Civil Engineering Reference
In-Depth Information
U
1
,
U
2
and
V
are calculated by integrating the
pressures over the areas of planes 1, 2 and 5,
respectively.
7.6.2 Computer programs for comprehensive
analysis
Appendix III lists equations that can be used to
carry out a comprehensive stability analysis of
a wedge using the input parameters discussed
in Section 7.6.1. These equations were origin-
ally developed by Dr John Bray and are included
in the third edition of
Rock Slope Engineering
(1981). This method of analysis has been used in
a number of computer programs that allow rapid
and reliable analysis of wedge stability. However,
there is a limitation to this analysis that should
be noted:
•
Shear strengths
—the slide planes (1 and 2) can
have different shear strengths defined by the
cohesion
(c)
and friction angle
(φ)
. The shear
resistance is calculated by multiplying the
cohesion by the area of slide plane, and adding
the product of the effective normal stress and
the friction angle. The normal stresses are
found by resolving the wedge weight in dir-
ections normal to the each slide plane (see
equations (7.3)-(7.5)).
•
External forces
—external forces acting on the
wedge are defined by their magnitude and
orientation (plunge
ψ
and trend
α
). The equa-
tions listed in Appendix III can accommodate
a total of two external forces; if there are
three or more forces, the vectors are added
as necessary. One external force that may
be included in the analysis is the pseudo-
static force used to simulate seismic ground
motion (see Section 6.5.4). The horizontal
component of this force would act in the
same direction as the line of intersection of
planes 1 and 2.
Wedge geometry.
The analysis procedure is
to calculate the dimensions of a wedge that
extends from the face to the point where planes
1, 2 and 3 intersect. The next step is to calculate
the dimensions of a second wedge formed by slid-
ing planes 1 and 2, the upper slope (plane 3) and
the tension crack (plane 5). The dimensions of the
wedge in front of the tension crack are then found
by subtracting the dimensions of the wedge in
front of the tension crack from the overall wedge
(see equations (III.54) to (III.57), Appendix III).
Prior to performing the subtractions, the program
tests to see if a wedge is formed and a ten-
sion crack is valid (equations (III.48) to (III.53),
Appendix III). The program will terminate if the
dip of the upper slope (plane 3) is greater than the
dip of the line of intersection of planes 1 and 2,
or if the tension crack is located beyond the point
where the point where planes 1, 2 and 3 intersect.
While these tests are mathematically valid, they
do not allow for a common geometric condition
that may exist in steep mountainous terrain. That
is, if plane 3 is steeper than the line of inter-
section of planes 1 and 2, a wedge made up
of five planes can still be formed if a tension
crack is located behind the slope face to create
a valid plane 5. Where this physical condition
exists in the field, an alternative method of ana-
lysis is to use Key Block Theory in which the shape
and stability condition of removal wedges can
be completely defined (Goodman and Shi, 1985;
Kielhorn, 1999; PanTechnica, 2002).
•
Bolting forces
—if tensioned anchors are
installed to stabilize the wedge, they are con-
sidered to be an external force. The ori ent-
ation of the anchors can be optimized to
minimize the anchor force required to pro-
duce a specified factor of safety. The optimum
anchor plunge
ψ
T
(
opt
)
and trend
α
T
(
opt
)
, with
respect to the line of intersection
(ψ
i
/α
i
)
, are
as follows (Figure 7.18(c)):
ψ
T
(
opt
)
=
(φ
average
−
ψ
i
)
(7.15)
and
α
T
(
opt
)
=
(
180
+
α
i
)
for
α
T
(
opt
)
≤
360
(7.16)
where
φ
average
is the average friction angle of
the two slide planes.
