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Figure 6 for the bridge with medium span length
and column height of 10.0 m. The review of the
drift data at each time step indicates that a log-
normal distribution can be fit very well to the
DDR response. It can be seen that both the me-
dian and standard deviation of the obtained dis-
tributions increase as the corrosion progresses
and the scatter plot tends to the higher DDRs.
this study, the damage limit states are assumed to
equal the ductility of 1.0, 2.0, 4.0, and 7.0 for the
slight, moderate, extensive, and complete damage
states, respectively. The estimation of these limit
states are beyond the scope of this chapter, but
the suggested values are in accordance with the
limit states available in the literature for similar
bridges (Hwang et al., 2000, Choi et al., 2004,
and Yang et al., 2009).
Under a ground motion excitation with the
peak ground acceleration of
PGA
i
(here i = 1,...,
60), a bridge sustains failure in a specific dam-
age state if its ductility is larger than the ductility
corresponding to that damage state. Depending
on whether or not the bridge sustains the state
of damage under different ground motions, the
parameters of each fragility curve (i.e., median,
c
k
and log-standard deviation,
ζ
k
) are estimated
using the maximum likelihood procedure given
in Shinozuka et al. (2000b). For the
k
-th damage
state (
k
= 1, 2, 3, and 4), the fragility curve is
developed following the formula below:
5. PROBABILISTIC LIFE-TIME
FRAGILITY ANALYSIS
Fragility analysis is considered as a powerful
tool for the probabilistic seismic risk assessment
of highway bridges. Through this analysis, a set
of fragility curves is developed to estimate the
conditional probability statements of the bridge
vulnerability as a function of ground motion in-
tensity measure. The damageability of the bridge
can be assessed by expert opinions (ATC, 1985),
empirical data from past earthquakes (Basoz and
Kiremidjian, 1999 and Shinozuka et al., 2000a),
and analytical methods (Mander and Basoz, 1999
and Shinozuka et al., 2000b). The current study
uses the later approach and defines four limit states
of damage. The definitions of damage states are
derived from HAZUS-MH (2007) and can be sum-
marized as: (at least) slight,
E
1
, (at least) moderate,
E
2
, (at least) extensive,
E
3
, and complete damage,
E
4
. Based on these damage states, the analytical
fragility curves of the bridge cases are generated
at different ages after the corrosion initiation time.
To perform fragility analysis, the column
curvature ductility is taken here as the primary
damage measure. The curvature ductility is
defined as the ratio of maximum column cur-
vature recorded from a nonlinear time-history
analysis to the column yield curvature obtained
from moment-curvature analysis. Following the
procedure given by Priestley et al. (1996), the
curvature ductility values of all the bridge cases
are calculated under the set of 60 ground motions
and then compared with damage limit states. In
ln(
PGA c
/
)
F PGA
(
|
ζ
,
c
)
=
Φ
i
k
(12)
k
i
k
k
ζ
k
where
F
k
is the probability of exceeding the
damage state of
k
and Φ[.] is the standard normal
distribution function. The fragility curves of the
intact two-span bridges with the column height
of 10.0 m, having a range of short, medium, and
long span lengths, are illustrated in Figure 7.
Additionally, the estimated median values (
c
k
) of
the fragility curves developed for all the bridges
under study are summarized in Table 4 for the
four damage states considering the intact bridge
conditions before the corrosion initiation time. For
the log-standard deviation (
ζ
k
), it is seen that dif-
ferent deviation values may result in intersecting
the fragility curves of different damage states. To
avoid any intersection, Shinozuka et al. (2000b)
suggest considering one common deviation value
for all the damage states. In this study, since the
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