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of axial forces were not signifi cant in the bridges considered in this study.
The effects of axial forces on the seismic response are expected to be more
important for the case of bridges with multi-column bents or bridges with
shorter span lengths. In the 3D analyses as the moment strength or yield
strength of column degrades, the stiffness also degrades accordingly such
that the yield displacements remain constant and the defi nition of ductility
remains consistent (Carr, 2009).
21.7.2 Damage states and developing the backbone curves
Several damage states were considered in the seismic evaluation of the
bridge under study including yielding, serviceability, bar-buckling and col-
lapse. The yielding and serviceability damage states for the bridge columns
were predicted using the method described by Priestley
et al.
(2007) that
involved the development and idealization of the moment-curvature
diagram. The serviceability limit state is usually exceeded when the con-
crete cover outside the confi ned core starts spalling (i.e., the maximum
compression strain in the unconfi ned concrete cover exceed 0.003 to 0.004)
or crack widths are larger than allowable limits (i.e., the maximum tensile
strain is larger than 0.015 for column elements; Priestley
et al.
, 2007).
However, it should be noted that the median compressive strain corres-
ponding to spalling based on available tests is reported as 0.008 with a
coeffi cient of variation of 0.45 by Berry and Eberhard (2007).
The ultimate strain in the steel bars corresponding to bar buckling from
the experimental equations by Berry and Eberhard (2007) was used to
compute the ultimate curvature of the columns and the corresponding drift
and ductility is defi ned as the point at which strength degradation begins
(i.e.,
θ
cap
in Fig. 21.2). The drift ratio at the bar-buckling damage state is
similar to that obtained for the 'Life Safety' limit state based on the recom-
mendations by Priestley
et al.
(2007) (more details available in Tehrani and
Mitchell, 2012a). The conservative limit of 0.1 on post-capping capacity
(Haselton
et al.
, 2007),
θ
pc
, was controlling for the bridge columns and was
adopted in this study due to lack of research. This limit is deemed to be
conservative for the bridge columns with spiral transverse reinforcement.
The bridge under study was designed and detailed to meet the code
requirements for ductile response, including capacity design concepts and
adequate support lengths at the abutments. The ductile columns contain
code-compliant spiral reinforcement to confi ne the concrete, to avoid shear
failure and to control buckling of the vertical reinforcing bars. For this
continuous bridge, with all other failure modes avoided, the fl exural response
governs the response of the bridge (i.e., sidesway collapse mechanism).
The collapse prediction is based on dynamic instability of the structure
(Vamvatsikos and Cornell, 2002). While the columns will undergo a ductile
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