Civil Engineering Reference
In-Depth Information
If the load case is wind buffeting as described in chapter 6.3, and the assumption of
negligible cross spectra between fluctuating flow components is adopted, then
(
ˆ
)
(
)
2
(
)
x
,
is given in Eq. 6.57 and
x
,
V
x
,
, see Eq. 6.40. Thus,
S
Δ
ω
S
Δ=
ω
I S
Δ
ω
qq
v
v
v
2
2
⎛
⎞
ρ
VB
ˆ
ˆ
ˆ
(
)
2
(
)
T
⎡
⎤
x
,
x
,
S
Δ
ω
=
⋅
B
⋅
I S
⋅
Δ
ω
⋅
B
(7.78)
⎜
⎟
qq
i
⎜
⎟
q
⎣
v
v
i
⎦
q
2
⎝
⎠
where
[
]
and
I
=
diag I
I
v
u
w
()
S
⎡
ω
⎤
ˆ
ui
(
)
Cx
,
0
⋅
Δ
ω
⎢
⎥
uu
i
ˆ
2
⎡
⎤
⎢
S
0
σ
⎥
ˆ
u
(
)
uu
S
Δ
x
,
ω
=
⎢
⎥
⎢
=
(7.79)
⎥
v
i
ˆ
()
S
ω
⎢
0
S
⎥
⎣
⎦
⎢
ˆ
wi
(
)
⎥
ww
0
Cx
,
⋅
Δ
ω
ww
i
⎢
2
⎥
σ
⎣
⎦
w
and where
ˆ
ˆ
(
)
(
)
Cx
,
and
Cx
,
are the reduced co-spectra defined in Eq.6.64.
Δ
ω
Δ
ω
uu
i
ww
i
Introducing
the
evenly
distributed
and
modally
equivalent
masses
(
)
(
)
T
T
mM
/
∫
φφ
x
and
mM
/
∫
φφ
x
(see Eq. 6.41), then the content of
=
⋅
=
⋅
i
i
i
i
j
j
j
j
L
L
on row
i
column
j
is given by
S
Q
R
2
2
3
3
⎛ ⎞
ρρ
BB V
⎛ ⎞
V
ˆ
()
2
()
S
J
ω
=
⋅
⋅
⋅
⎜ ⎟
⋅
ω
(7.80)
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
ˆˆ
i
ij
i
Q
ij
22
mmB
B
ω
ω
i
j
i
j
where
{
}
ˆ
ˆ
ˆ
T
()
⎡
2
(
)
⎤
T
()
∫∫
x
x
,
x
dx dx
φ BI S Bφ
⋅
⋅
⋅
Δ
ω
⋅
⋅
i
1
q
⎣
v
v
i
⎦
q
j
2
1
2
L
ˆ
exp
2
()
J
ω
=
(7.81)
ij
i
⎛
⎞ ⎛
⎞
T
T
∫
dx
∫
dx
⎜
φφ
⋅
⎟ ⎜
⋅
φφ
⋅
⎟
i
i
j
j
⎜
⎟ ⎜
⎟
⎝
⎠ ⎝
⎠
L
L
Single mode three component approach
In many cases where eigen-frequencies are well separated and flow induced coupling
effects are negligible a multi mode procedure as presented above may with sufficient
accuracy be replaced by a mode by mode approach. Then all modes are uncoupled, and
therefore, the covariance contributions between force components from different modes
