Civil Engineering Reference
In-Depth Information
estimates
C
ik
(
x
0
,
Θ
D
,
t
,
S
a
). To avoid conducting a reliability
analysis at each time
t
, Gardoni and Rosowsky (2011) developed seismic
fragility increment functions
,
Ĝ
F
,
k
(
t
) and
Ĝ
F
(
t
), for single column RC bridges.
In this case, the subscript
i
Θ
C
,
t
) and
D
ik
(
x
0
,
1 and is therefore omitted for simplicity in the
notation. The fragility increment functions provide estimates of the fragili-
ties at time
t
, given the fragilities at
t
=
0,
F
˜
k
(
t
,
S
a
) and
F
˜
(0,
S
a
), without
requiring any additional reliability analysis. The
Ĝ
F
,
k
(
t
) and
Ĝ
F
(
t
) are con-
structed such that estimates of the fragilities at time
t
can be obtained as
=
ˆ
ˆ
(
)
=
(
)
×
(
)
FtS
,
G
x
,
L
, ,
tS
F
0
,
S
k
a
F k
,
0
k
a
k
a
[19.9]
ˆ
ˆ
F
a
(
)
=
(
)
×
(
)
FtS
,
G
x
,,,
L
tS
0,
a
F
0
a
The fragility increment functions are functions of the environmental condi-
tions, the original material properties, a set of unknown model parameters,
Λ
,
t
, and
S
a
. The Akaike Information Criterion (AIC) (Akaike, 1978a,b)
and the Bayesian Information Criterion (BIC) (Schwarz, 1978) were used
to select accurate and parsimonious model forms for
Ĝ
F
,
k
(
t
) and
Ĝ
F
(
t
), and
the Bayesian updating rule was used to estimate the unknown model
parameters. The model selection and parameter estimation were carried out
using fragility data obtained by conducting a predictive reliability analysis
as described in Eq. 19.8 (Gardoni
et al.
2002) using probabilistic capacity
and demand models for corroding RC columns in the form shown in Eqs.
19.1 and 19.2 and calibrated by Choe
et al.
(2008, 2009).
The developed
Ĝ
F
,
k
(
t
) and
Ĝ
F
(
t
) are applicable within a wide range of
environmental conditions and material properties. The fragility increment
functions capture the effects of deterioration on the mean capacities and
mean demands, and of the increasing uncertainty over time in the probabi-
listic models for the corrosion initiation and the time-dependent corrosion
rate as it is refl ected in the probabilistic capacity and demand models used
to generate the fragility data.
As an example, Gardoni and Rosowsky (2011) used the developed
Ĝ
F
,
k
(
t
)
and
Ĝ
F
(
t
) to estimate the fragilities for deformation (
k
k
or
Λ
v
),
and combined deformation-shear failure modes of the example bridge
discussed in the previous section. Figure 19.7 shows the fragility estimates
for the example bridge versus
S
a
. In each subplot, the dashed line indicates
F
˜
k
(0,
S
a
) or
F
˜
(0,
S
a
), obtained by a reliability analysis following Gardoni
et al.
(2003), the dotted line indicates the predictive fragility
F
˜
k
(
t
=
d), shear (
k
=
=
150 years,
S
a
) or
F
˜
(
t
150 years,
S
a
) obtained by a reliability analysis following Choe
et al.
(2009) as discussed in the previous section, and the solid line indicates
F
ˆ
k
(
t
=
150 years,
S
a
) or
F
ˆ
k
(
t
150 years,
S
a
) obtained using the fragility incre-
ment functions using Eq. 19.9.
Overall the estimates obtained using the
fragility increment functions are in close agreement with the predictive
fragility estimates obtained from a traditional reliability analysis and only
=
=
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