Geology Reference
In-Depth Information
Figure 3. Portal frame
6.2 Performance Functions
and Neural Networks for
Reliability Estimates
to unity. Nevertheless, the lack of fit between the
approximation
F
(
X
) and the actual data
R
(
X
) can
be quantified by calculating the standard deviation
of the relative error,
The performance function
G
i
for the i
th
-limit state
can be written, in general, as
1
NP
Y T
Y
−
∑
σ
ε
r
=
(
k
k
)
2
(33)
NP
−
1
k
=
1
k
G
X
=
RLIM R
−
X
≅
RLIM F
−
X
(35)
( )
( )
( )
i
i
i
i
i
in which
Y
k
is the value calculated by the network,
T
k
is the target data from the nonlinear dynamic
analysis and
NP
= 450, the size of the database.
The neural network predictions can then be
corrected to account for the lack of fit, as follows:
Eqs.(36) through (43) show the performance
functions for the maximum relevant response
parameter and for three performance levels: op-
erational, life safety and collapse. Each of the
capacity terms in these functions show, in paren-
t
heses, t
he mean and the coefficient of variation
(
F
Y
( .
1
X
)
( .
1
X
)
=
+
σ
σ
=
σ
+
σ
RLIM COVRL
. In
G
11
,
u
y
indicates the mean
yield displacement for the frame, below which
the structure remains elastic.
,
)
i
i
m
N
F
Y
N
ε
ε σ
1
i
i
2
(34)
,
σ
are, respectively, the mean
value and the standard deviation calculated with
the corresponding neural network, and
σ
in which
Y
i
Y
i
Operational:
m
,
are the corresponding standard deviations of the
relative errors.
X
N
1
,
X
N
2
are two additional random
variables associated with the fit error and assumed
to be Standard Normals.
σ
•
Elastic
displacement
limit
ε
ε σ
G
( )
X
=
(
u
,
0 10
.
)
−
UMAX
( )
X
(36)
11
y
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