Civil Engineering Reference
In-Depth Information
are given by
(
)
T
i
∫
φ M φ
dx
ej
M
ae
L
ij
exp
μ
=
=
(4.71)
ae
ij
M
M
i
i
(
)
T
i
∫
φ K φ
dx
ej
K
ae
L
ij
exp
κ
=
=
(4.72)
ae
ij
2
2
M
M
ω
ω
i
i
i
i
(
)
T
i
∫
φ C φ
dx
ej
C
ae
L
ij
exp
ζ
=
=
(4.73)
ae
ij
2
ω
M
2
ω
M
i
i
i
i
Returning to Eq. 4.66, the response spectral density matrix (
N
by
mo
N
and contain-
ing single-sided spectra) is obtained from the basic definition of spectra as expressed
from the Fourier amplitudes, and thus, the following development applies:
mod
1
1
(
) (
*
)
T
(
)
⎡
⎤
ˆ
ˆ
()
*
T
S
ω
=
lim
a
⋅
a
=
lim
H a
⋅
H a
⎢
⎥
ˆ
ˆ
η
η
η
η
η
Q
Q
π
T
π
T
⎣
⎦
T
→∞
T
→∞
(4.74)
1
⎡
(
)
⎤
ˆ
ˆ
ˆ
ˆ
*
*
T
T
*
T
=⋅
H
lim
a
⋅
a
⋅
H
=⋅
H S H
⋅
⎢
⎥
ˆ
ˆ
ˆ
η
η
η
η
T
QQ
Q
π
⎣
⎦
T
→∞
where
S
is an
N
mod
by
N
mod
normalised modal load matrix
⎛
*
ˆ
1
⎞
⎡
a
⎤
⎜
Q
⎟
⎢
⎥
⎜
⎟
⎢
⎥
#
⎜
⎟
⎢
⎥
⎡
1
1
(
)
⎤
*
⎜
⎟
()
*
T
⎢
a
⎥
lim
lim
a
a
a
S
ω
=
a
⋅
a
=
⋅
⎢
""
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
⎥
Q
Q
Q
Q
⎜
i
Q
Q
Q
⎟
T
T
⎢
⎥
⎣
π
π
1
j
N
⎦
T
→∞
T
→∞
mod
⎜
⎟
⎢
#
⎥
⎜
⎟
⎢
⎥
*
ˆ
a
⎜
⎟
⎢
⎥
Q
N
⎝
⎣
⎦
⎠
mod
⎡
⎤
%
$
⎢
⎥
()
()
S
⇒
S
ω
=
⎢
⎢
ω
⎥
(4.75)
ˆ
ˆ
ˆ
Q
Q
ij
⎥
⎢
⎥
$
%
⎣
⎦
