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been found that the CMS can be used as an appropriate target response
spectrum for selecting ground motions as input for dynamic analysis (Baker,
2011). In the development of a CMS, some important aspect of records
including
M
,
R
and
will be considered from the deaggregation of seismic
hazard. Baker (2011) proposed a method for calculating the CMS, given by
Eq. [21.3]:
ε
(
)
+
(
)
(
)
(
)
μ
( )
=
μ
MRT
,,
ρ
TT
,
ε
T
σ
T
[21.3]
()
ln
ST
ln
ST
ln
S
i
i
1
1
ln
S
i
a
i
a
1
a
a
where an appropriate GMPE must be used to evaluate the mean and stan-
dard deviation of the natural logarithm of the spectral acceleration at the
vibration period
T
i
denoted by
ln
S
a
(
M
¯
,
R
¯
,
T
i
) and
μ
σ
ln
S
a
(
T
i
), respectively, in
which
M
¯
,
R
¯
, and
¯
(
T
1
) are the mean magnitude, mean distance and mean
epsilon at the considered period
T
1
, respectively. These mean values are
calculated from the seismic hazard deaggregation and
ε
(
T
i
,
T
1
) is the inter-
period correlation of spectral accelerations at vibration periods
T
i
and
T
1
.
The equation proposed by Baker and Cornell (2006b), Eq. [21.4], is used to
calculate the inter-period correlation.
ρ
π
T
T
T
⎡
⎢
⎤
⎥
⎛
⎜
⎞
⎟
(
)
=−
min
max
ρ
TT
,
1
cos
−
0 359
.
+
0 163
.
I
ln
ln
[21.4]
i
1
T
min
<
0 189
.
2
0 189
.
min
where
T
max
and
T
min
are the larger and the smaller of
T
i
and
T
1
, respectively,
and I
T
min=0.189
is an indicator function that equals 1 if
T
min
is less than 0.189 s
and equals 0 otherwise.
For measuring the match with the target spectrum, the sum of squared
errors (SSE) between the logarithms of the ground motion's spectrum and
the target spectrum is used as recommended by Baker (2011).
n
=
∑
[
()
−
()
]
2
SSE
=
ln
S
T
ln
S
T
[21.5]
a
j
a
CMS
j
j
1
where ln
S
a
(
T
j
) is the log spectral acceleration of the ground motion at
period
T
j
and ln
S
aCMS
(
T
j
) is the log CMS value at period
T
j
. The values of
SSE are computed for all records considered and the records with the small-
est SSE values will be selected. This approach is more effective if ground
motion scaling is allowed. To scale the ground motions, Baker (2011) recom-
mends scaling each ground motion so that its spectral acceleration at the
considered period,
S
a
(
T
1
), matches the target spectral acceleration from the
CMS,
S
aCMS
(
T
1
). Although the concept of scaling the records is sometimes
questioned, it has been observed that ground motions selected and scaled
to match the CMS produce displacements that are comparable to those
produced by unscaled ground motions, unlike scaling procedures using
other methods (Baker and Cornell, 2005; Luco and Bazzurro, 2007). Thus
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