Civil Engineering Reference
In-Depth Information
+∞
nuvw
sxyz
,,
,
1
⎧
⎪
⎨
=
(
)
(
)
i
−
ωτ
S
s
,
∫
Cov
s
,
e
d
Δ=
ω
Δ⋅
τ
τ
(3.37)
nn
nn
,
2
Δ=Δ
Δ Δ
π
⎪
⎩
f
f
f
−∞
but in wind engineering the frequency
f
(in Hz) is usually preferred, and then the dou-
ble sided cross spectra are defined by (see Eqs. 2.68 and 2.75)
+∞
nuvw
sxyz
=
,,
,
⎧
⎪
⎨
(
)
(
)
2
f
∫
−
πτ
S
sf
,
Cov
s
,
e
d
(3.38)
Δ=
Δ⋅
τ
τ
nn
nn
Δ=Δ
Δ Δ
,
⎪
⎩
f
f
f
−∞
()
The cross spectra are usually defined by the single point spectra,
, the coherence
S
f
n
(
)
(
)
function,
and the phase spectra,
, as shown in Eq. 2.87, i.e.
Coh
Δ
s f
,
ϕ
Δ
sf
,
nn
nn
nuvw
sxyz
,,
,
⎧
⎪
⎨
=
(
)
( )
(
)
[]
S
sf
,
S
f
Coh
sf
,
exp
i
ϕ
Δ=
⋅
Δ⋅
(3.39)
nn
n
nn
,
Δ=Δ
Δ Δ
⎪
⎩
f
f
f
Since the wind field is usually assumed homogeneous and perpendicular to the span of
the (line-like) structure, phase spectra may be neglected. It should however be acknowl-
edged that in structural response calculations spatial averaging takes place along the span
of the structure (see chapters 6.4 and 6.5), and then all imaginary parts cancel out and
only a double set of real parts remain. Taking it for granted that the single point spec-
trum
()
S
f
is known, it is then rather the normalised co-spectrum
n
(
)
Re
⎡
S
sf
,
⎤
nuvw
sxyz
,,
,
S
Δ
=
⎧
⎪
⎨
⎣
⎦
ˆ
nn
(
)
Co
s f
,
Δ
=
(3.40)
nn
()
,
Δ=Δ
Δ Δ
⎪
⎩
n
f
f
f
that is necessary to give special attention to in wind engineering. Some general expres-
sions occur in the literature. For a first approximation and under homogeneous condi-
tions, the following may be adopted
⎧
nuvw
sxyz
sxyz
,,
,,
,
=
⎛
⎞
f
s
⎪
V
Δ
⋅
ˆ
(
)
Co
s f
,
e p
⎜
c
⎟
Δ
=
−
⋅
=
(3.41)
⎨
⎪
nn
ns
()
f
f
f
⎜
⎟
⎝
f
⎠
,
Δ=Δ
Δ Δ
⎩
f
f
f
⎧
c
c
9
=
≈
uy
uz
⎪
⎪
f
f
(3.42)
c
c
c
c
6
where:
=
=
=
≈
⎨
⎪
ns
vy
vz
wy
f
f
f
c
3
≈
⎪
⎩
wz
f
