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example if it is desired to provide some constraint
on protection of equipments and contents in a floor,
some constraints such as limits on velocity or ac-
celeration of floors have to be added to the design
optimization problem. Evidently the formulation
of these particular constraints requires employing
a nonlinear dynamic analysis. It is noted that the
C
1
,
C
2
,
C
3
and
C
4
group of constraints may be
different for various levels of performances, and
they all can enter into the optimization problem.
The difference between research works on the
OPBSD is a consequence of their assumption on
the objective function, constraints and the solu-
tion procedure. Some examples of recent research
works in this ground that utilize the classical
optimization algorithms are discussed hereunder.
Many other researches have been excluded not
because of their scientific value, but because of
their weak relation to the subject of this chapter. For
example, Ganzerli et al. (2000), who were prob-
ably the first that incorporated pushover analysis
and the performance-based design concept, used
the idea of OPBSD for a one story one bay R/C
frame with pushover analysis. They minimized the
cost of R/C frame including costs of concrete and
reinforcements under plastic rotation constraints
at the ends of members. They considered perfor-
mance constraints at immediate occupancy, and
checked for the satisfaction of design conditions
in other levels. However, in their publication,
there was no sensitivity analysis and no explicit
design constraints. Therefore, their work is not
presented here in detail.
earthquake. The second phase involved mini-
mizing the cost of steel reinforcement subject to
constraints on inelastic displacements. Pushover
analysis was performed based on the assumption
that the fundamental mode of vibration was the
predominate response and did not change during
nonlinear behavior.
The details of the first phase that produces the
optimal dimensions of members for minor wind
or earthquake loadings was previously addressed
by Moharrami (1993) and is not the concern of
this chapter. The detail of the second phase, as
it is addressed by Zou and Chan, is presented
hereunder. To establish a nonlinear analysis pro-
cedure, the following assumptions were made for
the structural model of reinforced concrete frame.
1. All inelastic deformations occur at the plastic
hinges, which are located at the ends of each
frame member and, members are perfectly
elastic between the plastic hinges.
2. Beam -column connections are rigid zone,
and the plastic hinges are assumed to be
frictionless and have zero length.
As a requirement for the solution of nonlinear
equations of equilibrium, Eq.(14), the authors
expressed the pushover displacement in two parts
i.e. elastic and plastic (inelastic) displacements.
∆
=
∆
+
∆
,
(42)
j
j e
,
j p
In Eq.(42),
j
stands for the story number. Us-
ing the principle of virtual load method, Δ
j,e
and
Δ
j,p
can be obtained from Eq.(43) and Eq.(44)
respectively. It is to be noted that
f
s and
m
s are
internal forces and moments due to application
of a unit load in the direction in question, and
F
s
and
M
s are internal forces due to applied loads.
Example 1
The procedure of OPBSD of reinforced concrete
frames suggested by Zou and Chan (2005) is a
good example to be presented here. These au-
thors decomposed the optimal design process
into two single-criterion phases. The first phase
involved an elastic design optimization in which
the cost of concrete is minimized subject to elastic
spectral displacement constraints due to a minor
F f
EA
F f
GA
F f
GA
M m
GI
N
M m
EI
M m
EI
j
L
=
i
X X
Y Y
Z Z
X
X
∆
=
∫
+
+
+
+
Y
yj
+
Z
zj
dx
j
j
j
j
j e
0
i
1
X
Y
Z
X
Y
X
i
(43)
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