Geology Reference
In-Depth Information
formulation, all performance requirements against
future natural hazards are directly incorporated in
the objective function. Finally, the constraints in
Equation 2 may be incorporated into the definition
of admissible design space
Φ
, which leads to the
simplified expression:
cases, for example, skewed bridges, for which a
complete three dimensional model is required to
accurately capture their response under seismic
excitation. Each span of the bridge is modeled
here as a rigid body. For appropriately address-
ing at the design stage the pounding between
adjacent spans, soil-structure interaction charac-
teristics and the nonlinear behavior of isolators
and dampers, nonlinear dynamic analysis is
used (Makris & Zhang, 2004; Saadeghvaziri &
Yazdani-Motlagh, 2008; Zhang, Makris, & Delis,
2004). The pounding with the abutment and the
dynamic characteristics of the latter are incorpo-
rated in the analysis by modeling the abutment
as a mass (Saadeghvaziri & Yazdani-Motlagh,
2008) connected to the ground by a spring and
a dashpot, with stiffness and damping properties
that are related to the local soil conditions (Zhang
et al., 2004). The vibration behavior of the isola-
tors, the dampers and the pier is incorporated by
appropriate nonlinear models (Ruangrassamee &
Kawashima, 2003; Taflanidis, 2009; Taflanidis &
Beck, 2009a; Zhang et al., 2004). The pounding
between adjacent spans, or to the abutments, is
approximated here as a Hertz contact force with
an additional non-linear damper to account for
energy losses (Muthukumar & DesRoches, 2006).
A schematic of the bridge model with two
spans, supported by seismic isolators to the abut-
ments and to the intermediate pier is illustrated
in Figure 3. The two spans and abutments are
distinguished by using the convention right and
left for each of them. The gap between the two
spans is denoted by
x
o
and the gap between the
left or right span and the corresponding abutment
by
x
ol
or
x
or
, respectively. Let also
x
p
,
x
sl
,
x
sr
,
x
al
,
x
ar
, denote, respectively, the displacement relative
to the ground of the pier, the left and right span
of the bridge and the left and right abutment. The
total mass for the pier, the left and right span of
the bridge and the left and right abutment are
denoted, respectively, by
m
p
,
m
sl
,
m
sr
,
m
al
,
m
ar
.
This total mass includes both the weight of each
component as well as the live loads due to ve-
*
C
(3)
ϕ
=
arg min
( )
ϕ
ϕ
∈Φ
For evaluation of Equation 1 and the optimiza-
tion in Equation 3 an approach based on stochastic-
simulation will be discussed in detail later. Along
with the comprehensive risk quantification and
the bridge and excitation models described next
that establishes a versatile, end-to-end simulation
based framework for detailed characterization and
optimization of seismic risk. This framework puts
no restrictions in the complexity of the models
used allowing for incorporation of all important
sources of nonlinearities and adoption of advanced
models for characterization of the ground motion.
Thus, it allows an efficient, accurate estimation
of the seismic risk by appropriate selection of the
characteristics of the stochastic analysis. Next we
will discuss in detail the bridge and excitation
models used in this study, and then proceed to
the computational framework for performing the
optimization in Equation 3.
BRIDGE MODEL
For simplicity of the analysis, we will assume a
two-span (as in Figure 1), straight bridge, whose
fundamental behavior in the longitudinal direc-
tion can be adequately characterized with a planar
(two-dimensional) model. It is noted that the
bridge is assumed to be parallel to the direction
of the seismic wave propagation and also that it
does not fall in the category of long-span bridges.
The ideas discussed in the following sections,
related to probabilistic modeling and design, can
be directly extended, though, to more complex
Search WWH ::
Custom Search