Civil Engineering Reference
In-Depth Information
Finally, if the relevant mode only contains a cross sectional rotation component
θ
,
[
]
T
i.e.
00
, whose eigen-frequency is
, then
φ
=
φ
ω
θ
1
2
⎡
⎤
2
4
BV
J
⎛ ⎞
πω
ρ
ˆ
()
⎢
⎥
θ
σ
=
ω
φ
′
GI
−
φ
′′′
EI
(7.93)
⎜ ⎟
MM
(
) (
)
t
w
θθ
θ
θ
xx
4
mB
ω
⎢
⎥
1
−
κ
⋅
ζ
−
ζ
⎝ ⎠
θ
θ
⎣
ae
θ
ae
⎦
θ
θ
are defined in Eq. 6.31, and where
J
θ
where
m
θ
is defined in Eq. 6.27,
κ
and
ζ
ae
θ
ae
θ
is given in Eq. 6.29.
Example 7.3
Let us again consider the simply supported beam shown in Fig. 7.6, and as usual, let us for
simplicity assume that all cross sectional quantities are constants
a
long the
s
pan of the bridge, and
that it has a typical bridge type of cross section where
C
′
C
C
,
and
are negligible and
D
M
CDB C
′
. Let us set out to determine the covariance matrix associated with cross sectional
forces at spanwise positions
⋅
D
L
x
=
0
and
r
xL
=
/ 2
that is caused by resonant oscillations in a
r
chosen mode
a
a
⎡⎤ ⎡⎤
⎢⎥ ⎢⎥
φ
y
y
π
sin
x
φ
=
φ
φ
=
⋅
⎢⎥ ⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
i
z
z
L
a
θ
θ
i
whose eigen-frequency, eigen-damping-ratio and modally equivalent and evenly distributed mass
are
m
. The necessary calculations are given in Eq. 7.83, i.e.
ω
,
ζ
and
i
S
πω
⋅
i
ˆˆ
QQ
()
TT
ov
x
ii
C
=
⋅
T ββ T
⋅
⋅
⋅
(
) (
)
FF
r
i
i
R
i
41
−
κ
⋅
ζ
−
ζ
ae
i
ae
i
i
TT
S
where
and
T ββ T
are given in Eqs. 7.84 and 7.86. Since (see Ex. 7.2)
⋅
⋅
⋅
ˆˆ
Q
ii
i
i
D
C
B
⎡
⎤
2
0
⎢
⎥
D
ˆ
⎡
2
⎤
⎢
⎥
IS
0
ˆ
ˆ
2
(
)
uuu
B
=
0
0
C
BC
′
and
IS
⋅
Δ
x
,
ω
=
⎢
⎥
⎢
⎥
q
L
M
vv
i
ˆ
2
⎢
0
IS
⎥
⎢
⎥
⎣
⎦
′
www
⎢
⎥
⎢
⎥
⎣
⎦
then
