Civil Engineering Reference
In-Depth Information
where
{
}
(
)
(
)
Zt
,
Z
ˆ
Q
=
B
⋅
ψ v
⋅
(9.47)
i
i
Qi
00
n
0
n
where
i
=
1 or 2
and where
0
0
0
⎡
⎤
⎢
⎥
()
(
)
()
(
)
()
2
DC I
Z
DC
′
−
BC
I
Z
DC
′
−
BC
I
Z
⎢
⎥
Du i
D
Lv i
D
Lw i
⎢
⎥
2
()
()
(
)
()
(
)
VZ
L
ρ
⎡
⎤
()
2
BC I
Z
BC
′
DC
I
Z
BC
′
DC
I
Z
⎢
+
+
⎥
⎣
⎦
i
Lu i
L
Dv i
L
Dw i
B
=
Q
⎢
⎥
0
4
2
()
2
()
2
()
⎢
2
BC I
Z
BC I Z
BC I
Z
⎥
−
−
′
−
′
Mu
i
Mv
i
Mw
i
⎢
⎥
0
0
0
⎢
⎥
⎢
⎥
0
0
0
⎣
⎦
(9.48)
Thus,
()
()
ˆ
t
t
RBψ v
=
⋅
⋅
(9.49)
n
Q
n
n
n
where
()
Z
⎫
⎡
B
0
⎤
⎪
Q
1
0
B
=
⎢
⎥
Q
n
(
)
Z
0
B
⎪
⎢
⎥
⎣
Q
2
⎦
⎪
⎪
⎪
0
n
⎡
ψ 0
⎤
⎪
0
ψ
=
(9.50)
⎬
⎢
⎥
n
0 ψ
⎣
⎦
⎪
⎪
⎪
⎪
0
n
()
(
ˆ
Z
Z
⎡
v
⎤
01
ˆ
v
=
⎪
⎢
⎥
n
)
ˆ
v
⎪
⎣
⎦
02
n
⎭
and, as mentioned above, where indices 1 and 2 refers to element ends.
The motion induced (aerodynamic) load vector:
As shown in Eq. 5.8, the motion induced load will comprise two contributions, one
which is proportional to the element velocity and one which is proportional to its
dynamic displacements. Thus,
0
0
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
r
r
⎢ ⎥
y
⎢ ⎥
y
el
el
⎢ ⎥
⎢ ⎥
r
r
z
z
(
)
⎢ ⎥
⎢ ⎥
el
el
q
xt
,
=
c
+
k
(9.51)
ae
ae
ae
0
⎢ ⎥
0
⎢ ⎥
r
r
θ
θ
el
el
⎢ ⎥
⎢ ⎥
⎢ ⎥
0
⎢ ⎥
0
⎢ ⎥
⎢ ⎥
0
0
⎣ ⎦
⎣ ⎦
