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uncertainties in the ground motion and structural
properties, and at the same time minimizing the
total cost. This includes, in addition to the initial
cost of construction, that associated with repairs
following each of a series of earthquakes within a
given time interval. The preliminary design may be
the result of applying capacity design approaches,
according to existing Codes, in order to satisfy
the performance levels.
The subsequent sections of this Chapter discuss
the different aspects of the proposed design opti-
mization process, including: 1) the quantification
of the seismic hazard or possible ground motions
at the site; 2) the use of a structural dynamics
analysis model that represents, as best as possible,
the nonlinear response and the hysteretic energy
dissipation mechanisms; 3) the calculation of a
response database for the estimated ranges of
the different intervening input random variables
and design parameters; 4) a functional represen-
tation for the discrete databases, using neural
networks response surfaces; 5) implementation
of the response surfaces in the calculation of the
reliability levels achieved, at each performance
level, for specific values of the design parameters;
6) representation of these reliability levels by
neural networks in terms of the design param-
eters, in order to achieve an efficient calculation
of reliability constraint violations and, finally, 7)
an optimization algorithm that would search for
a minimum total cost objective under minimum
reliability constraints for each performance level.
Earthquakes show large ground motion un-
certainties which must be coupled with those in
the structural capacities, and the demands will
likely trigger a nonlinear structural response.
For each performance level, the formulation of
the limit-state functions requires the estimation
of maximum responses (e.g., maximum inter-
story drift) during the duration of the earthquake.
Since only discrete values can be obtained by
numerical analyses for specific combinations of
the variables and parameters, the responses need
to be given a continuous functional representation
for optimization and reliability estimation. Each
of the responses of interest can be approximately
represented by using response surfaces which,
when properly adjusted, can be used as substi-
tutes for the dynamic analysis (Hurtado, 2004).
Different forms for approximating surfaces have
been studied (Möller et al., 2009b), and neural
networks have been shown to offer advantages
of flexibility and adaptability. Regardless of the
type of response surface used, a major advantage
of the substitute is the computational efficiency
achieved with Monte Carlo-type simulations when
estimating probabilities of non-performance.
The approach chosen in the optimization prob-
lem must consider the presence of constraints,
the dimensionality and the form and number of
objective functions. Optimization methods can use
different approaches (Pérez López, 2005; Swisher
et al., 2000), some requiring the calculation of
gradients (steepest descent, conjugate gradients,
Newton or quasi-Newton schemes) and others, not
using gradients, implementing genetic or search
algorithms. This Chapter proposes an efficient
search-based algorithm which also accounts for
constraints given by specified minimum reliability
levels at each performance level. The optimiza-
tion approach presented in this Chapter is based
on previous work by the authors (Möller et al.,
2007, 2008, 2009a,c), and introduces additional
contributions as follows:
•
The optimization process is started from a
preliminary, deterministic seismic design;
•
The previous search-based algorithm by
the authors has been modified to mini-
mize the possibility of encountering local
minima;
•
The damage repair costs introduced into
the objective function take into account the
occurrence of multiple seismic events dur-
ing the life of the structure.
The optimization approach presented here is
illustrated with two performance-based design
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