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for selection. These parameters may be related to the global (e.g. drift ratio),
intermediate (e.g. moments, deformations at the bearings, and column cur-
vature ductility), or local (e.g. stresses) elements of the structure.
During the SAC project performed on steel moment frame buildings
(FEMA, 2000), PSDMs were defi ned as lognormally distributed (Cornell
et al.
, 2002). Therefore, the relation between EDPs and IMs can be pre-
sented with a linear dependence in log-log space. The recommended func-
tional form of the PSDMs is:
(
)
=×
(
)
++
ln
EDP
c
ln
IM
c
σ
PSDM
[20.8]
1
2
where
c
1
and
c
2
are the regression coeffi cients and
σ
PSDM
is the standard
deviation of the model. After EDPs and IMs are selected, regression analy-
ses are performed for each EDP-IM pair to estimate the regression coef-
fi cients
c
1
and
c
2
. Better predicting models (more effi cient models) are
selected according to the dispersion measures for the fi t (Mackie and Sto-
jadinovic, 2003). Recent studies (Nielson and DesRoches, 2007; Padgett
et al.
, 2008) found that the PGA is the most appropriate IM for highway
bridges and the PSDMs coupling several EDPs with the horizontal PGA
were presented.
20.4 Vector-valued probabilistic seismic hazard
assessment (VPSHA)
In addition to the common IMs (PGA and
S
a
(
T
1
)), more complex param-
eters like vector-based IMs have been examined by past researchers (Shome
and Cornell, 1999; Bazurro and Cornell, 2002). According to Baker and
Cornell (2005), using only
S
a
as the IM is not adequate to characterize the
ground motion intensity since there is a large variability in the structural
response due to different ground motions with the same
S
a
at a particular
period. The reason for that variability is the spectral shape; using
S
a
(
T
1
) as
IM does not take
S
a
values at other periods into account.
S
a
at other periods
has an impact on inelastic structures (due to period lengthening) and multi-
degree-of-freedom systems with multiple modal periods. Baker and Cornell
(2005) proposed
as a proxy for spectral shape and recommended the use
of vector-based IM, IM(
S
a
,
ε
ε
), coupling
S
a
(
T
1
) at the fundamental period of
the structure with
. If an EDP depends on two different IMs (IM
1
and IM
2
),
the rate of exceeding a specifi c value of EDP can be computed using the
conditional distribution of EDP given IM
1
and IM
2
. Therefore Equation
20.6 becomes:
ε
(
)
=
∫
(
)
ν
EDP
>
y IM
,
IM
f
EDP
>
y IM
,
IM
[20.9]
1
2
EDP
1
2
IM
2
(
)
(
)
×
f
IMIM
d
ν
IM
IM
1
1
2
2
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