Geology Reference
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during the design process. The basic mathematical
programming statement of the structural optimi-
zation problem is converted into a sequence of
explicit approximate primal problems. For this
purpose, the objective function and the reliability
constraints are approximated by using a hybrid
form of linear, reciprocal and quadratic approxi-
mations. An approximation strategy based on an
incomplete quadratic conservative approximation
is considered in the present formulation (Groen-
wold et al., 2007; Prasad, 1983). An adaptive
Markov Chain Monte Carlo procedure, called
subset simulation (Au and Beck, 2001), is used
for the purpose of estimating the failure prob-
abilities. The information generated by subset
simulation is also used to estimate the sensitivity
of the reliability constraints with respect to the
design variables. The above information is com-
bined with an approximation strategy to generate
explicit expressions of the objective and reliability
constraints in terms of the design variables. The
explicit approximate primal problems are solved
either by standard methods that treat the problem
directly in the primal variable space (Goldberg,
1989; Kovács, 1980; Scharage, 1989; Tomlin,
1970) or by dual methods (Fleury and Braibant,
1986; Haftka and Gürdal, 1992; Jensen and Beer,
2010). The proposed optimization scheme exhibits
monotonic convergence that is, starting from an
initial feasible design, the scheme generates a
sequence of steadily improved feasible designs.
This ensures that the optimal solution of each ap-
proximate sub-optimization problem is a feasible
solution of the original problem, with a lower
objective value than the previous cycle.
The structure of the chapter is as follows. First,
the design problem considering discrete sizing
type of design variables is presented. Next, the
structural and excitation models are discussed in
detail. The solution strategy of the problem in the
framework of conservative convex and separable
approximations is then discussed. This is followed
by the consideration of some implementation is-
sues such as reliability and sensitivity estimation.
Finally, two numerical examples that consider
structures with passive energy dissipation systems
as structural protective systems are presented.
FORMULATION
Consider a structural optimization problem de-
fined as the identification of a vector
{
y
of design
variables that minimizes an objective function,
that is
Minimize
f
({ })
y
(1)
subject to design constraints
h
({ })
y
≤
0
,
j
=
1
, ...,
n
(2)
j
c
where
h
j
represents a constraint function defined
in terms of reliability measures, and
n
c
is the
number of constraints. As previously pointed out
a pure discrete variable treatment of the design
problem is considered here. Thus, the side con-
straints for the discrete design variables are writ-
ten as
l
y Y
∈ =
{ ,
y l
=
1
, ...,
n
},
i
=
1
, ...,
n
(3)
i
i
i
i
d
where the set
Y
i
represents the available discrete
values for the design variable
y
i
,
listed in ascend-
ing order, and
n
d
is the number of design variables.
It is assumed that the available values are distinct
and they correspond to quantities such as cross
sectional areas, moments of inertia, etc. The par-
ticular quantity to be used depends on the problem
at hand. It is noted that alternative formulations
to the one proposed here should be considered
for cases where the possible values of the discrete
design variables are linked to a number of proper-
ties simultaneously, for example, section groups
in steel structures.
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