Civil Engineering Reference
In-Depth Information
1
(
)
()
*
T
where
S
ω
=
lim
a
is the cross spectral density matrix of the modal
⋅
a
RR
R
R
π
T
T
→∞
()
t
buffeting wind load.
R
is given in Eq. 9.111, and thus
N
N
()
T
()
T
∑
T
()
T
∑
T
()
⎡
ˆ
⎤
R Φ R Φ
t
t
AR Φ
t
ABψ v
t
=
=
⋅
⋅
=
⋅
⋅
⎣
(9.155)
nn
n Qnn
n
⎦
n
1
n
1
=
=
Taking the Fourier transform throughout Eq. 9.155 renders
N
()
T
∑
T
n
()
⎡
⎤
a
ω
=⋅
Φ
ABψ a
⋅
⎣
ω
(9.156)
Qn
v
ˆ
⎦
R
n
n
n
1
=
and thus, the cross spectral density matrix of the modal buffeting wind load is given by
⎧
T
⎫
N
N
⎡
⎤ ⎡
⎤
1
⎪
(
)
(
)
⎪
()
T
∑
T
*
ˆ
T
∑
T
lim
S
ω
=
Φ ABψ a
⋅
Φ ABψ a
⎨
⎬
⎢
⎥ ⎢
⎥
ˆ
nQnv
nQnv
RR
T
n
n
n
n
π
T
→∞
⎪
⎣
⎦ ⎣
⎦
⎪
n
1
n
1
=
=
⎩
⎭
NN
⎧
⎫
1
⎪
⎡
(
)
⎤
⎪
T
∑∑
T
*
ˆ
T
T
T
lim
=
Φ
ABψ
⋅
aa ψ BAΦ
⋅
⋅
(9.157)
⎨
⎬
⎢
⎥
nQn
v v
ˆ
mQ
m
n
T
n
m
m
π
⎪
⎣
T
→∞
⎦
⎪
⎩
⎭
nm
==
11
NN
⎧
⎫
⎪
⎪
ˆ
T
∑∑
T
⎡
()
T
T
⎤
=⋅
Φ
ABψ S
⋅
⋅
ω
⋅
ψ BA
⋅
⋅
Φ
⎨
⎬
⎪
⎭
n
⎣
Qn
vv
ˆˆ
mQ
⎦
m
n
m
⎪
⎩
nm
11
==
1
(
)
ˆ
()
*
T
where
S
ω
=
lim
a
⋅
a
is the cross spectral density matrix of the reduced
vv
ˆˆ
v
ˆ
nm
ˆ
π
T
T
→∞
turbulence velocity vector, previously developed in chapter 9.6, see Eq. 9.124. Thus
()
*
()
()
T
()
*
()
T
()
T
()
⎡
⎤
S
ω
=
H
ω
⋅
S
ω
⋅
H
ω
=
H
ω
⋅
Φ S
⋅
ω
⋅
Φ H
⋅
ω
⎣
RR
⎦
ηη
η
RR
η
η
η
(9.158)
⎛
NN
⎞
⎧
⎫
(
)
⎪
⎪
ˆ
ˆ
ˆ
*
T
∑∑
T
⎡
1 2
1 2
T
T
⎤
T
=
H Φ
⎜
ABψ SCo S ψ BAΦ H
⋅
⋅
⎟
⎨
⎬
η
⎜
nnn n mm
mmm
⎟
η
⎣
⎦
⎪
⎪
⎝
⎩
⎭
⎠
nm
11
==
ˆ
ˆ
n
ˆ
m
(
)
()
()
s
,
where
Co
Δ
ω
,
S
ω
and
S
ω
are defined in Eqs. 9.120 and 9.121.
nm
nm
()
()
()
()
Finally, since
r Φη
, then
t
=⋅
t
a
ω
=⋅
Φ a
ω
, and thus, the cross spectral
r
η
density matrix of the displacement response quantities is given by
