Environmental Engineering Reference
In-Depth Information
Two different definitions of the factor of safety against wedge falling have been reported
in the literature, each with its own rationale. The first appeared in Sofianos et al. (1999) and
Brady and Brown (2006), for example. It is the ratio of the pull-out resistance of the wedge
to the weight of the wedge and was expressed as follows:
2
MH
W
0
(9.9)
FS
=
,
where
Mf
=
( ,, ,
φα
i
kk
)
and
H
=
fpRK hR
(, ,
,
/
)
1
s
n
0
0
In the above equations,
W
is the weight of the wedge, α is the semiapical angle of the
wedge, ϕ and
i
are the effective friction and dilation angles of the joints,
k
s
and
k
n
are the
shear stiffness and normal stiffness of the joints,
R
is the radius of the tunnel,
p
and
K
0
are
the vertical
in situ
stress and the coefficient of the horizontal
in situ
stress,
h
is the clear height
of the wedge (measured from the tunnel crown), and θ is the angle denoted in
Figure 9.14
.
The second definition is similar in principle to that which has been long and widely used
in soil and rock slope stability analysis, in the
Unwedge
program of
Rocscience.com
, and
in Asadollahi and Tonon (2010), for example. It is the ratio of the available shear strength
to the shear strength required for equilibrium. In this context of tunnel roof wedge, this
definition was given in Asadollahi and Tonon (2010) as follows (assuming the dilation angle
of the joints
i
= 0):
S
NW
2
cos
sin
α
(9.10)
FS
=
,
where
Nf Hk
=
(
,, ,,)
φα
k
and
S
=
fH
(
,, ,,
φα
k
k
n
)
2
0
s
n
0
s
2
α
+
where
N
and
S
are normal and shear forces, and other symbols as defined for
Equation 9.9.
Low and Einstein (2013) showed that the two definitions,
Equations 9.9
and
9.10,
can be
recast in terms of
N
,
W
, α, and ϕ, as follows:
Limiting wedgeweight
Actual wedgeweight
2
N
tancos
φα α
W
−
2
N
sin
tan an
(
φα
α
/
−
1
FS
=
=
=
(9.11)
1
W
/
2
N
in
)
Maximum availableresisting forces
Downward drivingforces
2
N
NWWN
tancos
sin
φα
α
tan an
(
φα
α
FS
2
=
=
=
(9.12)
2
+
1
+
/
2
in
)
The “Limiting wedge weight” in
Equation 9.11
means the wedge weight at limiting equi-
librium, that is, the wedge weight that just causes failure. It is negative if ϕ < α.
similarities, and differences between
FS
1
and
FS
2
are rendered much more transparent in
9.11
—which is mathematically equivalent to
Equation 9.9
—can be very large and positive
if
W/N
is small and ϕ > α, and negative if ϕ < α.
The two definitions as given by
Equations 9.11
and
9.12
,
and hence
Equations 9.9
and
where the
FS
1
contours (solid lines) of 1, 10, 20, and 30 are shown together with the
FS
2
contours (dashed lines) of 1.0, 1.15, 1.30, 1.48, and 1.60. The
FS
1
= 1.0 and
FS
2
= 1.0 con-
tours coincide perfectly. For the input values given in the figure caption, the factors of safety
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