Geoscience Reference
In-Depth Information
Ifthe losses,however, are correlated, thevariance is given by
n
E
[
L
total
]
=
E
[
L
i
]
(19.11a)
i
=
1
⎡
⎤
n
n
n
⎣
⎦
2
2
σ
L
total
=
1
σ
L
i
+
ρ
L
i
L
j
σ
L
i
σ
L
j
(19.11b)
i
=
i
=
1
j
=
1
j
=
i
where
ρ
ij
is the correlation between loss
L
i
at sites
i
and loss
L
j
at site
j
within the
system.
The challenge in evaluating eq. (19.11) is in estimating the correlations
ρ
ij
. Recent
researchbyLeeandKiremidjian(2006)hasdemonstratedthatthelossesatpairsofbridge
sites are correlated through ground motion and bridge damage. In the following subsec-
tions we briefly summarize their results.
2.4. GROUND MOTION CORRELATION
In their formulation, pairs of ground motion are modeled as jointly normally distrib-
uted random variables with unit median conditioned on the magnitude and distance for
that earthquake, and covariance matrix
{
Li
,
Lj
}
defined in terms of the earthquake error
2
s
, and uncorrelated residual error
r
aris-
ε
l
, distance dependent correlated site error
ε
ε
ing from the attenuation model for the study region. The error terms
are
assumed to be mutually uncorrelated zero-mean normally distributed random variables.
With these assumptions, Lee and Kiremidjian provide the following formulation for the
ground motion correlation forall pairs of siteswhen
i
{
ε
l
,
ε
s
,
ε
r
}
=
j
:
v
ε
i
,ε
j
2
s
e
−
(
)
e
r
ij
/
r
o
Co
Var
ε
j
=
σ
+
σ
ρ
ε
i
,ε
j
=
(19.12)
√
Var
σ
e
+
σ
r
+
σ
s
(ε
i
)
As can be seen from eq. (19.12), the correlation decays with distance where
r
0
is the
standardized distance. The standardized distance
r
0
represents the distance below which
the correlation becomes 1. As the distance
r
ij
between sites
i
and
j
increases above the
value of
r
0
, theexponential termin eq. (19.12) approaches tozero.
2.5. DAMAGECORRELATION
Correlation of damage between bridges of similar designs, material properties, construc-
tion methods and site characteristics can be expected to be relatively high when these
bridges are subjected to ground motions from the same earthquake. In most applica-
tions, bridges are grouped by engineering bridge classes (HAZUS, 2000). Correlation of
