Civil Engineering Reference
In-Depth Information
limited displacement; sliding planes on which the
strength would be sensitive to movement include
smooth, planar joints or bedding planes with no
infilling. Second, where the slope is a topographic
high point and some amplification of the ground
motions may be expected. In critical situations, it
may also be advisable to check the sensitivity of
the slope to seismic deformations using Newmark
analysis as discussed in Section 6.5.5.
The factor of safety of a plane failure using
the pseudo-static method is given by modifying
equation (6.4) as follows (assuming the slope is
drained,
U
=
V
provided that
k
V
<k
H
(NHI, 1998). Furthermore,
equation (6.32) will only apply when the ver-
tical and horizontal components are exactly in
phase. Based on these results, it may be acceptable
to ignore the vertical component of the ground
motion.
6.5.5 Newmark analysis
When a rock slope is subject to seismic
shaking, failure does not necessarily occur when
the dynamic transient stress reaches the shear
strength of the rock. Furthermore, if the factor of
safety on a potential sliding surface drops below
1.0 at some time during the ground motion it
does not necessarily imply a serious problem.
What really matters is the magnitude of perman-
ent displacement caused at the times that the
factor of safety is less than 1.0 (Lin and Whitman,
1986). The permanent displacement of rock and
soil slopes as the result of earthquake motions
can be calculated using a method developed by
Newmark (1965). This is a more realistic method
of analyzing seismic effects on rock slopes than
the pseudo-static method of analysis.
The principle of Newmark's method is illus-
trated in Figure 6.12 in which it is assumed that
the potential sliding block is a rigid body on a
yielding base. Displacement of a block occurs
when the base is subjected to a uniform horizontal
acceleration pulse of magnitude
ag
and duration
t
0
. The velocity of the block is a function of the
time
t
and is designated
y(t)
, and its velocity at
time
t
is
y
. Assuming a frictional contact between
the block and the base, the velocity of the block
will be
x
, and the relative velocity between the
block and the base will be
u
where
=
0):
cA
+
(W(
cos
ψ
p
−
k
H
sin
ψ
p
))
tan
φ
W(
sin
ψ
p
+
k
H
cos
ψ
p
)
FS
=
(6.30)
The equation demonstrates that the effect of the
horizontal force is to diminish the factor of safety
because the shear resistance is reduced and the
displacing force is increased.
Under circumstances where it is considered that
the vertical component of the ground motion will
be in phase with, and have the same frequency,
as the horizontal component, it may be appro-
priate to use both horizontal and vertical seismic
coefficients in stability analysis. If the vertical
coefficient is
k
V
and the ratio of the vertical to
the horizontal components is
r
k
(i.e.
r
k
=
k
V
/k
H
),
then the resultant seismic coefficient
k
T
is
r
k
)
1
/
2
k
T
=
k
H
(
1
+
(6.31)
acting at an angle
ψ
k
atn
(k
V
/k
H
)
above the
horizontal, and factor of safety is given by
=
cA
+
(W(
cos
ψ
p
−
k
T
sin
(ψ
p
+
ψ
k
)))
tan
φ
u
=
x
−
y
(6.33)
FS
=
W(
sin
ψ
p
+
k
T
cos
(ψ
p
+
ψ
k
))
(6.32)
The resistance to motion is accounted for by the
inertia of the block. The maximum force that
can be used to accelerate the block is the shear-
ing resistance on the base of the block, which
has a friction angle of
φ
◦
. This limiting force
is proportional to the weight of the block
(W)
Study of the effect of the vertical component
on the factor of safety has shown that incor-
porating the vertical component will not change
the factor of safety by more than about 10%,
