Civil Engineering Reference
In-Depth Information
Introducing Eq. 9.106, then
1
1
(
)
*
T
()
⎡
(
)
(
)
⎤
*
T
lim
lim
S
ω
=
a
⋅
a
=
H a
⋅
H a
r
r
r
⎢
rR
rR
⎥
T
T
⎣
⎦
π
π
T
T
→∞
→∞
(9.109)
1
(
)
*
*
TT
*
T
=⋅
H
lim
a
⋅
a
⋅
H
=⋅
H
S
⋅
H
r
R
R
r
r
RR
r
T
π
T
→∞
1
(
)
()
*
T
S
lim
a
a
where
ω
=
⋅
is the cross spectral density matrix of the buffeting
RR
R
R
π
T
T
→∞
wind load. The response covariance matrix may then be obtained simply by integration
throughout the frequency domain, i.e.
2
1
⎡
⎤
Cov
Cov
Cov
σ
"
"
"
1
i
1
j
1
N
r
⎢
⎥
⎢
# %#
#
#
⎥
⎢
⎥
2
Cov
"
σ
"
Cov
#
⎢
⎥
i
1
i
ij
∞
⎢
⎥
()
∫
d
Cov
=
#
# % #
#
=
S
ωω
(9.110)
⎢
⎥
rr
rr
⎢
⎥
0
2
Cov
"
Cov
"
σ
#
j
1
ji
j
⎢
⎥
⎢
⎥
#
% #
"""""
⎢
⎥
2
Cov
σ
⎢
⎥
⎣
N
1
N
⎦
r
r
What then remains is to develop
S
into an expression of known quantities. As shown
RR
in Eqs. 9.49 and 9.76
N
N
()
∑
T
()
∑
T
ˆ
()
t
t
⎡
t
⎤
R
=
A
⋅
R
=
A
⋅
⎣
B ψ v
(9.111)
nn
n Qnn
n
⎦
n
=
1
n
=
1
and thus a Fourier transform of the load vector will render
N
()
∑
T
()
⎡
⎤
a
ω
=
A
⋅
⎣
B ψ a
ω
(9.112)
ˆ
⎦
R
n
Q
n
v
n
n
n
=
1
where
T
()
⎡
⎤
a
ω
=
⎣
aaa aaa
(9.113)
v
ˆ
u
ˆ
v
ˆ
w
ˆ
u
ˆ
v
ˆ
w
ˆ
⎦
n
1
1
1
2
2
2
n
contains the Fourier amplitudes of the reduced turbulence components at the nodes of
element
n
(i.e. at element ends 1 and 2, see Eqs. 9.45 and 9.50). The cross spectral
density matrix of the load is then given by
