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of perturbation based approach for evaluation of
nominal response and its dispersion in this regard
may be found in Chen, Song, & Chen, (2007).
It has been numerically shown that the error in
estimation by linear perturbation approximation
goes up as the amplitude of uncertainty of the
interval variables increases. Thus, the approach
will be applicable for systems having small de-
gree of uncertainty so that the linear perturbation
based analysis will be satisfactory. An extension
to the linear perturbation analysis is to use higher
order perturbation method. Stochastic simulation
approach may be also applied for larger ampli-
tude of uncertainty (Datta, 2010). However, the
stochastic simulation will require assumptions
of pdf functions. One may choose conservative
uniform distribution for this purpose. This aspect
needs further study and it is beyond the scope of
the present chapter.
is the vector comprising of displacements of the
MDOF system due to ground motion u t
g ( ) at
base of the system. { } is the influence coefficient
matrix. The displacement of the system subjected
to ground motion, 
( ) = ( )
u t
u

e
i
ω can be as-
t
ω
g
g
{
} =
{
}
( )
( )
e i
t
sumed as Y
ω ω , where
H Y ( { } is the complex frequency response func-
tion (FRF) vector. Using this, the equation of
motion can be expressed in the frequency domain
as following:
t
H Y
F ω .
(8)
(
)
{
} = − [
{
} = ( )
{
}
[
K
] [
2
M
] + [
i
C H
]
( )
M L
]{ } ( )
u g
 i.e.
D
( )
H
( )
ω
ω
ω
ω
ω
ω
Y
Y
is the dynamic stiffness
matrix, F ( { } is the forcing vector and u g ( )
is the Fourier amplitude of ground motion. In Eq.
(8), H Y ( { } is a function of u and can be ex-
plicitly re-written as,
In the above, D ( )
Stochastic Structural
Optimization (SSO)
}
(9)
{
} = (
{
}
{
} = (
1
{
(
)
(
)
)
(
)
)
(
)
D
ω
,
u H
ω
,
u
F
ω
,
u
or
H
ω
,
u
D
ω
,
u
F
ω
,
u
Y
Y
As already discussed, the SSO procedure under
random earthquake load involves solution of a
nonlinear optimization problem involving sto-
chastic performance measure. The constraint in
a generic form can be stated as, the probability
of not to exceed a given threshold of a stochastic
response measure in a given time period, must be
greater than some prescribed minimum value. The
related formulation of stochastic dynamic analysis
to obtain the constraint of the related optimization
problem is briefly presented here.
The dynamic equilibrium equation of a linear
multi degree of freedom (MDOF) system under
seismic excitation can be written as,
For a linear dynamic system, the power spec-
tral density (PSD) function S
of any
response variable Y(t) can be readily obtained as
(Lutes, & Sarkani, 1997; Datta, 2010),
(
)
ω ,
u
YY
T
=
{
}
{
}
(
)
(
)
( )
(
)
*
S
ω
,
u
H
ω
,
u
S u u
ω
H
ω
,
u
YY
Y
 
Y
g g
(10)
{
} is the complex conjugate
(
)
Where, H
*
ω
u
Y
{
} and S u g g
  ( ) is the known PSD
function of stochastic ground motion.
The records of the ground motion at site are
necessary for realistic seismic reliability analy-
sis. However, in scarcity of sufficient data for
statistical descriptions, various statistical models
are developed to describe the stochastic ground
motion process. The simplest such stochastic
(
)
of H
Y ω ,
u
{
} + [
{
} + [

{
} = − [
[
]
( )
]
( )
]
( )
]{ } ( )
M Y
t
C Y
t
K Y
t
M L
u t
g

(7)
Where, [ ] , [ ] and [ ] are the global mass,
damping and stiffness matrix, respectively. Y ( )
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