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is the vector that specifies the stochastic excita-
tion. The corresponding multidimensional prob-
ability integrals involved in the estimation of the
probabilities of failure include more than two
thousand random variables in this case. Thus, as
in the previous example the estimation of the
failure probability for a given design represents
a high-dimensional reliability problem. The reli-
ability-based optimization problem is written as
indicated. The objective function is normalized
by its value at the initial design. From Table 2 and
figure 9 it is observed that the dimensions of the
column elements at the final design of the system
without the vibration control devices are greater
than the corresponding elements of the model
with the devices, as expected. The total weight of
the unprotected model increases almost 40
%
with
respect to the weight of the enforced model. This
result highlights the beneficial effects of the vibra-
tion control devices in protecting the structural
system. For illustration purposes Figure 10 shows
a typical displacement-restoring force curve of
one of the U-shaped flexural plates at the final
design. The nonlinear incursion of the dissipator
is clear from the figure.
It is found that the process converges in less
than four iterations in this case and therefore the
design process takes few excursion probability
and sensitivity estimations. This computational
cost is substantially different for the case of direct
optimization. In that case the number of excursion
probability and sensitivity estimations increases
dramatically with respect to the proposed ap-
proach. In direct optimization the excursion prob-
abilities and their sensitivities need to be esti-
mated for every change of the design variables
during the optimization process. As in the previ-
ous example the method generates a series of
steadily improved feasible designs that moves
toward the final design. That is, the design process
has monotonic convergence properties.
Min
f d
({ })
(54)
subject to
P
i
({ })
d
≤
P
*
=
10
−
3
i
=
1 2 3 4
,
,
,
(55)
F
F
with side constraints
d
D i
,
1
, ...,
4
∈
=
(56)
i
i
where the set
D
i
represents the available discrete
values for the design variable
d
i
. The set
D
i
is
defined as
D
i
=
{
}
. , . , , .
m. It is
seen that each of the design variables can be
chosen from a discrete set of 36 diameters. There-
fore, the discrete set of design variables involves
more than
10
6
possible combinations. The opti-
mization problem is solved by the sequential
approximate optimization strategy discussed in
previous sections.
0 40 0 41
0 75
Final Designs
CONCLUSION
The initial and the final designs of the structural
system are presented in Table 2. The correspond-
ing iteration history of the optimization process
in terms of the objective function is shown in
Figure 9. Note that the initial design is an inte-
rior point in the design space, as it is required by
the algorithm (see first step of the Solution Scheme
section). For comparison the final design of the
system without vibration control devices is also
A general framework for reliability-based design
optimization of a class of stochastic systems in-
volving discrete sizing type of design variables
has been presented. The reliability-based design
problem is formulated as an optimization prob-
lem with a single objective function subject to
multiple reliability constraints. The high compu-
tational cost associated with the solution of the
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