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which is equal to one. Considering the definition
of failure probability in terms of the performance
function, the linear expansion of the performance
function in (34), and the approximation of the
failure probability function given in (38), the
partial derivatives of the
j
-th transformed failure
probability function can be expressed as (Jensen
et al., 2009)
1
(37)
Ω
F
({ })
y
=
{{ },{ }
θ
ξ
|
D y
({ },{ },{ })
θ
ξ
≥
}.
j
j
where
D y
= −
1
The failure probability function, evaluated at the
current design
{ }
({ },{ },{ })
θ
ξ
κ
({ },{ },{ }).
y
θ
ξ
j
j
y
k
, is approximated locally as
an explicit function of the normalized demand
around
D
j
*
=
1
as
∂
h
t
({ })
y
ψ
+
ψ δκ
∆
y
ψ
1
e
−
e
0
1
lj
l
0
j
≈
×
lim
*
≈
∈ −
ψ
+
ψ
(
D
−
1
)
P D y
({ },{ },{ })
k
θ
ξ
≥
D
*
e
,
0
1
j
∂
y
P
({ })
y
k
∆
y
∆
y
→
0
j
j
(38)
l
l
F
l
k
{ } {
y
=
y
}
j
*
D
[
1
ε
,
1
+
ε
]
=
ψ δκ
1
,
l
= …
1
,
,
n
j
jl
d
(41)
where
D
j
*
is a threshold of the normalized demand
(in the neighborhood of one) and
ε
represents a
small tolerance. The coefficients
ψ
0
and
ψ
1
can
be calculated by least squares with samples of the
performance function
κ
j
(or normalized demand
function
D
j
) generated at the last stage of subset
simulation (Valdebenito and Schuëller, 2011). The
gradient of the
j
-th transformed failure probabil-
ity function at
{ }
where
δκ
jl
is the
l
-th element of the vector
{
δκ
j
and all other terms have been previously defined.
It is noted that the previous approach for estimat-
ing the gradients of the failure probability functions
requires a single reliability analysis plus the
evaluation of the performance functions in the
vicinity of the current design. Validation calcula-
tions have shown that this approach is quite ef-
ficient for estimating the sensitivity of failure
probability functions with respect to design vari-
ables (Jensen et al., 2009; Valdebenito and
Schuëller, 2011).
},
y
k
(see Equation (26)) is given
by
∂
P
({ })
y
t
∂
h
({ })
y
1
F
j
=
×
j
∂
y
P
({ })
y
k
∂
y
l
F
l
k
k
{ } {
y
=
y
}
j
{ } {
y
=
y
}
1
,
,
l
= …
n
d
NUMERICAL EXAMPLES
(39)
Example No.1
where
n
d
is the total number of design variables.
On the other hand, the gradient of the
j
-th failure
probability function can be estimated by means
of the limit:
Description
Consider the reliability-based optimization of the
two-story frame structure under earthquake load-
ing shown in Figure 2. The floor masses are
m
i
=
10
7
kg,
i
=
1 , ,
and the initial linear inter-
story stiffnesses are
K
1
∂
P
({ })
y
P
({ }
y
k
+
{ ( )}
δ
l
∆
y
)
−
P
({ })
y
k
F
F
l
F
=
lim
,
j
j
j
∂
y
∆
y
∆
x
→
0
l
l
l
{ } {
y
=
y
k
}
l
= …
1
,
,
n
d
=
9 0 10
.
×
9
N/m and
(40)
9
= ×
.
N/m. A 3
%
of critical damping
is assumed in the model. The damping ratio is
treated as uncertain and modeled as a log-normal
K
2
7 0 10
where
{ ( )}
δ l
is a vector of length
n
d
with all
entries equal to zero, except for the
l
-th entry,
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