Civil Engineering Reference
In-Depth Information
where
S
and
w
S
are defined in Eq. 3.39. (A transition between spectral density
descriptions using
f
rather than
ω
as the frequency variable is shown in Eq. 2.68.)
Since spatial averaging will eliminate any complex parts of the cross spectra, Eq. 6.63
may for all practical purposes be replaced by
uu
()
S
ω
⎫
ˆ
ˆ
u
(
)
2
(
)
S
Re
S
x
,
/
Co
x
,
=
⎡
Δω σ
⎤
=
⋅
Δω
⎪
⎣
⎦
uu
uu
u
uu
2
σ
⎪
⎬
⎪
u
(6.64)
()
S
ω
ˆ
ˆ
w
(
)
2
(
)
S
Re
S
x
,
/
Co
x
,
=
⎡
Δω σ
⎤
=
⋅
Δω
⎣
⎦
ww
ww
w
ww
⎪
⎭
2
σ
w
where
ˆ
and
ˆ
Co
Co
are the reduced
u-
and
w-
component co-spectra (see Eq. 3.40).
uu
ww
Example 6.3
Let us again (similar to example 6.2) consider a simply supported horizontal beam type of bridge
with span
L
500
m
z
50
m
=
that is elevated at a position
=
, but now we set out to calculate the
f
x
L
2
dynamic response at
=
associated with the two mode shapes
r
T
T
⎡
0
0
⎤
⎡
00
⎤
φ
=
⎣
φ
and
φ
=
⎣
φ
1
⎦
2
⎦
1
2
0.8
2.0
rad s
/
with corresponding eigen-frequencies
ω
=
and
ω
=
. As can be seen,
φ
1
2
contains only the displacement component in the across wind vertical direction while
φ
only
sin
x L
contains torsion. Let us for simplicity assume that
φφ
==
π
. Thus, the aim of this
z
θ
1
2
example is to calculate the corresponding dynamic response quantities
σ
and
σ
at
r
z z
rr
θθ
x
L
2
Cov
θ
between them. It is taken for granted that the chosen mean
wind velocity settings are well below any instability limit, such that any changes to resonance
frequencies may be ignored. Again, it is assumed that the cross section is close to a flat plate with
the following static load coefficient properties:
=
and the covariance
rr
z
r
D
CC
B
C
0
C
′
=
5
C
0
C
′
1.5
′
=
=
=
and
D
L
M
M
(Quantifying the drag coefficient is obsolete since
y
direction response is not excited.)
Let us also assume that the entire span is flow exposed, i.e.
L
=
L
, and adopt the following
exp
wind field properties:
