Civil Engineering Reference
In-Depth Information
solving equation (4.14) for a range of minor prin-
cipal stress values defined by
σ
t
<σ
3
<σ
3 max
,
as illustrated in Figure 4.22. The fitting process
involves balancing the areas above and below
the Mohr-Coulomb plot. This results in the fol-
lowing equations for the angle of friction
φ
that boundary. The failure propagates from this
initiation point into the biaxial stress field and
it eventually stabilizes when the local strength,
defined by equation (4.14), is higher than the
induced stresses
σ
1
and
σ
3
. Most numerical mod-
els can follow this process of fracture propagation
and this level of detailed analysis is very import-
ant when considering the stability of excavations
in rock and designing support systems.
However, for slope stability, failure is ini-
tiated along a sliding surface within the slope
where the rock is subject to a biaxial stress field
and it is useful to consider the overall behavior
of a rock mass rather than the detailed failure
propagation process described earlier. This leads
to the concept of a global “rock mass strength”
and Hoek and Brown (1997) proposed that this
could be estimated from the Mohr-Coulomb
relationship:
and
cohesive strength
c
(Figure 4.23):
sin
−
1
6
am
b
(s
+
m
b
σ
3n
)
a
−
1
φ
=
6
am
b
(s
+
m
b
σ
3n
)
a
−
1
+
a)(
2
+
a)
+
2
(
1
(4.24)
c
=
(σ
ci
(
1
−
a)m
b
σ
3n
(s
+
m
b
σ
3n
)
a
−
1
)
+
2
a)s
+
(
1
(
1
+
a)(
2
+
a)
+
a))
(4.25)
1
+
(
6
am
b
(s
+
m
b
σ
3n
)
a
−
1
)/((
1
×
+
a)(
2
σ
3 max
/σ
ci
.
Note that the value of
σ
3 max
, the upper
limit of confining stress over which the rela-
tionship between the Hoek-Brown and the
Mohr-Coulomb criteria is considered, has to be
determined for each individual case. Guidelines
for selecting these values for slopes are presented
in Section 4.5.5.
The Mohr-Coulomb shear strength
τ
, for a
given normal stress
σ
, is found by substitution
of these values of
c
where
σ
3n
=
2
c
cos
φ
1
σ
cm
=
(4.28)
sin
φ
−
with
c
and
φ
determined for the stress range
σ
t
<
σ
3
<σ
ci
/
4 giving the following value for the rock
mass strength
σ
cm
:
+
s)
a
−
1
σ
cm
=
σ
ci
(m
b
+
4
s
−
a(m
b
−
8
s))(m
b
/
4
2
(
1
+
a)(
2
+
a)
(4.29)
and
φ
into the following
equation:
4.5.5 Determination of
σ
3 max
The issue of determining the appropriate value
of
σ
3 max
for use in equations (4.24) and (4.25)
depends upon the specific application. For the
case of slopes, it is necessary that the calculated
factor of safety and the shape and location
of the failure surface be equivalent. Stability
studies of rock slopes using Bishop's circular
failure analysis for a wide range of slope geo-
metries and rock mass properties have been
carried out for both Generalized Hoek-Brown
and Mohr-Coulomb criteria to find the value
of
σ
3 max
that gives equivalent characteristic
curves. These analyses gave the following rela-
tionship between
σ
3 max
τ
=
c
+
σ
tan
φ
(4.26)
The equivalent plot, in terms of the major and
minor principal stresses, is defined by
2
c
cos
φ
1
sin
φ
1
+
σ
1
=
σ
3
sin
φ
+
(4.27)
−
1
−
sin
φ
4.5.4 Rock mass strength
The uniaxial compressive strength of the rock
mass
σ
c
is given by equation (4.18). For under-
ground excavations, instability initiates at the
boundary of the excavation when the compressive
strength
σ
c
is exceeded by the stress induced on
the rock mass strength
