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the probabilistic seismic demand model (PSDM) approach where the
results of nonlinear dynamic analyses for a specifi c structure are used to
evaluate behavior of important EDPs in terms of the IM levels. The main
idea of these studies was to develop PSDMs for particular structures and
to provide the annual frequency of exceeding a given structural engineering
demand measure
y
conditioned on IM, as:
(
)
=
∫
(
)
×
(
)
[20.6]
ν
EDP
>
y IM
f
EDP IM
d
ν
IM
EDP
IM
where
f
ED
P
(
EDP
|
IM
) is the probabilistic seismic demand model for a
particular EDP and IM. PSDMs represent the connection between IMs and
EDPs as shown in Fig. 20.1 and provide information about the probability
of exceeding the pre-determined critical levels of EDPs for a particular
class of structures. These models may be used as risk-based design tools,
since they present the variability in the structural demand parameters for
specifi ed ground motion intensities. Moreover, when coupled with PSHA,
PSDMs can be used to compute structural demand hazard curves (Mackie
and Stojadinovic, 2003). PSDMs may be incorporated into the hazard inte-
gral to directly estimate the annual probability of exceeding a certain EDP
(Gülerce and Abrahamson, 2010). For an EDP that depends on only a scalar
ground motion IM, the hazard integral for the EDP can be written as:
⎧
⎪
⎪
∫
∫
∫
()(
)()
⎫
⎪
⎪
N
⋅
f
Mf
,
R f
ε
min
M
R
ε
(
)
=
ν
EDP
>
y
[20.7]
M
R
ε
(
)
ˆ
[
]
(
)
P EDP
>
y EDP IM M R
,, ,
ε
σ
ddd
MR
ε
ln
EDP
where
ED
ˆ
P
(
IM
(
M
,
R
,
lnEDP
is the standard
deviation of ln(EDP) for a given IM (EDP is modeled as lognormal variate).
This approach combines the site-specifi c ground motion hazard with the
structural responses of interest from nonlinear dynamic analyses of the
given structure. The fi nal result of the hazard integral given in Equation
20.7 is a structural demand hazard curve representing the annual probabil-
ity of exceeding a specifi ed value of EDP.
ε
)) is the median EDP and
σ
20.2.3 Damage measure
The last step of the PEER-PBEE approach is to evaluate the distribution
of DMs with respect to the EDP levels. In current practice and building
codes, damage measures are typically not continuous, rather a set of discrete
damage states (FEMA-273, 1997; Miranda and Aslani, 2003). To assess the
extent of incurred damage, fragility functions are assigned for discrete
damage states, which provide the probability of exceeding a damage state
for a given EDP level (represented by the third chain in Fig. 20.1). With
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