Geology Reference
In-Depth Information
The characteristics of the models for the seis-
mically isolated bridge and for future earthquake
excitations, described in the next two sections are
not known with absolute certainty. Uncertainties
may pertain to (i) the properties of the bridge
system, for example, related to stiffness or damp-
ing characteristics and traffic loads; to (ii) the
variability of future seismic events, i.e., the mo-
ment magnitude or the epicentral distance; to (iii)
the predictive relationships about the character-
istics of the excitation given a specific seismic
event, for example duration of strong ground
motion or peak acceleration; or to (iv) parameters
related to the performance of the system, for
example, thresholds defining fragility of system
components. A probability logic approach pro-
vides a rational and consistent framework for
quantifying all these uncertainties and explicitly
incorporating them into the system description.
In this approach, probability can be interpreted
as a means of describing the incomplete or miss-
ing information (Jaynes, 2003) about the system
under consideration and its environment, repre-
senting the seismic hazard, through the entire
life-cycle (Taflanidis & Beck, 2009a).
To formalize these ideas denote the vector of
controllable system parameters, referred to
herein as design variables, be
ϕ ∈ ⊂ ℜ
signer criteria (an example is discussed in the
illustrative application considered later). The
conventions that lower values for
h
(
φ
,
θ
) corre-
spond to better performance, i.e. it is associated
with risk and ultimately corresponds to a risk
measure, is used herein.
For addressing the uncertainty in
θ
a prob-
ability density function (PDF)
p
(
θ
), is assigned to
it, quantifying the relative likelihood of different
model parameter values. This PDF incorporates
our available knowledge about the structural
system and its environment into the respective
knowledge, and should be selected based on this
knowledge (Jaynes, 2003). In this setting, the
overall performance, i.e. seismic risk, is described
by the following probabilistic integral that cor-
responds to the expected value of
h
(
φ
,
θ
)
=
∫
C
( )
ϕ
h
( , ) ( )
ϕ
θ
p
θ θ
d
(1)
Θ
Different selections for
h
(
φ
) lead to different
characterization for seismic risk; for example, if
h
(
φ
,
θ
)=
C
in
(
φ
,
θ
)+
C
lif
(
φ
,
θ
), where
C
in
(
φ
,
θ
) cor-
responds to the initial cost and
C
lif
(
φ
,
θ
) to the
additional cost over the lifetime due to repairs or
downtime, then risk corresponds to life cycle cost
(Taflanidis & Beck, 2009a), if
h
(
φ
,
θ
)=
I
F
(
φ
,
θ
),
where
I
F
(
φ
,
θ
) is the indicator function for some
event
F
(one if
F
has occurred and zero if not),
then risk corresponds to the probability of unac-
ceptable performance (Taflanidis & Beck, 2009b).
The probabilistically-robust stochastic design
(Taflanidis & Beck, 2008) is then established
by selecting the design variables that minimizes
seismic risk
C
(
φ
)
n
, where
Φ denotes the admissible design space with vol-
ume
V
Φ
. These variables are related to the adjust-
able characteristics of the damper application, for
example viscosity properties or damper size. Let
θ Θ
Φ
∈ ⊂ ℜ
n
, denote the augmented vector of
model parameters where Θ represents the space
of possible model parameter values. Vector θ is
composed of all the model parameters for the
individual structural system, excitation, and per-
formance evaluation models indicated in Figure
2. The seismic performance of the bridge, for
specific design φ and model description θ, is
characterized by the performance measure
h
θ
{
}
0
(2)
ϕ
*
=
arg min
C
( ) |
ϕ
f
c
( )
ϕ
≥
ϕ
∈
Φ
Where arg stands for “argument that mini-
mizes” and
f
c
(
φ
) is a vector of deterministic
constraints, related, for example, to location or
space constraints for the dampers. Note that in this
n x n
, which ultimately quan-
tifies seismic utility or risk according to the de-
→ ℜ
+
( , ) :
ϕ
θ
ℜ
ϕ
θ
Search WWH ::
Custom Search