Incremental dynamic analysis (IDA) applied to seismic risk assessment of bridges - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 572

(.) is the cumulative normal distribution function, S a50% is the
median capacity determined from IDA, and

where Φ

β RTR is the record to record

variability in the IDA results.
In the IDA results, only the uncertainty due to record-to-record vari-
ability is considered explicitly. Therefore it is necessary to modify the fragil-
ity curves developed using the IDA results to include the uncertainty due
to other sources of uncertainty as discussed before (i.e, β MDL ).

The modifi ed fragility curve then can be computed using the following
equation:

β DR , β TD and

( )

() −

C

⎡
⎢

ln x ln S ⎤

⎥

a 50 %

( ) =

P failure S = x Φ [21.7]

a

β

TOT

where β TOT is the total uncertainty as defi ned by Eq. [21.1]. While the

median collapse intensity is unchanged, additional uncertainty causes a
large increase in the probability of collapse at the lower IM levels such as
that corresponding to the maximum considered earthquake (MCE).

21.7 Case study for a continuous 4-span bridge
In this section, an application of IDA for the probabilistic seismic perfor-
mance and risk assessment will be provided in more detail for a 4-span
continuous reinforced concrete bridge located in Montreal.

21.7.1 Bridge properties and modelling assumptions
The structure studied includes a regular 4-span bridge supported by single
columns similar to that shown in Fig. 21.4, except that the column heights
and diameters were 7 m and 2 m, respectively. The span length is 50 m and
the superstructure consists of a box girder with a uniform dead load of
200 kN/m. The bridge is assumed to be unrestrained against transverse
movement at the abutments (i.e., it is assumed that the bridge is either
designed for free transverse movements at the ends or the bridge shear keys
at the abutments are expected to fail at high seismic intensity levels). The
bridge was designed based on the 2010 CHBDC provisions (CSA, 2010)
except that the design spectrum given in the 2010 NBCC (NRCC, 2010)
was adopted. The design spectra in the 2010 NBCC correspond to 2% prob-
ability of exceedance in 50 years. The bridges were designed for Montreal
and soil class C (i.e., V S30 555 m/s) with an importance factor I of 1.0. The

concrete compressive strength and the yield stress of the reinforcing bars
were taken as 40 and 400 MPa, respectively. The minimum reinforcement
ratio was controlling in all cases, when either a force modifi cation factor of
R

=

= 3 or R = 5 was used in design, and therefore all bridge columns in this