i
and plunge
ψ
i
of the line of inter-
section of planes
A
and
B
can be determined on
the stereonet, or calculated using equations (7.1)
and (7.2) as follows:
(R
A
+
R
B
)
tan
φ
W
sin
ψ
i
FS
=
(7.3)
where
R
A
and
R
B
are the normal reactions
provided by planes A and B as illustrated in
Figure 7.4, and the component of the weight act-
ing down the line of intersection is
(W
sin
ψ
i
)
.
The forces
R
A
and
R
B
are found by resolving
them into components normal and parallel to the
direction along the line of intersection as follows:
R
A
sin
β
tan
−
1
tan
ψ
A
cos
α
A
−
tan
ψ
B
cos
α
B
α
i
=
tan
ψ
B
sin
α
B
−
tan
ψ
A
sin
α
A
(7.1)
ψ
i
=
tan
ψ
A
cos
(α
A
−
α
i
)
=
tan
ψ
B
cos
(α
B
−
α
i
)
(7.2)
2
ξ
R
B
sin
β
2
ξ
1
1
−
=
+
(7.4)
where
α
A
and
α
B
are the dip directions, and
ψ
A
and
ψ
B
are the dips of the two planes.
Equation (7.1) gives two solutions 180
◦
apart; the
correct value lies between
α
A
and
α
B
.
R
A
cos
β
2
ξ
R
B
cos
β
2
ξ
1
1
−
+
+
=
W
cos
ψ
i
(7.5)
(a)
½
(b)
N
Plane B
Plane A
½
R
A
R
B
Face
W
cos
i
Direction of
sliding
(c)
i
W
cos
i
W
W
sin
i
Figure 7.4
Resolution of forces to calculate factor of safety of wedge: (a) view of wedge looking at face
showing definition of angles
β
and
ξ
, and reactions on sliding planes
R
A
and
R
B
; (b) stereonet showing
measurement of angles
β
and
ξ
; (c) cross-section of wedge showing resolution of wedge weight
W
.
