Civil Engineering Reference
In-Depth Information
(
)
ˆ
T
i
∫
dx
φ C φ
⋅
⋅
ae
j
C
2
B
m
ω
ρ
ae
L
ij
exp
i
ζ
=
=
⋅
ae
ij
(
)
2
2
4
T
ω
M
∫
φφ
⋅
dx
i
i
i
i
i
L
(6.66)
*
2
*
∫
∫
φφ
H dx
+
B
φφ
A dx
i
j
1
i
j
2
zz
θθ
2
B
m
ρ
L
exp
=
⋅
(
)
4
2
2
2
∫
φφφ
++
dx
i
y
z
θ
i
i
i
L
*
*
m
is defined in Eq. 6.41. If
*
where
H
and
A
, are given in Eq. 5.37 and where
H
and
*
A
are taken as modal constants and independent of span-wise position, then
a
ζ
becomes diagonal due to the orthogonal properties of the mode shapes, i.e.
diag
⎡ ⎤
ζ
=
ζ
(6.67)
ae
⎣ ⎦
ae
i
where
*
2
2
*
2
∫
∫
H
φ
dx
+
B A
φ
dx
1
i
2
i
z
θ
2
ρ
B
m
L
exp
ζ
=
⋅
(6.68)
(
)
ae
i
4
2
2
2
∫
dx
φφφ
++
i
i
i
i
y
z
θ
L
ˆ
()
This implies that
N
by
mo
N
diagonal matrix. In vortex shedding
induced vibration problems it is usually not essential to include the along wind load
effects. The load vector may then be reduced to
is an
H
ω
mod
η
T
(
)
[
]
xt
,
0
q
q
θ
q
=
(6.69)
z
and the corresponding Fourier transform is
T
(
)
x
,
⎡
0
a
a
⎤
a
ω
=
⎣
(6.70)
q
q
q
⎦
z
θ
The cross sectional load spectrum is defined by (see Eq. 4.78)
⎡
⎤
⎡
⎤
00
0
00
0
⎢
⎥
⎢
⎥
1
1
(
)
(
)
*
T
⎢
*
*
⎥
x
,
lim
lim
0
a
a
a
a
0
0
S
S
S
Δ=
ω
a a
=
=
⎢
⎥
q
qq
qq
qq
qq
qq
π
T
π
T
⎢
zz
z
θ
⎥
⎢
zz
z
θ
T
→∞
T
→∞
⎥
SS
⎢
*
*
⎥
⎢
0
aa
aa
⎥
⎣
qq
qq
⎦
⎣
qq
qq
⎦
z
θ
θ
θ
z
θ
θ
θ
(6.71)
