Geoscience Reference
In-Depth Information
bridgeswithinabridgeclasscanbeclearlyseensincethesebridgesarepresumedtoper-
form in a similar manner in an earthquake. Partial between bridge class correlation can
be explained by the seismic design requirements of bridges built under the same design
criteria. Data required for the estimation of damage correlation are difficult to find if at
allavailable.Thereforeanequi-correlatedassumptionismadewherethecorrelation
ρ
Di
,
Dj
is equal to a constant
ρ
D
. When two sites are subjected to the same ground motions,
i.e.
u
i
u
j
, and the bridges at the two sites are in the same engineering class, then the
bridgesareconsideredtobeperfectlycorrelatedandthecorrelationcoefficientis
=
ρ
D
=1.
Thedamagetotwobridgesofthesameengineeringclassisconsideredtobeuncorrelated
if the ground motions at the two sites are different, i.e.
ρ
D
=0for
u
i
=
u
j
. For these
twospecialcases,LeeandKiremidjian(2006)developclosedformsolutionsforthejoint
probability density function of damage of pairs of bridges in a system. For partially cor-
related bridges closed form equation does not exist and the joint probability of damage
needstobeevaluatednumerically.Theyproposeanumericalmethodforestimatingthese
probabilities. The reader is referredtotheir paper for further detail.
2.5.1. Probability distributions of the structural loss for multiple sites and single event
In general, the first terms in eqs. (19.8) and (19.9) can be expanded to explicitly show
thedamagemeasure
DM
,engineeringdemandparameter
EDP
andintensitymeasure
IM
conditionalprobabilitydensityfunctions.Therealchallengeisinevaluatingtheprobabil-
itydensityfunction(PDF)oflossforallbridgesinthenetworksystemforagivenevent.
Thechallengeisfurtherincreasedwhenallpossibleeventsareconsidered.Inthissection
we develop theaggregated lossfrom structural damage for asingle event.
Foragivenevent
Q
j
,
1,2,...,
N
,thetotallossresultingfromdamagetocomponents
(bridges) of the network is the sum of all the losses. Since the loss of each component is
a random variable with its own distribution, the sum of the losses is a convolution of the
individual probability density functions. That is,
j
=
L
total
=
L
1
+
L
2
+
L
3
+···+
L
n
(19.13)
f
L
total
=
f
L
1
⊗
f
L
2
⊗
f
L
3
⊗···⊗
f
L
n
(19.14)
where
L
total
is the total lossforasetof
n
bridges
L
i
isthe lossforbridge
i
,
i
=
1,2,...,
n
foragiven event
Q
j
f
isthe PDF ofa random variable
⊗
isthe symbol for convolution
In the above equations the subscript referring to the event
j
is dropped for simplicity of
notation.Usingthewellknownpropertythattheconvolutioninthetimedomainbecomes
multiplicationinthefrequencydomain,wecancomputetheprobabilitydensityof
L
total
