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the optimization problem. As the MLSM based
approximation technique can capture the actual
response even beyond the sampling zone, the
convergence is faster than the conventional LSM
based techniques (Kim, Wang, & Choi, 2005).
min
, and
β
are taken as 0.99, 10 sec and
3
σ
Y
(
σ
Y
is the
maximum RMS displacement), respectively. The
MLSM based adaptive RSM technique as dis-
cussed previously is used to approximate the
constraint function. In the present numerical study,
the second-order polynomial without cross terms
is considered. The DOE is constructed consider-
ing the centre and axial points following the
Saturated Design (SD) method (Bucher, & Macke,
2005). However, the points chosen are at the
nominal value (
x
i
) of the input variable (xi) and
at axial points xi =
x
i
± hi
∆
x
i
, where hi is a
positive integer. For each input variable six axial
points (hi=1, 2, … 6) are considered on each axis
taking
∆
x
i
as 5% of
x
i
to cover the different
amplitude levels of uncertainty. Thus, for the
present study, the number of required training
points with respect to dimension of the input vec-
tor (N) is: (6x2N+1) i.e. the total number of
sampling points is sixty one with five input vari-
ables. As more axial points are considered than
that required by the SD method, the design be-
comes a redundant design (Bucher, & Macke,
2005). To study the effectiveness of the MLSM
based adaptive RSM, the computed responses are
compared with that obtained by the conventional
LSM based RSM. The maximum RMS displace-
ment of the frame is shown for a wide range of
column size in Figure 3. The results obtained by
the direct random vibration analysis are also shown
in the same figure for ease in comparison. It can
be readily observed from the figure that the pre-
dicted responses by the MLSM based RSM better
matches with the direct random vibration analysis
results. The error using the conventional LSM
based RSM drastically increases beyond the range
of data points which are used in the DOE to con-
struct the metamodel. During the iteration stage
of any gradient-based optimization algorithm, the
DVs may take values outside the sampling range
Unless mentioned specifically
R
T
Numerical Study
A three storied concrete building frame (as shown
in Figure 2) subjected to earthquake motion is
taken up to elucidate the effectiveness of the
proposed RDO approach. The frame structure is
idealized by fifteen two-nodded plane frame ele-
ments having three degrees of freedom at each
node. Both the DVs and DPs are considered to
be of UBB type and are described by their respec-
tive dispersions,
∆
u
i
, representing the maximum
possible ranges of variations expressed in terms
of the percentage of the corresponding nominal
values. The necessary information with associ-
ated notations are summarized in Table 1. The
deterministic dimension L and h are taken as 6.0
m and 4.0 m, respectively. The natural frequency
(
ω
g
) and the damping ratio (
ξ
g
) of the soil layer
are considered to be 18.85 rad/sec and 0.65, re-
spectively. The peak ground acceleration (
x
g
,max
)
is taken as 0.2g, where g is the acceleration due
to gravity.
The objective function f(u) considered in the
present study is the weight of the frame. Apart
from the stochastic constraint, as discussed ear-
lier, a size constraint (g2) limiting the ratio between
the depth and width of the beam members is also
considered in the SSO formulation. The SSO
formulation under earthquake load with deter-
ministic DVs and DPs can be defined as:
x
=
{
}
find
b
d
b
to minimize:
f
=
(
9
hb
2
+
6
Lb d
)
ρ
c
b
b
c
b b
c
2
1
σ
σ
1
2
β
σ
log
(
1
/
mi
R
)
such that:
g
(
x z
,
)
:
Y
Y
exp
−
−
n
≤
0
1
π
T
Y
g
2
x z
(
,
)
:
d
−
3
b
≤
0
0.1 ≤
b b d
c
,
,
≤
1 2
b
b
b
b
(30)
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