Civil Engineering Reference
In-Depth Information
The mean square integral response up to the time instant
t
= corre-
sponding to a specific frequency band with
a
j
j
=σ is obtainable as well
from the following equation:
i
2
K
EW xab
a
[
(
,)]
∑
j
ψ
gj
i
ξ
()
t
=
(2.17)
tb
=
i
j
i
=
0
At this stage we are interested in finding out the total expected energy of
the ground motion process. This is obtained by using the Parseval identity
in Equation (2.14) in conjunction with the following orthogonal relation:
∞
∫
ψωψωω=δδ
−∞
ˆ
()
ˆ
*
()
d
(2.18)
ab
,
ab
,
jk
il
j
i
j
l
The orthogonal relation will be explained later in the chapter. The total
expected energy of the process thus will take the following expression:
∞
∞
∫
∑∑
Kb
a
∫
2
2
2
ˆ
EX
[|
( |]
ωω=
d
EWxab
(,)
ψ ωω
()
d
(2.19)
ψ
gj
i
ab
j
,
l
j
i
j
−∞
−∞
denotes the Fourier transform of the ground
motion process,
x
g
(
t
). The expectation of the squared wavelet coefficients
[
In the above equation, ω
X
()
2
gj i
correspond to nonoverlapping energy bands for different
j
values. Thus, the expected energy of
EW xab
(,)]
ψ
in the frequency band corre-
xt
g
()
sponding to
a
j
is given as
∞
∞
∫
∑
Kb
a
∫
j
2
,)]|
ˆ
2
2
(2.20)
EX
[|
( |]
ωω=
d
EW xab
[
(
ψ ωω
( |
d
ψ
gj
i
ab
j
,
l
j
i
−∞
−∞
The above equation may also be rewritten as
∞
∑
Kb
a
∫
j
2
2
EX
[|
( |]
ωω=
d
EW xab
[
(
,)]
(2.21)
ψ
gj
i
j
i
−∞
so that
∞
∞
∑
∫
∫
2
j
2
EX
[|
( |]
ωω=
d
E F
[|
( |]
ω
d
ω
(2.22)
j
−∞
−∞
