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T
b1
and T
b2
, respectively, which can be calculated
by Equation (12); and the yield displacement,
v
y
,
which can be calculated for different isolators by
experiments. Also, the ratio of k
b2
/ k
b1
, defined as
γ
, can be obtained from experiments.
In this study, the values of
v
y
and
γ
are assumed
to be about 2.5 cm and 0.142, respectively, given
in references (Matsagar & Jangid, 2004; Rodellar
& Manosa, 2003). Finally, damping ratio of elas-
tic phase
ξ
b
1
and that of the plastic phase
ξ
b
2
are
defined by Equation (13).
•
P
c
=0.25
•
P
m
=0.01
In order to apply the multi-objective genetic
algorithm optimizer, first using a single objective
GA, each of the objective functions is optimized
and the results are shown in Figure 3 for dis-
placement of the building top story and that of
the base isolators. Then, using these results the
Pareto-optimal front diagram is obtained (Figure
4), from which the following optimal values are
evaluated for bilinear isolators (Pourzeynali &
Zarif, 2008):
T
=
2
π
(
m k
)
,
T
=
2
π
(
m k
)
b
s
b
b
s
b
1
1
2
2
(12)
k
=
k
/
k
=
0 40
.
,
m
=
m m
/
=
0 35
.
,
0
b
1
1
0
b
1
ξ
=
0 19
.
,
ξ
=
0 50
.
b
b
ξ
=
c T
/ (
4
π
m
)
,
ξ
=
c T
/ (
4
π
m
)
(13)
1
2
b
b
b
s
b
b
b
s
1
1
2
2
Table 5 shows the controlled responses of the
building stories for four selected earthquakes;
ensemble average responses for 18 reference
earthquakes; as well as, ensemble average reduc-
tion ratios for the same reference earthquakes
calculated for all stories of the building. It is seen
Here also the same parameters of the linear
isolators have been chosen for GA optimizer:
•
Number of initial populations = 25
•
Number of generations = 300
Figure 3. Performance of the genetic algorithm with single objective function: (a) Maximum displace-
ment of bilinear base isolator; (b) Maximum displacement of the isolated building top story
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