Performance-Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 39

Table 4. Target minimum reliability levels (Paulay and Priestley, 1992)

Performance level Pf annual β annual Pf β

Operational 2 x 10 -2 2.054 0.10101 x 10 0 1.276

Life safety 2 x 10 -3 2.878 0.10010 x 10 -1 2.326

Collapse 2 x 10 -4 3.540 0.10001 x 10 -2 3.090

Table 5. Target minimum reliability levels proposed

Performance level Pf annual β annual Pf β

Operational 1 x 10 -2 2.326 0.50252 x 10 -1 1.642

Life safety 1 x 10 -3 3.090 0.50025 x 10 -2 2.576

Collapse 1 x 10 -4 3.719 0.50003 x 10 -3 3.291

6.6 Optimization Sensitivity to
Specified Minimum Target
Reliabilities

expanded, indicating that at least a local minimum
had been found.
The different initial conditions lead eventually
to essentially the same optimum total cost, but
the solutions contain different optimal combina-
tions of the individual parameters. This implies
that the cost function may have different local
minima, but that in this case these minima are
approximately of the same magnitude, with the
largest difference in total cost being only of the
order of 2.1%. However, the difference in the
column steel reinforcement ratio, for example,
between using initially either the preliminary
design or the Under-design 1, is closer to 81%.
The final optimum results are shown in Table 9. It
should be emphasized that all these different solu-
tions satisfy the minimum reliability constraints,
and that Table 9 implies a trade-off between the
optimal depth of the beam, for example, and the
optimal amount of steel used. If the constraints
had been formulated in terms of specified narrow
reliability intervals, rather than just only minima,
the number of optimum design possibilities would
have been correspondingly narrowed.

In this section we discuss the influence that the
definition of the minimum reliability constraints
may have in the optimum solution. Figure 7 shows
the evolution of the total cost either for the target
reliabilities (β) in Table 4 (Paulay and Priestley,
1992) or for those proposed in Table 5. In both
cases the starting configuration is the preliminary
design. Figure 8 shows higher total costs as a result
of imposing higher target reliability constraints,
as expected.
At the optimum solution, the constraints for
the different performance levels are met but not
to the same degree. For example, for the minimum
targets in Table 4, β 1 = 1.276, β 2 = 2.326 and β 3 =
3.090, the final (optimum) reliability levels in
Table 7 show β 1 = 1.697, β 2 = 2.422 and β 3 = 3.102.
Thus, the constraint is just met for the collapse
performance level, and satisfied with a greater
margin for performance levels 1 and 2.
Similar results are obtained using the proposed
target reliabilities from Table 5 ( β 1 = 1.642, β 2 =
2.576 and β 3 = 3.291). In this case, the reliability
levels achieved at the optimum solution are β 1 =
1.691, β 2 = 2.749 and β 3 = 3.296.