Performance-Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 26

D. For each variable combination, determina-
tion of the mean maximum structural re-
sponse and corresponding standard deviation
over the set of earthquake records, followed
by an approximation of these means and
standard deviations by explicit, continuous
neural network response surface functions
F i ( X ).
E. Using the mean and standard deviation
for each maximum response to represent
its corresponding cumulative probability
distribution by a lognormal. In turn, with
these distributions, implementing the neural
networks from d), to calculate the probabili-
ties of non-performance or the associated
reliability indices β j . for any given set of
design parameters.
F. For a set of parameters x d , chosen within their
bounds, approximation of the discrete results
from e) by means of neural network response
surface functions for the reliabilities β j ( x d ).
Using these approximations directly, during
the optimization, to calculate the reliability
achieved at each performance level for a
particular vector x d , in order to efficiently
evaluate constrain violations.

that the development of a response database is the
task involving the greatest computational effort.
A nonlinear model for reinforced concrete, based
on bar elements, has been discussed elsewhere
(Möller, 2001; Möller and Foschi, 2003) and
has been used in the first application example
presented here in Section 7.
The reliability and, ultimately, the optimization
results, are conditional on the adequacy of the
analysis model, and a proper “model error” vari-
able must be included in the random set X . Some
of the issues to be addressed in the formulation of
the dynamic analysis relate, for example, to how
the hysteretic energy dissipation is represented.
This issue is considered in the second application
example in Section 8, a soil/structure interaction
problem which highlights the dependence of the
optimization results on the assumptions made in
the analysis model.

3. RESPONSE REPRESENTATION
BY NEURAL NETWORKS

3.1 Neural Networks

Neural networks are algorithms for the transmis-
sion of information between input and output.
This technique has been used here to represent the
structural response databases. Because the neural
networks literature is quite extensive, only a brief
description of the type of network used here is
included for completeness.
Several input parameters are assumed to oc-
cupy individual “neurons” in an “input layer” and,
similarly, several outputs can form the “output
layer” containing the “output neurons”. The in-
formation between input and output is assumed to
flow through intermediate or hidden neurons, the
strength of the information between two neurons
j and i being given by weight parameters W ji . The
architecture of the network is shown in Figure
1, for N inputs and K outputs. The information
received by a hidden or output neuron is modi-

2. NONLINEAR DYNAMIC
ANALYSIS MODEL

The optimization requires the development of a
response database in terms of the random variables
X , for each earthquake record considered. This
calculation implies the application of a nonlinear
dynamic analysis and a step by step integration of
the equations of motion over the duration of the
earthquake. The focus of this Chapter is not on
the details of the particular structural nonlinear
dynamic analysis used. Many modeling techniques
have been proposed, aiming at achieving an ad-
equate balance between accuracy and simplicity.
It is important to emphasize, however, that each
approach has an associated modeling error, and