Performance-Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 40

Table 6. Initial costs and reliabilities, preliminary design

Costs [$] Performance level β

Initial C 0 ( x d ) 2839 Operational 1.837

Repair C d ( x d ) 1113 Life safety 2.846

Total 3952 Collapse 3.489

6.6 Optimization Sensitivity to the
Assumed Damage-Cost Relationship

We analyze now the effect of changing the
more significant constraints, 2 and 3, using as a
base the minimum targets specified in Table 4.
Thus, β 2 and β 3 are each individually changed by
± 10%, with the corresponding results shown in
Figure 8.
When β 2 is decreased by 10%, the results
are identical to those shown in Figure 7, since
constraint 2 is more easily met and constraint 3
remains the determinant one. On the other hand,
if β 2 is increased by 10%, constraints 2 and 3 now
become jointly important, leading to different
combinations of the design parameters but still
with similar minimum total cost.
If the target β 3 is increased by 10%, the total
cost for the optimum solution also increases as
constraint 3 was already the most important. On the
other hand, if β 3 is decreased by 10%, constraints
2 and 3 now have a similar influence, resulting
again in a similar minimum total cost but with a
different combination of design parameters.

Finally we consider the effect on the optimization
of the form used for the cost-damage relationship
(Eq.(29)). While maintaining b = 1, the relationship
is modified by changing the level of damage index
DIES LIMIT at which the structure is considered to be
destroyed and needs to be replaced. Thus, from the
value assumed for the base results, DIES LIMIT = 0.6,
two different possibilities were considered: 0.4 or
0.8. In the first, the cost increases more rapidly as
a function of damage, while the reverse occurs in
the second case. The results are shown in Figure
9, with corresponding differences of 15% and -7%
with respect to the minimum total cost for the base
relationship. However, the optimum values for the
design parameters remain unaffected.

Table 7. Optimum results starting from the preliminary design

Optimal
value

Performance
level

Design parameter Costs [$] β

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

57.0 Initial C 0 ( x d, ) 2601 Operational 1.697

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

40.3 Repair C d ( x d ) 1020 Life safety 2.422

x d (3) = X

( 6 = ρ

0.00617 Total 3621 Collapse 3.102

span

x d (4) = X

( 7 = ρ

0.01028

end

x d (5) = X

( 8 = ρ

0.02130

col