Environmental Engineering Reference
In-Depth Information
where
s
a
is the actual settlement for a future case. To avoid negative values in settlement, it
is reasonable to assume that α is lognormally distributed.
brated can be denoted as θ = {μ
α
, σ
α
}. Assuming that α is independent from
I
i
, and thus is
independent from
s
p
, then the mean and variance of
s
a
can be determined, respectively, as
follows (Ang and Tang 2007):
µ
=
Es
[]
= =⋅
Es
[
α µµ
α
]
(4.37)
a
a
p
p
2
2
2
2
2
2
2
and
σ
=
ar
[]
s
= =⋅ +⋅ +⋅
Var
[
α µσσµσσ
α
s
]
(4.38)
a
a
p
p
α
p
α
p
The COV of
s
a
, denoted as δ
a
, can then be computed as
(
)
0.
2
2
2
2
2
2
(4.39)
δσµµσσµσσ
=
/
=
⋅ +⋅ +⋅
/
(
µ
⋅
µ
)
a
a
a
α
p
α
p
α
p
α
p
4.5.2 Calibration database
To calibrate the model bias factor α, a database consisting of 64 case histories was compiled
and summarized in Juang et al. (2013). Among the 64 cases, 21 were obtained from the 1989
Loma Prieta, California, earthquake; 19 case histories were obtained from the 1999 Kocaeli,
Turkey, earthquake; and 24 case histories were from the 1999 Chi-Chi, Taiwan, earthquake.
The database consists of 32 liquefaction-induced free-field settlement observations and 32
liquefaction-induced building settlement observations. The settlement behaviors of the two
groups of observations are different because the building settlement observations are compli-
cated by the soil-structure interaction. Interested readers are referred to Juang et al. (2013)
for further discussions. For our purposes here, the 32 observations on liquefaction-induced
free-field settlement, the details of which are summarized in
Table 4.7,
were used for our
model calibration. In this database, observations of the post-liquefaction ground settlement
may be categorized as either a fixed value or a range. How these two types of data can be
used together for model calibration will be explained in the following section.
4.5.3 Maximum likelihood estimation of
statistics of model bias factor
Consider a general situation where the database consists of
m
+
n
case histories of lique-
faction-induced settlement, where
m
is the number of cases with a fixed-value settlement
observation and
n
is the number of cases in which the settlement encompasses a range of
values. Thus, in a special situation when
n
= 0, the database will consist of only cases with
fixed-value observations; and when
m
= 0, the database will consist of only cases with range
observations. In this section, we deal with a database of both fixed-value observations and
range observations. In the following, we will use
i
and
j
as indexes for a fixed-value observa-
tion and a range observation, respectively. For a fixed-value observation,
d
i
= {
s
i
}. Assuming
s
i
follows lognormal distribution, the chance to observe a fixed-value observation
s
i
is
2
1
1
2
ln
s
λ
−
i
i
f
(|)
d
θ
=
f s
(
|
θ
)
=
exp
−
(4.40)
i
i
ξ
2
πξ
s
i
ii
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