Civil Engineering Reference
In-Depth Information
1
2
. The load spectrum is shown for various relevant values of
b
, which
is the parameter that controls the narrow-bandedness of the process. The reduced co-spectrum (see
Eq. 5.34)
VB
ˆ
where
σ
=
ρ
σ
q
q
z
2
z
⎡
2
⎤
⎛
2
x
⎞
⎛
x
⎞
Δ
Δ
ˆ
(
)
⎢
⎥
Co
Δ=
x
cos
⋅
exp
−
⎜
⎟
⎜
⎟
q
z
3
λ
D
⎢
3
λ
D
⎥
⎝
⎠
⎝
⎠
z
z
⎣
⎦
at various values of λ is shown in the top right hand side diagram. It is this parameter that
control the spanwise coherence (and thus, the length scale) of the vortices. The characteristic
“lock-in” effect associated with vortex shedding induced dynamic response is controlled by the
aerodynamic damping parameter
a
K
. Establishing data of the mean wind velocity variation of
a
K
will in general require wind tunnel experiments. As indicated in example 6.4 above, such
data may often be fitted to an expression of the following type:
n
m
−
⎡
−
⎤
⎛ ⎞
⎛ ⎞
V
V
⎢
⎥
K
=⋅
2.6
K
⋅
⎜ ⎟
⋅
exp
−
⎜ ⎟
a
az
⎜ ⎟
⎢
⎜ ⎟
⎥
z
0
V
V
⎝ ⎠
R
⎝ ⎠
R
⎢
⎥
z
z
⎣
⎦
(
)
where
V
=
ωπ
D
2
St
is the resonance velocity (see Eq. 5.32) and
K
is the value at the apex
R
z
a
z
of the
K
variation. See the lower left hand side diagram in Fig. 6.12, where
n
=
6
and
m
=
8
.
a
z
Let us again consider a simply supported horizontal beam type of bridge with span
L
=
500
m
that is elevated at a position
f
z
= . Let us investigate the response variation with the mean
wind velocity at various levels of structural eigen-damping. It is assumed that the entire span is
flow exposed, i.e.
exp
L
= , and the expression for
a
K
given above is adopted. Let us allot the
following values to the remaining constants that are necessary for a numerical calculation of
(
50
)
xL
2
σ
=
:
r
r
z
ρ
(kg/m
3
)
B
(m)
D
(m)
St
m
(kg/m)
ˆ
q
z
b
a
K
(rad/s)
ω
σ
λ
z
z
a
1.25
20
4
4
0.8
0.1
0.9
0.15
0.4
1.2
0.2
10
Since
m
is constant along the span, then the modally equivalent and evenly distributed mass
mm
=
. The dynamic response is given in Eq. 6.91, i.e.:
z
z
1/2
⎡
⎤
(
)
σ
xL
=
2
σ
ˆ
1
ρ
BD
2
D
λ
(
)
r
r
q
⎢
⎥
z
z
g
VV
,
=
⋅
⋅
⋅
⋅
(
)
z
72
7/4
2
⎢
⎥
z
D
2
m
St
π
bL
ζζ
−
z
z
z
ae
z
⎣
⎦
where
3/2
2
⎡
⎤
⎛ ⎞
1
VV
/
⎛
−
⎞
V
1
(
)
R
z
⎢
⎥
gV V
,
⎜ ⎟
exp
=
⋅
−
⎜
⎟
z
⎜
⎟
⎜ ⎟
z
V
⎢
2
b
⎥
⎝
⎠
⎝ ⎠
R
z
z
⎣
⎦
D
ω
π
z
V
V
5.1
m s
The resonance mean wind velocity
is given by:
=
≈
.
R
z
R
z
2
St
