Civil Engineering Reference
In-Depth Information
⎡
⎤
⎡
⎤
%$
%$
⎢
⎥
⎢
⎥
κ
=
κ
and
ζ
=
ζ
(6.50)
⎢
⎥
⎢
⎥
ae
ae
ij
ae
ae
ij
⎢
⎥
⎢
⎥
$%
$%
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
are given by
(
)
ˆ
T
i
∫
dx
φ K φ
⋅
⋅
ae
j
K
2
B
m
ρ
ae
L
ij
exp
κ
=
=
⋅
(6.51)
(
)
ae
ij
2
2
M
T
ω
∫
dx
φφ
⋅
i
i
i
i
i
L
(
)
ˆ
T
i
∫
dx
φ C φ
⋅
⋅
ae
j
C
2
ω
B
m
ρ
ae
L
ij
exp
i
ζ
=
=
⋅
(6.52)
(
)
ae
ij
2
2
4
M
T
ω
∫
dx
φφ
⋅
i
i
i
i
i
L
Fully expanded versions of these expressions are given by
⎡
⎢
κ
(
ae
ij
*
*
*
*
*
*
∫
P
H
BA
P
H
BA
=
φφ
+
φφ
+
φφ
+
φφ
+
φφ
+
φφ
yy
4
zy
6
θ
y
6
yz
6
zz
4
θ
z
4
2
⎢
⎣
i
j
i
j
i
j
i
j
i
j
i
j
B
m
ρ
L
exp
2
i
⎡
⎤
)
(
)
⎤
*
*
2
*
2
2
2
+
φφ
BP
+
φφ
BH
+
φφ
B A
dx
∫
φ
+
φ
+
φ
dx
⎢
⎥
y
3
z
3
3
y
z
θ
θ
θ
θ
⎥
⎦
θ
i
j
i
j
i
j
i
i
i
⎢
⎥
⎣
⎦
L
(6.53)
⎡
⎢
ζ
(
ae
ij
*
*
*
*
*
*
∫
P
H
BA
P
H
BA
=
φφ
+
φφ
+
φφ
+
φφ
+
φφ
+
φφ
yy
1
zy
5
θ
y
5
yz
5
zz
1
θ
z
1
2
i
j
i
j
i
j
i
j
i
j
i
j
⎢
⎣
B
m
ρ
L
exp
4
i
⎡
⎤
)
(
)
⎤
*
*
2
*
2
2
2
BP
BH
B A
dx
∫
dx
+
φφ
+
φφ
+
φφ
φ
+
φ
+
φ
⎢
⎥
y
2
z
2
2
⎥
⎦
y
z
θ
θ
θ
θ
θ
i
j
i
j
i
j
i
i
i
⎢
⎥
⎣
⎦
L
(6.54)
As mentioned above, the normalised modal load matrix
S
(
N
by
N
) is given in
mod
mod
()
(
)
Eq. 4.75. Its content
S
, containing the cross sectional load matrix
x
,
, is
ω
S
Δ
ω
ˆˆ
Q
ij
qq
defined in Eq. 4.77 (and 4.78). Based on the buffeting load expressions in chapter 5.1 it
is now only
q
S
that remains for further expansion. Recalling from Eq. 6.32 that the
