Optimal Seismic Performance- Based Design of Reinforced Concrete Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 227

the design return period of 50 years. This building
frame is assumed to be a Class B building in this
example and the coefficients of expected failure
loss are shown in Table 1. Other conditions stay
the same as Case B.
Figure 9 presents a Pareto optimal set, which
provides the best compromise between the initial
structure cost, f 1 , and the damage loss, f 2 . The
minimum structure cost, f 1 , for the single objec-
tive optimal design problem given in Eqs. (36)-
(39), i.e., corresponds to Point C0 in Figure 8
where f 1 = US $4184 and the corresponding dam-
age loss, f 2 , is found to be US $7522 . The optimal
steel reinforcement ratios corresponding to Point
C0 are taken as initial steel ratios of members in
the subsequent multi-objective inelastic design
optimization (see Eqs. (33)-(35)) and the optimal
Pareto set curve can then be generated. Besides,
US $4184 at Point C0 is the minimum value of
the initial structure cost, f 1 , satisfying design
constraints at this Pareto curve; correspondingly,

US $7522 at Point C0 is the maximum value of
the damage loss, f 2 .
Firstly, it is observed from Figure 9 that an
improvement in the damage loss leads to an in-
crease in the initial material cost of the structure.
From the Pareto optimal set, a designer can have
an overview of the tradeoffs. Secondly, any point
in the Pareto optimal set can be selected as an
optimum design, which represents a designer's
preference. A designer has many choices rather
than only one and he/she may finally choose a
particular solution, which he/she thinks, should
be realized. Thirdly, the life-cycle cost can be
further written as F

1 1 2 2 , where ϕ 1 and
ϕ 2 are the weighting factors and their magnitudes
are dependent on a designer's decision. A preferred
balance of the initial and damage losses can
therefore be obtained. It is worth noting that dif-
ferent values of ϕ 1 and ϕ 2 will lead to different
minimum values of the life-cycle cost.
Figure 6 lists the optimal steel reinforcement
ratios at Points C 0 and C*, respectively. The steel

= ϕ f + ϕ f

Figure 9. A Pareto optimal set of Case C