Seismic vulnerability analysis of a complex interconnected civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 479

where the conditional mean vector and the conditional covariance and cor-
relation matrices are:

( ) ln s − μ M ,,, RFV

( ) = ( ) +

1 ln s

m MRFV ,,, m MRFV ,,, C

1

ln s ln s ln s ln s ln s ln s

2 1 2 2 2 1

σ

2

s

1

[18.8]

CC

ln s ln s ln s ln s

C = C −

21 12

[18.9]

ln s ln s ln s ln s ln s

22 1 22

2

σ

s

1

R = D

− 1

C D

− 1

[18.10]

ln s ln s ln s ln s ln s ln s

ln s ln s ln s ln s ln s ln s

22 1 22 1

22 1 22 1

with:

σ s

2

C

= ⎡

⎤
⎥ =

ln s ln s

C

12

DR D [18.11]

1

⎢

ln ss ln ln ss s ln ln l nnlnln s s s

C C

ln s ln s ln s ln s

21 22

σ 0

⎡ ⎤

s

1

⎢
⎢
⎢

⎥
⎥
⎥

D ss =

[18.12]

ln ln

0 σ

⎣ ⎦

s n

1 ρ

⎡ ⎤

ln s ln s

1 n

⎢
⎢
⎢

⎥
⎥
⎥

R ss = [18.13]

ln ln

ρ 1

⎣ ⎦

ln s ln s

n 1

The standard deviations in the above equations are total standard devia-
tions

2 2 . Models for the within-site correlation coeffi cient are
available in the literature, e.g. for spectral accelerations at different periods
as in Inoue and Cornell (1990) with the simple form (valid for periods
between 0.1 s and 4.0 s):

σ = στ +

T

( )

ρ ln =− 1033 . ln TT [18.14]

ss ln j i

i j

18.6 Performance metrics

This section describes metrics to quantitatively measure the performance
of an infrastructure and its components. These metrics often express either
the comparison of a demand with a capacity quantity, as a difference or
ratio, the consequence of a mitigation action, or the assembled conse-
quences of all damages (the 'impact'). In general, for an infrastructure
(SOS), they can be categorised according to the three hierarchical
levels into:

￿ component-level metrics;
￿ system-level metrics;
￿ infrastructure-level metrics.