Civil Engineering Reference
In-Depth Information
Of all the eigen-values that may be extracted from Eq. 8.5 the main focus is on the
one that represents the stability limit at the lowest mean wind velocity, i.e. it is the
lowest
V
) that has priority.
Cases of structural behaviour close to a stability limit may in general be classified
according to the response type of displacement that develops. The problem of
identification is greatly simplified if the impedance is taken directly from the
characteristic behaviour of each stability problem as known from full scale or
experimental observations. For a bridge section there are four types of such behaviour.
First, there is the possibility of a static type of unstable behaviour in torsion, called static
divergence. Second, there is the possibility of a dynamic type of unstable behaviour in
the across wind vertical (
z
) direction, called galloping. Third, there is a possible unstable
type of dynamic response in pure torsion, and finally, there is the possibility of an
unstable type of dynamic response in a combined motion of vertical displacements and
torsion, called flutter. Thus, it is always either
(and corresponding
ω
cr
r
r
,
r
θ
or both that are the critical
response quantities. It is then only necessary to search for the instability limits associated
with the two most onerous modes,
φ
and
φ
with corresponding eigen-frequencies
ω
1
and
ω
, of which one contain a predominant
φ
component and the other contain a
2
z
predominant
φ
component. Therefore, the impedance matrix may be reduced to
⎧
2
⎡
(
)
⎤
⎡
κ
κ
⎤
/
0
10
ωω
⎡ ⎤
⎪
ae
ae
ˆ
r
1
(
)
11
12
⎢
⎥
E
ω
,
V
=
−
−
⎢
⎥
⎨
⎢ ⎥
r r
η
2
01
κ
κ
⎢
(
)
⎥
⎢
⎥
0
/
⎣ ⎦
ωω
⎪
⎣
ae
ae
⎦
⎣
⎩
21
22
⎦
r
2
(8.7)
⎫
⎡
ζζ
−
−
ζ
⎤
/
0
⎡
ωω
⎤
⎪
1
ae
ae
r
1
11
12
+
2
i
⋅
⎢
⎥
⎬
⎢
⎥
0
/
ωω
−
ζ
ζ
−
ζ
⎢
⎥
⎣
⎦
⎪
r
2
⎣
ae
2
ae
⎦
⎭
21
22
where (see Eqs. 8.3 and 8.4)
2
⎡
κ
()
V
⎛
ω
⎞
⎢
(
ae
ij
i
*
*
*
*
*
∫
P
H
BA
P
H
=
φφ
+
φφ
+
φφ
+
φφ
+
φφ
⎜
⎟
⎜
⎟
⎢
yy
4
zy
6
θ
y
6
yz
6
zz
4
2
i
j
i
j
i
j
i
j
i
j
B
m
ω
ρ
⎝
⎠
⎣
i
L
exp
2
i
⎡
⎤
)
(
)
⎤
*
*
*
2
*
2
2
2
BA
BP
BH
B A
dx
∫
dx
+
φφ
+
φφ
+
φφ
+
φφ
φ
+
φ
+
φ
⎢
⎥
z
4
y
3
z
3
3
⎥
⎦
⎢
y
z
θ
θ
θ
θ
θ
θ
i
j
i
j
i
j
i
j
i
i
i
⎥
⎣
⎦
L
(8.8)
