Civil Engineering Reference
In-Depth Information
T
⎧
⎡
⎤
a
....
a
....
a
a
=
⎪
η
⎣
η
η
η
⎦
1
i
N
where:
(4.58)
⎨
T
⎡
⎤
⎪
a
....
a
....
a
a
=
⎣
Q
Q
Q
Q
⎦
⎩
1
i
N
Since the assumption of a modal frequency domain motion induced load proportional to
and in phase with structural displacement, velocity and acceleration is adopted, then
a
is given by
Q
ae
(
)
2
i
a
=−
M
ω
+
C
ω
+
K
⋅
a
(4.59)
ae
ae
ae
η
Qae
M
C
and
K
are
where
,
N
by
N
matrices
ae
ae
ae
mod
mod
⎡
⎤
⎡
⎤
⎡
⎤
%
$
%$
%$
⎢
⎥
⎢
⎥
⎢
⎥
M
C
K
(4.60)
M
=
C
=
K
=
⎢
⎥
⎢
⎥
⎢
⎥
ae
ae
ij
ae
ae
ij
ae
ae
ij
⎢
⎥
⎢
⎥
⎢
⎥
⎢
$
%
⎥
⎢
$%
⎥
⎢
$%
⎥
⎣
⎦
⎣
⎦
⎣
⎦
whose elements on row
i
column
j
are given by
⎡
⎤
M
T
⎡
⎤
φ M φ
⋅
⋅
ae
ij
i
ae
j
⎢
⎥
⎢
⎥
⎢
⎥
T
C
∫
⎢
⎥
dx
=
φ C φ
⋅
⋅
(4.61)
ae
i
ae
j
⎢
⎥
ij
⎢
⎥
L
⎢
⎥
exp
T
i
⎢
⎥
φ K φ
⋅
⋅
K
⎢
⎥
⎣
ae
j
⎦
ae
ij
⎣
⎦
where
a
M
,
a
C
and
a
K
are three by three motion dependent cross sectional load
coefficient matrices
%$
%$
%$
⎡
⎤
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
M
=
⎢
m
C
=
⎢
c
K
=
⎢
k
(4.62)
ae
nm
⎥
ae
nm
⎥
ae
nm
⎥
⎢
$%
⎥
⎢
$%
⎥
⎢
$%
⎥
⎣
⎦
⎣
⎦
⎣
⎦
n
⎫
y z
,,
and where:
=
θ
.
⎬
⎭
m
