Civil Engineering Reference
In-Depth Information
2
()
x
∞
⎫
φ
2
ˆ
2
()
i
r
()
()
x
∫
H
S
d
σ
=
ω
⋅
ω
ω
⎪
⎪
⎪
⎬
r
i
r
i
2
Q
i
K
i
0
(4.31)
2
()
⎡
∞
∞
⎤
⎪
φ
x
2
2
ˆ
ˆ
i
r
()
()
( )
()
H
0
∫
S
d
S
∫
H
d
≈
⋅
⎢
⋅
ωω
+
ω
⋅
ω
ω
⎥
⎪
i
i
i
2
Q
Q
K
i
i
⎢
⎥
⎪
⎣
⎦ ⎭
i
0
0
It is in the following taken for granted that
−
1
ˆ
()
⎡
⎤
H
01
=−
κ
=
1
(4.32)
i
⎣
ae
i
⎦
i.e. that
K
=
0
at
ω =
0
. (This is an obvious assumption as the structure is not in
ae
i
0
motion at
ω =
.) Introducing
∞
⎫
()
2
∫
Sd
ωωσ
=
⎪
⎪
⎪
⎬
⎪
Q
Q
i
i
0
(4.33)
∞
2
πω
ˆ
()
i
∫
Hd
ωω
=
i
(
)
⎪
41
−
κζ
⎪
⎭
0
ae
tot
i
i
where
ζ
=−
ζ
ζ
, the following is obtained
tot
i
ae
i
i
2
()
⎡
⎤
S
()
πω
ω
x
K
⎡
φ
⎤
⎢
i
i
Q
2
2
2
i
r
2
⎥
i
σσ σ
≈+=
⋅
σ
+
(4.34)
⎢
⎥
⎢
(
)
r
B
R
Q
i
i
i
⎥
i
⎢
⎥
41
−
κζ
⎣
⎦
⎣
i
ae
tot
⎦
i
i
4.3 Single mode three component response calculations
In this second approach it is assumed that the eigen-frequencies are still well spaced out
on the frequency axis, but that each mode shape contain three components, i.e. the dis-
placements
φ , and the rotation φ , as illustrated in Fig. 4.5. Adopting the same
assumptions regarding motion induced load effects as presented in chapter 4.2 above, the
total cross sectional load is given by
φ
and
y
z
(
)
(
)
xt
,
xt
, , , ,
(4.36)
q q
=
+
q
rrr
tot
ae
T
(
)
(4.37)
where:
q
xt
,
=
⎣
⎡
q
q
q
θ
⎤
⎦
y
z
is the flow induced part of the load, and
T
(
)
xt
,,,,
⎡
q
q
q
θ
⎤
q
rrr
=
⎣
(4.38)
⎦
ae
y
z
ae
is the motion induced part.
