Civil Engineering Reference
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modelling. In the abutment model, the effects of all important abutment
parameters, such as strength and stiffness due to back-fi ll soil, abutment
wing wall and back wall, piles, bearing pads, shear keys and gaps, are con-
sidered in the response.
The effects of expansion joints can be considered using gap elements in
the structural models. For example, for the case of the continuous bridges,
the expansion joints are typically situated at the end abutments and can be
considered by means of gap elements in the abutment models. If cap beams
are present, then the cap beams and beam-column joints are designed as
capacity protected elements (design to transmit the maximum expected
resistance of the columns) and therefore are modelled as elastic elements.
The drift capacity at bar buckling damage state can be used to defi ne the
drift at which strength degradation begins (i.e.,
cap
in Fig. 21.2). A study by
Berry and Eberhard (2007) provides some empirical equations to estimate
the drift capacity of bridge columns at different damage states, including
cover spalling and bar buckling, which can be used in structural modelling
and prediction of the capacity of columns. Very few test results are available
to calibrate the post-capping stiffness of bridge columns. Haselton
et al.
(2007) developed an equation to estimate
θ
pc
and recommended a conserva-
tive upper limit of 0.1 for the post-capping chord rotation of columns which
is controlling for most columns designed using modern seismic codes.
Although this equation was developed for columns with rectangular cross-
sections with rectangular ties, it may also be conservatively used for columns
with circular cross-sections containing code conforming spiral reinforce-
ment. This drift ratio can be used to defi ne the post-capping stiffness,
K
c
, as
shown in Fig. 21.2. A summary of different methods to estimate the drift
capacity of bridge columns at different damage states is available in Tehrani
(2012) and Tehrani and Mitchell (2012a).
θ
21.4 Sources of uncertainty
Many sources of uncertainty contribute to total variability in predicted
responses using IDA (ATC-63, 2008):
• Record-to-record (RTR) uncertainty. This uncertainty is due to vari-
ability in the seismic response of structures to different ground motion
records. The variability in response is mainly due to different frequency
content and characteristics of the records, variability in the hazard char-
acterization of the ground motions records, and different duration of
records.
• Design requirements-related (DR) uncertainty. Such uncertainty is
associated with the quality of the design requirements.
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