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.
The same FS 1 is obtained whether computed from Equation 9.9 or 9.11 , and the same
FS 2 is obtained whether computed from Equation 9.10 or 9.12. Nevertheless, the rationales,
similarities, and differences between FS 1 and FS 2 are rendered much more transparent in
Equations 9.11 and 9.12 than in Equations 9.9 and 9.10. That FS 1 can be negative when
ϕ < α is also readily appreciated from Equation 9.11 . One may note that FS 1 by Equation
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
9.10 , are mathematically equivalent when FS 1 = FS 2 = 1. This is shown in Figure 9.15a ,
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
at the mean-value point α = 25° and ϕ = 35° are FS 1 = 30.1 and FS 2 = 1.48, but Figure 9.15a
 
 
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