Civil Engineering Reference
In-Depth Information
×
m
identify matrix, this is because the means
U
i
and
U
j
is uncorrelated, for
i
j
.
Substituting Eq. [12.22] into Eq. [12.20] and Eq. [12.21], the mean vector
and covariance matrix of
Z
can be reduced to
≠
t
t
[12.23]
MA0BB d
=+=
S
=
AIAAA
=
Z
ZZ
Because
Y
and
Z
are both joint normal distribution, if the mean vector
and covariance matrix of
Z
matches the mean vector and covariance matrix
of
Y
,
Y
and
Z
will have the same joint normal distribution. Equations
[12.24] and [12.25] show the selection of
A
and
B
such that
Z
will have the
same joint normal distribution as
Y
.
MM BM
Z
=⇒=
[12.24]
Y
Y
[
(
)
]
t
t
[12.25]
SS S S
ZZ
=
⇒ =
AA
⇒=
A
chol
YY
YY
YY
where chol is the Choleski factorization of any square and positive defi nite
matrix.
Because the entry of the covariance matrix is not bounded, the computa-
tion might cause numerical instability. The Choleski factorization of
Σ
YY
is
calculated using Eq. [12.26].
S
=
DR D
⇒
AA
t
=
DR D
⇒
A
[12.26]
YY
YYYY
YYYY
[
(
)
]
t
[
(
)
]
t
=
chol
DR D
=
D
chol
R
=
DD
YY
YYYY
Y
YY
where
D
Y
is a diagonal matrix with standard deviations of random variables
Y
along its diagonal and
R
YY
is the correlation coeffi cient matrix of random
variables
Y
. By substituting Eqs. [12.24] and [12.26] into Eq. [12.19], the
correlated EDP vector can be generated using Eq. [12.27].
ZDLUM
YY
=
+
[12.27]
Y
Finally, the generated
EDP
vectors are transformed to the lognormal
distribution,
W
, by taking the exponential of
Z
. Figure 12.6 summarizes the
process of generating correlated
EDP
vectors.
12.3
Application: seismic performance assessment of
high-rise buildings
This section shows an example of applying the performance-based assess-
ment procedure to compare the seismic performance of a high-rise building
designed using three design approaches. Effectiveness of the design methods
is assessed using the repair costs associated with anticipated earthquake
ground shaking.
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