Geoscience Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
0.01
0.1
1
0
0.5
1
1.5
2
0.01
0.1
1
10
Yield Coefficent
Fundamental Period (s)
Sa(1.5T s ) (g)
(a)
(b)
(c)
)
onthe (a)yield coefficient, (b) initialfundamental period, and (c) spectral acceleration
at 1.5 timesthe initialfundamental period (Bray and Travasarou, 2007)
Fig. 14.11. Dependence of theprobability ofnegligible displacement
(
D
<
1cm
being brought near to the mean period of most ground motions. However, this proba-
bility increases sharply as the slope's period continues to increase as it is now moving
awayfromtheresonancecondition.Aprobitregressionmodelwasusedforthisanalysis
(Green, 2003), and the selection of the functional form for modeling the probability of
occurrence of “zero” displacement was guided by thetrends shown inFigure 14.11.
In the case where a non-negligible probability of “nonzero” displacement is calculated,
the amount of “nonzero” displacement needs to be estimated. A truncated regression
model was used as described in Green (2003) to capture the distribution of seismic dis-
placement,giventhat“nonzero”displacementhasoccurred.Theestimationofthevalues
of the model coefficients was performed using theprinciple of maximum likelihood.
5.5. EQUATIONS FOR ESTIMATING SEISMIC DEVIATORIC DISPLACEMENTS
As mentioned, the model for estimating seismic displacement consists of two discrete
computationsof:(1)theprobabilityofnegligible(“zero”)displacementand(2)thelikely
amount of “nonzero” displacement. The model for computing the probability of “zero”
displacement is
P
(
D
=
“0”
) =
1
1
.
76
3
.
22ln
(
k y )
5 T s ))
0
.
484
(
T s )
ln
(
k y ) +
3
.
52ln
(
S a (
1
.
(14.4)
where P
is the probability (as a decimal number) of occurrence of “zero”
displacements, D istheseismicdisplacementintheunitsofcm,
(
D
=
“0”
)
isthestandardnormal
cumulativedistributionfunction(i.e.,NORMSDISTinExcel), k y istheyieldcoefficient,
T s is the initial fundamental period of the sliding mass in seconds, and S a (
1
.
5 T s )
is the
spectral acceleration of the input ground motion at aperiod of 1
5 T s inthe units of g .
This first step can be thought of as a screening analysis. If there is a high probability
of “zero” displacements, the system performance can be assessed to be satisfactory for
.
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