Civil Engineering Reference
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T
1
= 1.09 s
0.1
Hazard curve
Fitted curve
0.01
-1.35
y
= 2.59e-0.5
x
0.001
2
= 0.997
R
0.0001
0.00001
0.01
0.1
1
S
a
(
T
1
)
21.11
Seismic hazard curve for Montreal, site class C and
T
=
1.09 s.
where
f
R
(
IM
) is the probability density function of capacity in terms of IM
and
H
(
IM
) is the conventional hazard curve. This can be numerically com-
puted by Eq. [21.9] using the fragility and hazard curves.
F
R
is the fragility
function developed for a damage state (e.g., cover-spalling or collapse) and
spectral acceleration,
S
a
,
is used as intensity measure, IM.
∑
[
(
(
)
]
()
λ
DS
=
FS
−
FS
HS
[21.9]
R
a
R
a
a
i
i
−
1
i
All
a
S
i
On the other hand, if some reasonable approximations are made, Eq. [21.8]
can be analytically integrated assuming that the IM values of capacity are
lognormally distributed and that the IM hazard curve can be approximated
by fi tting a straight line in the log-log space,
H
(
IM
)
k
0
IM
−
k
, (Shome and
Cornell, 1999; Cornell
et al.
2002). Based on these assumptions the integra-
tion in Eq. [21.8] will result in Eq. [21.10]:
=
1
2
⎡
⎢
⎤
⎥
(
)
⋅
(
)
2
C
λλ
=
S
exp
k
β
[21.10]
DS
a
50
%
TOT
Therefore, the mean annual rate of exceeding a damage state,
λ
DS
, can be
estimated using Eq. [21.10] given the median capacity of the structure at
the damage state, S
C
a50%
and the total uncertainty,
β
TOT
, which are determined
through IDA. The values of
k
0
and
k
in Fig. 21.11 are estimated as 2.59e-5
and 1.35, respectively, at
T
1
1.09 s. The probability of exceeding a damage
state in the next
T
years can then be estimated using the Poisson distribu-
tion, given in Eq. [21.11].
=
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