Performance-Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 30

Figure 2. Schematic for the design parameters' optimization process

either satisfy the reliability constraints or not.
Should the preliminary design parameters x d 0 not
satisfy the reliability constraints, a search is made
according to the following steps, Figure 2(b),
until one combination x d is found satisfying the
constraints. This combination is then taken as the
initial anchor.
Next, n combinations of x d are chosen randomly
in the neighborhood of x d 0 , Figure 2(c), choosing
each of the design parameters, x di , within the inter-
val x d 0i ± r 1 x d 0i (initially r 1 = 0.1). The reliability
constraints are verified for each combination,
choosing a new combination if the constraints are
not satisfied. The total cost is calculated for the
n combinations, keeping the one corresponding
to the lowest cost. If this cost is lower than the
one corresponding to the anchor, then the corre-
sponding combination becomes the new anchor
and the process is re-initiated. The work for this
Chapter used n = 100.
If none of the n combinations leads to a cost
lower than the one corresponding to the anchor, the
search is densified by choosing more combinations
within the interval, until one is found leading to a
lower cost. The work for this chapter restricted the
number of combinations to a maximum of n = 500.

If still no combinations are found leading to
a cost lower than that for the anchor, the search
factor radius r 1 is amplified to r 2 = r 1 + Δ r , with
Δ r = 0.3 used in this work. A number m of com-
binations x d is chosen within the amplified search
volume, satisfying the reliability constraints, and
maintaining the same density as in the previous
search domain corresponding to the radio r 1. Of
the m combinations (maximum of 2000), the
first that is associated with a cost lower than that
for the anchor is retained and used as the new
anchor. This sequence is repeated three times in
order to reduce the possibility of encountering a
local minimum.
The search process is stopped when no com-
bination is found, satisfying the constraints, with
a cost lower than for the anchor.

5. OBJECTIVE FUNCTION:
THE TOTAL COST

Costs associated with the occurrence of earth-
quakes and associated damage have been discussed
by Lagaros et al. (2010). Here, the total cost C ( x d )
in Eq.(1) includes the initial construction cost