Geology Reference
In-Depth Information
The PDF f DIES (DIES) in Eq.(30) is calculated
by obtaining first, and then differentiating, the
cumulative distribution of DIES , using simulation
and the neural networks for the response DIES .
This is represented by a Beta distribution, given
that the damage index is bounded by 0 and 1.
In general, an earthquake does result in addi-
tional costs beyond those limited to the structure:
losses associated with contents or non-structural
components, cost for interruption of service,
insurance and losses for injury or casualties.
These additional costs can be taken into account
within the format proposed here, but have not
been considered in the applications shown next
in this Chapter.
The nonlinear dynamic analysis was then
carried out for each of the combinations and for
each sub-combination, obtaining a database for
the following response parameters:
UMAX: The maximum displacement at
the top of the portal
DIST: The maximum inter-story drift
DILO: A maximum local damage index
DIES: A maximum global damage index
Thus, to each of the response results R i ( i =
1, 450) obtained for the 450 combinations, cor-
responded NS = 10 results R ki ( k = 1, 10) for the
set of sub-combinations. These results can be used
to obtain the mean and the standard deviation of
each R i over the set of secondary variables (differ-
ent earthquake records and hysteretic properties):
6. APPLICATION EXAMPLE1:
OPTIMIZATION OF A SIMPLE
PORTAL FRAME
NS
NS
1
1
)
R
R
(
R
R
2
=
σ
=
i
i
k
R
k
NS
NS
1
i
i
i
k
=
1
k
=
1
6. 1 Random Variables,
Response Databases, and Neural
Network Representations
(31)
The data for these statistics form two databases,
each with NP = 450 entries. Both databases are
represented by neural networks with the main
variables as input.
The training of the networks used the OPT
algorithm, with a subsequent attempt to improve
the solution by using a gradient-based approach
as described in Secion 4.3. This additional step
only produced a small improvement in the results.
The agreement between the data T and the neural
network outputs Y , are shown in Figure 4 for the
response DIST as a typical case. To quantify the
goodness of the regression, the linear correlation
coefficient ρ YT is calculated,
The structure chosen for this application example
is a simple reinforced concrete portal frame, shown
in Figure 3.
The main input variables for the determination
of the response databases are shown in Table 1
with their corresponding bounds.
Within the bounds, 450 variable combinations
were generated by experimental design (Zhang,
2003). For each combination, 10 sub-combinations
were generated for the following secondary vari-
ables: (a) a set of random phase angles to gener-
ate an artificial earthquake record (Möller, 2001;
Shinozuka, 1967), with the resulting record then
scaled to the peak acceleration a G included in the
particular combination; and (b) the concrete and
steel strength as they affect the variability in the
parameters for the hysteretic relationship moment-
curvature for beam and columns cross-sections
(Möller et al., 2006).
σ
σ σ
)
1
NP
YT
ρ
=
where
σ
=
(
Y Y T
) (
T
YT
YT
k
k
NP
1
k
=
1
Y
T
(32)
The results in Figure 4 show a very good
agreement and a correlation coefficient very close
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