Performance-Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 25

examples. The first is for a simple, reinforced con-
crete portal frame, for which the design parameters
are the column and beam dimensions, as well as
the steel reinforcement ratios. The optimality of
the preliminary design is discussed, as well as the
efficiency of the proposed optimization process.
This example also considers the sensitivity of the
optimal solution to the assumed relationship be-
tween calculated damage and repair costs, as well
as to the reliability levels prescribed as minimum
constraints for each of the performance levels.
Reliability estimates and optimization results
are conditional on the analysis models used. The
second example is the optimization for the permis-
sible pile-cap mass carried by a pile foundation.
This main objective in this example is to discuss
the importance of the analysis model formulation
in the final optimization results. Two different
models are used, based on two different formula-
tions for hysteretic energy dissipation.

in which β j ( x d ) are the reliability indices achieved
with the design parameters x d for each of the three
performance levels j : “operational”, “life safety
or controlled damage” and “collapse”, with β jT
being the corresponding prescribed minimum
targets over the design life T D .
The design parameters x d could be, for example,
either dimensions, steel reinforcement ratios, or
statistical parameters for some of the random
variables in the problem (e.g., the mean value of
the required steel yield strength).
Given a procedure to verify reliability constrain
compliance, as described in Section 2.2, an opti-
mization technique is then applied to minimize the
total cost while satisfying the minimum reliability
indices for each performance level. The optimiza-
tion algorithm proposed here is a gradient-free,
search-based approach, as described in Section 5.

1.2 The Methodology for
Reliability Evaluation

1. THE OPTIMIZATION
FORMULATION

The optimization process requires the calcula-
tion of reliability levels achieved with the design
parameters x d . This is implemented through the
following steps, all prior to the optimization proper.
These steps, to be addressed in the following sec-
tions, include:

1.1 General Objective

The aim of the optimization considered here is
to find values for the design parameters, grouped
in a vector x d , to minimize the objective function
C ( x d ). This function is the total cost, and it is given
by the sum of the initial construction cost C 0 ( x d )
and the damage repair cost C d ( x d ), required after
the occurrence of earthquakes during the service
life of the structure. That is,

A. Definition of the random variables X in the
problem, and their corresponding upper and
lower bounds;
B. Selection of combinations for the variables
X within their corresponding bounds, using
design of experiments;
C. Construction of a structural response data-
base: for each variable combination in b),
and for each record in a set of earthquakes
considered likely to occur at the site, determi-
nation of the maximum structural responses
R i ( X ) that enter in the formulation of the
performance functions, using a nonlinear
dynamic time-step analysis;

C ( x ) = C ( x ) + C ( x ) (1)

d 0 d d d

The minimum total cost must correspond to a
structure that also satisfies minimum reliability
requirements for different performance levels:

β ( x ≥ ) β , j = 1 2 3 , , (2)

j d jT