Geoscience Reference
In-Depth Information
The annualized risk for the system from all possible events that occur with rate
ν
i
is
expressed in the following equation:
⎨
⎩
⎬
⎭
alle
v
ents
ν
i
∗
v(
L
)
=
l
s
f
L
s
|
Q
(
l
s
|
Q
)
dl
s
+
l
n
f
L
n
|
Q
(
l
n
|
Q
)
dl
n
(19.9)
all net
w
or k
components
where
L
s
isthestructural lossof the components
Q
isthescenario event
L
n
isthe loss due tonetwork disruption
f
istheprobability density function of the random variable
ν
is theannual rate of occurrence of an event or the rate of
DV=total loss L
Equation19.9cannotbeexpressedinclosedformandisevaluatednumericallyorthrough
simulation. For large networks, the analytical complexity can be challenging and com-
putational run-time can be excessive. Several methods have been proposed for efficient
computationofthemultipleintegralsimplicitlycontainedineq.(19.9)througheqs.(19.1)
and(19.2).Alsoimplicitinthisequationistheaggregationoflossfromallnetworkcom-
ponents. Thisaggregation isfurther discussed in thenext section.
2.3.2. Point estimates of the structural loss for multiple sites and single event
Transportation planners and bridge engineers are usually interested in risk estimates that
are applicable to multiple bridges in order to make decisions for retrofitting strategies or
planningnewrouts.Inthissection,wewillgeneralizethetwomethodsfortheestimation
of the loss at a single site and apply them to a set of bridges. In the development that
follows the dependence on the event Qis dropped tosimplifythe notation.
The loss from
n
components in a network is the sum of random variables. According to
probability theory, the sum of the expected values of the loss of all the components will
be equal to the expected value of the total loss. The variance of the total loss is equal
to the sum of the variances, under the assumption that the damage of the components is
uncorrelated. The equations follow:
all bridges
{
E
(
total loss
)
=
E
(
l
i
)
}
all bridges
σ
(19.10)
2
2
σ
=
i
where
E
(
l
i
)
is theexpected value of the lossat a single site
σ
i
is the variance of the lossat a single site
