Geoscience Reference
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and the shear wave velocity profile of the sliding block was developed using the rela-
tionship that shear wave velocity
is proportional to the fourth-root of the vertical
effective stress. The sliding block height
(
V
s
)
was increased until the specified value of
T
s
wasobtained.Forcommon
T
s
valuesfrom0.2to0.7s,anotherreasonablecombination
of
H
andaverage
V
s
wereusedtoconfirmthattheresultswerenotsignificantlysensitive
totheseparametersindividually.Fornonzero
T
s
values,
H
variedbetween12and100m,
and the average
V
s
was between 200 and 425m/s. Hence, realistic values of the initial
fundamental period and yield coefficient for awiderange of earth/waste fills were used.
(
H
)
5.4. FUNCTIONAL FORMS OF MODEL EQUATIONS
Situations commonly arise where a combination of earthquake loading and slope prop-
erties will result in no significant deformation of an earth/waste system. Consequently,
the finite probability of obtaining negligible (“zero”) displacement should be modeled
as a function of the independent random variables. Thus, during an earthquake, an earth
slope may experience “zero” or finite permanent displacements depending on the char-
acteristicsofthestronggroundmotionandtheslope'sdynamicpropertiesandgeometry.
As discussed in Travasarou and Bray (2003b), seismically induced permanent displace-
ments can be modeled as a mixed random variable, which has a certain probability mass
at zero displacement and a probability density for finite displacement values. Displace-
mentssmallerthan1cmarenotofengineeringsignificanceandcanforpracticalpurposes
be considered as negligible or “zero.” Additionally, the regression of displacement as a
functionofagroundmotionintensitymeasureshouldnotbedictatedbydataatnegligible
levels of seismicdisplacement.
Contrary to a continuous random variable, the mixed random variable can take on dis-
crete outcomes with finite probabilities at certain points on the line as well as outcomes
over one or more continuous intervals with specified probability densities. The values of
seismicdisplacementthataresmallerthan1cmarelumpedto
d
0
=
1cm.Theprobability
density function of seismicdisplacement is then
)
f
D
(
f
D
(
d
)
=˜
p
δ
(
d
−
d
0
)
+
(
1
−˜
p
d
)
(14.3)
where
f
D
(
d
)
isthedisplacementprobabilitydensityfunction;
p
istheprobabilitymassat
˜
istheDiracdeltafunction;and
f
D
(
D
=
d
0
;
δ
(
d
−
d
0
)
d
)
isthedisplacementprobability
density function for
D
d
0
.
The predictive model for seismic displacement consists of two discrete steps. First, the
probabilityofoccurrenceof“zero”displacement(i.e.,
D
>
≤
1cm)iscomputedasafunc-
tion of the primary independent variables
k
y
,
T
s
, and
S
a
(
. The dependence of the
probability of “zero” displacement on the three independent variables is illustrated in
Figure 14.11. The probability of “zero” displacement increases significantly as the yield
coefficientincreases,anddecreasessignificantlyasthegroundmotion'sspectralaccelera-
tionatthedegradedperiodoftheslopeincreases.Theprobabilityof“zero”displacement
decreases initially as the fundamental period increases from zero, because the slope is
1
.
5
T
s
)
