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3. MULTIOBJECTIVE DESIGN
OPTIMIZATION PROBLEM
the steel reinforcement cost
f
1
, as given in Eq.
(11).
3.1 Objective Function
3.2 Expected Future Structural Loss
f
2
In seismic performance based design, the total
life-cycle cost of structures should be considered
and the final design should be established based
on a good balance between the initial structural
cost and its loss expectation in the design life
period. Due to the fact that the life-cycle cost
involves the consideration of construction cost
and damage loss which are inherently conflicting,
i.e., the decrease of one increases the other, it is
necessary to formulate the design optimization
as a multi-objective optimization problem. In
this research study, the initial construction cost
of an RC building frame is estimated simply in
terms of the material costs of the concrete mem-
ber sizes and steel reinforcements. In contrast,
the establishment of an approximate but rational
cost function to explicitly represent the expected
damage loss of building structures in terms of
sizing design variables remains one of the chal-
lenging difficulties.
If the topology of a building structural system
is predefined, the design objective is to minimize
the multi-objective life-cycle cost of the structure.
Using
F
to denote the life-cycle cost function
including the initial structure cost
f
1
and ex-
pected future damage loss
f
2
, the multi-objective
function can be expressed as
The structural failure losses due to an earthquake
attack consist of direct loss and indirect loss. The
direct loss is the cost of repair or replacement of
structural members, contents, non-structural
components and equipment, and so on. The indi-
rect loss may include the losses associated with
structural malfunction, injuries and fatalities,
psychological and political influence, etc. The
accurate estimation of both direct and indirect
losses is generally a very complex task involving
not only engineering analysis but also many
other issues. The failure of each structural perfor-
mance may lead to a different failure loss.
Herein, Li's work (1998) is introduced as an
example of derivation of the expected future
structural loss
f
2
for convenient discussion in the
following. According to the work by Li (1998),
the total loss expectation can be defined by the
summation of the product of the occurrence prob-
ability of earthquake and the system failure loss.
That is, for
N
r
performance levels, the loss ex-
pectation function
f
2
can be stated as
N
r
∑
f
=
P L
r
×
(31)
2
r
r
=
1
where
r
denotes the seismic design level and
r
;
P
r
is the occurrence probabil-
ity of an earthquake for the rth seismic design
level, which can be determined from specified
code requirements;
L
r
is the structural failure loss
including direct and indirect losses under the rth
seismic design level. In Li's study (1998), the
damage loss of a structure corresponding to dif-
ferent discrete levels of performance criteria is
expressed in terms of the maximum inter-story
drift index under minor, moderate and severe
=
1 2
,
, ...,
N
r
=
{
}
Minimize:
F
f
,
f
(30)
1
2
In order to facilitate the numerical solution of
the optimization problem, the implicit objective
function in Eq. (11) needs to be first expressed
explicitly in terms of the design variables. As
presented previously, the initial structure cost
f
1
of an RC framework is simply assumed to be the
summation of the concrete material cost
f
1
and
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