Geology Reference
In-Depth Information
Box 2.
N
t
N
t
f
f
=
∑
x t
∫
∫
( )
=
Re
( ) cos(
τ
ω τ ϕ τ
−
)
d
+
C x t
;
( )
=
Re
( )(
τ
t
−
τ
)
cos(
ω τ ϕ τ
i
−
)
d
+
C t C
+
g
i
i
i
1
g
i
i
1
2
i
1
i
=
1
0
0
(8)
Finally, the problem of deriving critical future
earthquake loads on inelastic structures can be
posed as determining the optimization variables
y
N
f
∞
∑
∫
C
=
0
;
C
= −
Re
( ) cos(
τ
ω τ ϕ τ
−
)
d
2
1
i
i
i
i
=
1
0
ϕ ϕ ϕ
such that the
damage index DIPA is maximized subjected to
the constraints of Eq (10). The solution to this
nonlinear constrained optimization problem is
tackled by using the sequential quadratic program-
ming method (Arora, 2004). The following con-
vergence criteria are adopted:
=
{ ,
R R
, ...,
R
N
,
,
, ...,
}
t
(9)
1
2
1
2
N
f
f
The constraints of Eq (7) can be expressed in
terms of the variables
R
,
ϕ
=
1 2
,
i
,
, ...,
N
(see
i
i
f
Box 3)
Here
i
= −1
. To quantify the constraints
quantities E, M1, M2, M3,
M
4
( )
ω
and
M
5
(
ω
it
is assumed that a set of Nr earthquake records
denoted by
v
|
f
−
f
|
≤
ε
;
|
y
−
y
|
≤
ε
(12)
=
1 2
are available for
the site under consideration or from other sites
with similar geological soil conditions. The values
of energy, PGA, PGV and PGD are obtained for
each of these records. The highest of these values
across all records define E, M1, M2 and M3. The
available records are further normalized such that
the Arias intensity of each record is set to unity
( ),
t
i
,
, ...,
N
j
j
−
1
1
i j
,
i j
,
−
1
2
gi
r
Herein,
f
j
is the objective function at the
j
th
iteration,
y
i,j
is the
i
ith optimization variable at the
j
th iteration and
ε
1
,
are small quantities to be
specified. The structure inelastic deformation is
estimated using the Newmark β-method which is
built as a subroutine inside the optimization pro-
gram. The details of the optimization procedures
involved in the computation of the critical earth-
quake and the associated damage index are shown
in Figure 2. Further details can also be found in
Abbas (2006).
It may be emphasized that the quantities
µ
( )
∞
∫
v
2
1 2
/
(i.e.,
[
( )
t dt
]
=
Arias, 1970), and are
1
,
gi
0
v
gi
=
1
. The bounds
M
4
(
ω
and
M
5
(
ω
are obtained as:
N
r
denoted by
{
}
i
t
M
( )
ω
=
E
max |
V
( ) |;
ω
M
( )
ω
=
E
min |
V
( ) |
ω
and
E t
H
( )
do not reach their respective maxima
at the same time. Therefore, the optimization is
performed at discrete points of time and the op-
timal solution
4
gi
5
gi
1
≤ ≤
i N
1
≤ ≤
i N
r
r
(11)
ω
=
1 2
denotes the Fou-
rier transform of the ith normalized accelerogram
v
Here
V
( ),
i
,
, ...,
N
gi
r
*
*
*
*
*
*
*
t
y
=
[
R R
,
, ...,
R
N
,
ϕ ϕ
,
, ...,
ϕ
]
gi
( )
. The bound
M
4
(
ω
has been considered
earlier (Shinozuka, 1970, Takewaki, 2001, 2002).
The lower bound was considered by Moustafa
(2002) and Abbas & Manohar (2002).
t
1
2
1
2
N
f
f
is the one that produces the maximum DIPA across
all time points. The critical earthquake loads are
characterized in terms of the critical accelerations
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