Civil Engineering Reference
In-Depth Information
where
Θ
=
(
Θ
C
,
Θ
D
) and
(
)
=
(
)
−
(
)
g
x
,,
Q
S
C
x
,
Q
D
x
, ,
Q
S
[19.5]
ik
0
a
ik
0
C
ik
0
D
a
Similarly, the fragilities at time
t
can expressed as
(
)
=
[
(
)
≤
]
Ft SPg
Q
,,
x
, ,,
Q
t S t S
0
,
ik
a
ik
0
a
a
{
}
[19.6]
(
)
=
[
(
)
≤
]
Ft SP
Q
,,
∪∪
g
x
, ,
Q
ttS
,
0
tS
,
a
ik
0
a
a
ik
where
(
)
=
(
)
−
(
)
g
x
,
Q
, ,
t S
C
x
,
Q
,
t
D
x
,
Q
, ,
t S
[19.7]
ik
0
k
a
ik
0
C
ik
0
D
a
In addition, predictive fragility estimates
F
˜
ik
(
t
,
S
a
) and
F
˜
(
t
,
S
a
) (Gardoni
et
al.
2002) can be constructed to incorporate the epistemic uncertainty in the
models parameters
Θ
as
(
)
=
(
)
(
)
∫
FtS
,
F
Q
, ,
tSf
Q Q
d
ik
a
ik
a
Q
[19.8]
(
)
=
∫
(
)
(
)
FtS
,
F
Q
, ,
tS f
Q Q
d
a
a
Q
The predicted fragilities are the expected value of
F
ik
(
Θ
,
t
,
S
a
) or
F
(
Θ
,
t
,
S
a
)
over the distribution of
.
The analyses are carried out using OpenSees. This is a comprehensive,
open-source, object-oriented fi nite element software that has reliability and
response sensitivity capabilities (Haukaas and Der Kiureghian 2005). Choe
et al.
(2008, 2009) and Zhong
et al.
(2012a) extended OpenSees with the
implementation of probabilistic models for corrosion initiation, corrosion
rate, and loss of reinforcement area. In addition, Choe
et al.
(2007) and
Zhong
et al.
(2012b) developed closed-form approximations to the fragility
that do not require any specialized reliability software.
Θ
Reliability of an example RC bridge subject to deterioration
This section shows the results of the methodology described above as it is
applied to the example RC bridge already introduced. The numerical results
of the reliability analysis are taken from Choe
et al.
(2009). Figure 19.5
shows the iso-probability lines of
F
˜
ik
(
t
,
S
a
) and
F
˜
(
t
,
S
a
) as a function of
t
and
S
a
, where the iso-probability lines are contour lines along which
F
˜
ik
(
t
,
S
a
)
and
F
˜
(
t
,
S
a
) have constant values. Figure 19.5(a) shows the iso-probability
lines considering the deformation failure mode only (solid lines) and con-
sidering that either the deformation or the shear demands might exceed
the corresponding capacities (dashed lines). Figure 19.5(b) shows the iso-
probability lines considering the shear failure mode only. It can be observed
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