Performance-Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 38

Table 3. Initial values for the design parameters

Table 6. It is seen that the reliability levels for this
design already satisfy the minimum values shown
in Tables 4 and 5. Starting from these preliminary
parameters, the optimization process was applied,
as described, using the target minimum reliabili-
ties from Table 4 (Paulay y Priestley, 1992). The
optimization took into account the bounds for
the design parameters (minimum and maximum
cross-sectional dimensions and steel reinforce-
ment ratios) specified in Table 1 and used in the
development of the response databases.
The final design parameters obtained are shown
in Table 7, with the final total cost and the reli-
ability levels achieved for each of the performance
levels.
It is interesting to compare the optimum solu-
tions obtained when the optimization is started
from conditions other than the preliminary design.
With this purpose, Table 8 shows five different
initial combinations of design parameters, two of
them “over-dimensioned in comparison to the
preliminary design” and two “under-dimen-
sioned”.
The evolution of the total cost during the op-
timization, in terms of the number of steps, is
shown in Figure 6. Each step corresponds to one
evaluation of total cost with all reliability con-
straints being satisfied. This figure shows that the
optimization converges to approximately the same
final total cost, regardless of the choice for the
initial design parameters. Those corresponding
to the preliminary design provide a cost solution
which is already quite close to the optimum,
showing the adequacy of the methodology em-
ployed for the preliminary estimation.
When the optimization starts from an under-
designed combination, the first cost that is calcu-
lated is controlled by the requirement to satisfy
the reliability constraints. It can be observed in
Figure 6 that the number of steps needed to achieve
the optimal solution varies, implying a varying
number of anchor combinations and process
repetitions. In each case, no lower total cost was
found when the optimization search radius r 1 was

Design parameter Initial value

x d (1) = X ( 3 = h b [ cm ]

50

x d (2) = X ( 5 = h c [ cm ]

45

x d (3) = X

( 6 = ρ

0.00804

span

x d (4) = X

( 7 = ρ

0.01143

end

x d (5) = X

( 8 = ρ

0.03148

col

6.4 Optimization Constraints:
Minimum Reliability Levels

Minimum target reliabilities for each performance
level are specified in terms of tolerable annual
probabilities of non-performance, Pf annual . Using
a Poisson arrival process, with a mean arrival
rate ν, the exceedence probability Pf annual can be
converted to a probability of non-performance P f
for the earthquake event, and finally expressed
as a target event reliability index β :

≅ − −

1

Pf = − 1 . exp − ν P → P → β Φ ( P )

anual f f f

(48)

A mean arrival rate ν = 0.20, for earthquakes
with magnitudes M ≥ 5, was used for the city of
Mendoza, Argentina. Table 4 shows annual and
event target probabilities as recommended by
Paulay and Priestley (1992). In this application,
more stringent targets were also considered, as
shown in Table 5, to evaluate the sensitivity of
the optimization outcome to the levels used for
the targets.

6.5 Optimization Results

The total cost and the reliability levels correspond-
ing to the initial preliminary design are shown in