Optimal Seismic Performance- Based Design of Reinforced Concrete Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 213

responses can be checked against acceptability
limits on both global system levels (such as the
lateral load stability and the inter-story drift) and
local element levels (such as the element strength
and the sectional plastic rotation) (ATC-40 1996).
When the responses of a structure do not meet the
targeted performance level, the structure needs to
be resized and the design process repeated until
a solution for the desired performance level is
reached. In general, the determination of the satis-
factory performance response that fulfils both the
system level response and element level response
requires a highly iterative trial-and-error design
procedure even with the aid of today's finite ele-
ment analysis software.

element to be used to control the inelastic perfor-
mance of the structure. Based on these consider-
ations, in the second phase, the tension steel re-
inforcement ratio, ρ i , and the compression steel
reinforcement ratio, ′ ρ i , of rectangular cross sec-

tions are taken as design variables, for a flexural
concrete building having i =1, 2,?, N i members
and 2 N i plastic hinges (assuming one hinge at
each end of a member). The width B i and depth
D i of the cross section are fixed in this phase.
If the topology of a building structural system
is predefined, the design objective of the reinforced
concrete framework in the second inelastic design
phase is to minimize the steel reinforcement cost
f 1 , which can be expressed in terms of steel re-
inforcement design variables:

2.2.2 Design Optimization Problem and
Explicit Drift Formulation

Minimize: steel cost

N i

∑ (

+ ′

′

f = w L ρ L ρ ) (10)

While the concrete material plays an important
role in controlling the elastic displacement re-
sponse of a RC building, the steel reinforcement
can have a significant effect on the inelastic dis-
placement and ductility of the RC building beyond
the linear elastic limit. Moreover, when an RC
structure works in the inelastic stage, steel rein-
forcement is generally the more cost-effective

1 s si si i si i

i = 1

where w si is the cost coefficient for steel reinforce-
ments; and L si and ′ L si are the lengths of the

lower and upper steel reinforcements for member
i . Herein, only the longitudinal reinforcement of
member sections is considered as design variables
and the transverse reinforcement is assumed to

Figure 1. Nonlinear analysis procedure