Civil Engineering Reference
In-Depth Information
considered necessary for a sufficiently accurate solution. It follows from Eqs. 7.84 -
7.87 that
⎡
⎤
σ
′′′
EI
⎡
⎤
φ
VV
yz
yy
⎢
⎥
⎢
⎥
1
⎢
⎥
σ
EI
⎢
⎥
φ
′′′
VV
z
y
zz
2
⎡
⎤
⎢
2
⎥
⎢
⎥
3
BV
J
⎛ ⎞
ρ
πω
ˆ
⎢
⎥
⎢
i
σ
′
GI
′′′
EI
⎥
=
φ
−
φ
(7.90)
⎢
⎥
⎜ ⎟
MM
ii
(
) (
)
xx
θ
t
θ
w
4
mB
ω
⎢
⎥
⎢
1
−
κ
⋅
ζ
−
ζ
⎥
⎢
⎥
⎝ ⎠
i
i
ae
i
ae
⎣
⎦
σ
i
i
′′
EI
φ
⎢
⎥
⎢
⎥
MM
z
y
yy
⎢
⎥
⎢
⎥
σ
φ
′′
EI
⎢
⎥
⎢
⎥
⎣
MM
⎦
⎣
yz
⎦
zz
i
i
where
ˆ
i
J
is given in Eq. 7.86. The total variances and covariance coefficients are then
given by Eqs. 7.81 and 7.88.
Single mode single component approach
In some cases a single mode single component approach will suffice. The necessary
calculations are then further reduced. Let us first consider a single mode that only
00
T
contains an along wind
y
component, i.e.
⎡
⎤
⎦
, and whose eigen-frequency
φφ
=
⎣
y
is
. Then the necessary calculations are reduced to
ω
y
1
2
⎡
⎤
2
⎡
′′′
EI
⎤
⎡
σ
⎤
φ
3
⎛ ⎞
πω
BV
J
mB
VV
ρ
⎢
⎥
y
z
yy
ˆ
()
y
⎢
⎥
⎢
⎥
=
⋅
⎜ ⎟
⋅
ω
⋅
⋅
(7.91)
(
) (
)
⎜ ⎟
yy
⎢
⎥
4
σ
ω
⎢
⎥
⎢
⎥
′′
EI
1
φ
−
κ
⋅
ζ
−
ζ
⎝ ⎠
⎣
MM
⎦
y
y
⎢
⎥ ⎣
⎦
zz
y
z
ae
y
ae
⎣
⎦
y
y
where
are defined in Eq. 6.24, and where
ˆ
y
m
is defined in Eq. 6.20,
J
κ
and
ζ
y
ae
y
ae
y
is given in Eq. 6.22 (see also Eq. 6.19).
Similarly, if the relevant mode only contains an across wind
z
component, i.e.
[
0
T
]
0
, whose eigen-frequency is
, then
φ
=
φ
ω
z
z
1
2
⎡
⎤ ⎡
⎤
⎡
σ
⎤
2
φ
′′′
EI
3
BV
J
mB
⎛ ⎞
πω
ρ
VV
z
y
zz
ˆ
()
⎢
z
⎥ ⎢
⎥
⎢
⎥
=
⋅
⋅
ω
⋅
⋅
(7.92)
⎜ ⎟
(
) (
)
z
z
σ
4
⎢
⎥
⎢
ω
⎥
⎢
⎥
1
′′
EI
−
κ
⋅
ζ
−
ζ
φ
⎝ ⎠
MM
⎣
⎦
z
z
yy
⎣
ae
z
ae
⎦
⎣
z
y
⎦
z
z
are defined in Eq. 6.31, and where
ˆ
J
where
m
is defined in Eq. 6.27,
κ
and
ζ
ae
z
ae
z
is given in Eq. 6.28.
