Civil Engineering Reference
In-Depth Information
tude scale factor and therefore have
1. On the other hand, intensity
measures, such as
IA
, scale quadratically with scale factor (i.e.
α
=
2), and
signifi cant duration,
D
s
(e.g. Bradley, 2011a, equation 1), is independent of
amplitude scale factor (i.e.
α
=
0). Hence, in the situation considered here,
where the unscaled intensity measure
IM
j
of the
m
th ground motion can
be computed,
IM
j
m
,unscaled
and the target value is defi ned by the value for
which the GCIM distributions are constructed (i.e.
IM
j
α
=
im
j
), then the
required scale factor for ground motion
m
is given by rearranging Equation
(4.12) to obtain:
=
1 α
=
⎛
im
⎞
⎟
j
SF
[4.13]
⎜
m
IM
m
,unscaled
j
Note that
SF
m
is independent of the individual realized ground motion
values,
IM
nsim
, and therefore the scale factor for each prospective ground
motion need only be computed once (for
IM
j
im
j
). As a result, each
amplitude-scaled ground motion will have a unique intensity measure
vector
IM
m
, with elements obtained from:
=
(
)
α
IM
i
m
=
IM
m
,unscaled
SF
[4.14]
m
i
Hence, the intensity measure values (which scale analytically with scale
factor) of all ground motions can be pre-computed (before executing the
ground motion selection algorithm), which signifi cantly reduces computa-
tional demands.
4.5.4 Representativeness of the selected motions
compared to the GCIM distributions
Because ground motion selection is desired for a fi nite number of
N
gm
ground motions, then a comparison of the appropriateness of the
N
gm
ground motions as representative of
IM
i
|
IM
j
(for all
i
) must be done so using
statistical goodness-of-fi t tests. The majority of ground motion intensity
measures of engineering interest are continuous variables, and the statistical
test for such cases is presented here, while tests for discrete variables can
be found elsewhere (Bradley, 2010a).
For continuous intensity measure variables, the adequacy of a particular
suite of ground motions with respect to a pre-defi ned theoretical distribu-
tion of a single
IM
i
can be assessed by the KS goodness-of-fi t test (e.g. Ang
and Tang, 2007, p. 293-296). The KS test measures the absolute difference
between the theoretical cumulative distribution function (CDF) and the
empirical distribution function (EDF) of the sample, which is mathemati-
cally given by:
(
)
−
(
)
D
=
max
F
im im
S
im
[4.15]
IM
IMi IM
|
i
j
N
i
i
j
gm
IM
i
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