Optimal Seismic Performance- Based Design of Reinforced Concrete Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 223

structure is to be minimized subject to the elastic
spectral drift constraints under the minor earth-
quake loading condition. The unit construction
cost of concrete is assumed to be US$90/m3,
including the concrete material cost and labour
cost. Elastic inter-story drift constraints are taken
into account with an allowable inter-story drift
ratio limit of 1/450. The initial sizes are arbi-
trarily chosen to be 350 × 350mm for the columns
and 200 × 350mm for the beams. Size bounds are
defined as 350~1000mm for the depths and the
widths of the columns, 200~350mm for the widths
of the beams and 350~450mm for the depths of
the beams.
In the inelastic phase of the optimization, the
design objective is to minimize the steel reinforce-
ment cost subject to the performance-based in-
elastic drift constraints under the severe earthquake
loading condition. The unit construction cost of
steel reinforcement is assumed to be US$960/
tonne including the steel material cost and the
labour cost. Inelastic inter-story drift constraints
are considered with an allowable inter-story drift
ratio limit of 1/100 and the P-Delta effect is con-
sidered in the example. Initial reinforcement ratios
are calculated based on the strength requirements
of members after the elastic phase design process.
Such strength-based reinforcement ratios are
taken as the lower bounds for the inelastic design
process. Their upper bounds are assumed to be
6.0% for columns and 4.0% for beams. For sim-
plicity, symmetrical arrangement of steel rein-
forcement is assumed such that ρ

structure. During the pushover analysis, the lateral
loads are applied incrementally in proportion to the
initial loads but the gravity loads, shown in Figure
4(a), are assumed to be fixed. The design process
is deemed to converge when the difference in the
structure costs for two successive design cycles
is within 0.5% and when the difference between
the active inter-story drift value and its allowable
limit at the performance point is within 0.5%.
Figure 5 presents the optimal design history
for both the elastic and inelastic drift optimiza-
tion processes. In the elastic optimal design,
rapid and steady convergence of the concrete cost
from the initial US$1528 to the finial US$2205
after 6 design cycles has been found. The rapid
convergence can be explained by the fact that the
member force distribution for such structures is
somewhat insensitive to changes in member size.
In contrast, the inelastic optimal design process
converges quite slowly but steadily in 11 design
cycles. Relatively slow, but steady, convergence is
inevitable due to the need of maintaining a small
change in the steel reinforced ratios during the
nonlinear design optimization process. However,
the OC design method is able to achieve a smooth
and steady convergence to the optimal design as
evidently shown in Figure 5.
In the elastic design optimization subject to
elastic inter-story drift constraints, the total mate-
rial cost only includes the concrete cost, which is
minimized from the arbitrary, initial US$1528 to
the final US$2205. In the inelastic design opti-
mization subject to inelastic inter-story drift
constraints, the initial total cost US$3898 consists
of the steel reinforcement cost of US$1693 cal-
culated based on code-specified strength require-
ments and the concrete cost of US$2205, which
is to be fixed in this inelastic phase of the opti-
mization. Since initial violations in inelastic inter-
story drift constraints are found in the strength-
based design, an increase in the steel reinforcement
is necessary, resulting in a final total construction
cost of US$4184. In addition, it is found from
Case B that there is a relatively large increase of

i = ′ . Flex-
ural moment hinges and axial-moment hinges are
assigned to the end locations of the beams and
columns, respectively. The ultimate plastic hinge
rotation, θ U , is assumed to be 0.02 radian for the
moment hinges on the beams and 0.015 radian
for the axial-moment hinges on the columns.
Initial applied lateral loads applied in push-
over analysis are shown in Figure 4(a), which is
proportional to the product of the story mass and
the first mode shape of the elastic model of the

ρ