Civil Engineering Reference
In-Depth Information
and
0
0
0
0
0
0
⎡
⎤
⎢
⎥
()
()
()
000
−Φ
DC
s
00
⎢
DD
θ
⎥
⎢
000
−Φ
BC
s
00
⎥
2
V
ρ
LL
θ
()
s
k
=
(9.59)
⎢
⎥
ae
0
2
2
000
BC
s
00
′ Φ
⎢
⎥
MM
θ
⎢
⎥
0
0
0
0
0
0
⎢
⎥
0
0
0
0
0
0
⎢
⎥
⎣
⎦
Thus, the motion induced load
(
)
ae
x t
,
q
may be obtained from history integration
⎧
0
0
⎫
⎡
⎤
⎡
⎤
⎪
⎢
⎥
⎢
⎥
⎪
(
)
(
)
rx
,
rx
,
τ
τ
⎪
⎪
⎢
⎥
⎢
⎥
y
y
el
el
⎪
⎪
⎢
⎥
⎢
⎥
(
)
(
)
t
rx
,
τ
rx
,
τ
d
d
⎪
⎪
z
z
(
)
( )
⎢
⎥
()
⎢
⎥
el
el
x t
,
∫
s
s
d
q
=
c
⋅
+
k
⋅
τ
(9.60)
⎨
⎬
ae
ae
ae
(
)
(
)
0
d
⎢
⎥
0
d
⎢
⎥
τ
rx
,
τ
rx
,
τ
τ
⎪
⎪
0
θ
θ
⎢
el
⎥
⎢
el
⎥
⎪
⎪
⎢
0
⎥
⎢
0
⎥
⎪
⎪
⎢
⎥
⎢
⎥
⎪
⎪
0
0
⎣
⎦
⎣
⎦
⎩
⎭
It is also a physical requirement that if the structural motion is harmonic then loads
obtained from indicial functions must be equal to those obtained from aerodynamic
derivatives. It is shown in Appendix D how this requirement may be used to determine
indicial functions from known aerodynamic derivatives.
Thus, if Eq. 9.51 is applicable (i.e. a quasi-steady load hypothesis is adopted), then
the motion induced load associated with element ends
i
=
1 or 2
is given by
L
L
⎧
⎫
()
() ()
() ()
t
Z
t
Z
t
Q
=
c
⋅
d
+
k
⋅
d
(9.61)
⎨
⎬
ae
ae
i
i
ae
i
i
i
n
2
0
2
0
⎩
⎭
n
where
i
= .(the
Z
dependency comes from the variation of
V
with the elevation of element ends 1 and 2.)
In this case the aerodynamic load vector at the level of element
n
is given by
1 or 2
d
is the is the six by one displacement vector at element ends
(
)
Zt
,
⎡
R
⎤
()
ae
1
(
)
( )
(
)
( )
R
t
=
=
c
Z
,
Z
⋅
d
t
+
k
Z
,
Z
⋅
d
t
(9.62)
⎢
⎥
ae
(
)
ae
12
n
ae
12
n
n
Zt
,
n
n
R
⎣
⎦
ae
2
n
T
()
[
]
t
where
d
=
d
d
and
n
1
2
n
