Civil Engineering Reference
In-Depth Information
12
⎛
⎞
ˆ
ˆ
()
⎜
( ) ( )
(
)
⎟
2
J
ω
=
∫∫
φ
x
⋅
φ
x
⋅
o
Δ
x
,
ω
dx dx
∫
φ
dx
θ
θ
1
θ
2
ww
1
2
θ
⎜
⎟
L
L
⎝
⎠
exp
then the
r
,
r
and
r
θ
response spectra are given by (see Eq. 4.30, 6.18, 6.19, 6.28 and 6.29)
2
2
⎡
⎤
()
2
⎛ ⎞
S
BD
V
ω
ρ
ˆ
ˆ
(
)
⎢
(
)
()
()
⎥
u
Sx
,
x
CI
H J
ω
=
φ
⋅
⋅
⎜
⎜ ⎟
⋅
⋅
ω
⋅
ω
⋅
r
r
y
r
D
u
y
y
2
y
⎢
mB
⎥
ω
σ
⎝ ⎠
y
y
u
⎣
⎦
2
2
⎡
⎤
()
3
S
BV
⎛ ⎞
ω
ρ
ˆ
ˆ
(
)
(
)
()
()
w
⎢
⎥
Sx
,
x
CI
′
H J
ω
=
φ
⋅
⋅
⋅
ω
⋅
ω
⋅
⎜ ⎟
r
r
z
r
L
w
z
z
2
z
2
mB
⎢
ω
⎥
σ
⎝ ⎠
z
z
⎣
⎦
w
2
2
⎡
⎤
()
4
S
BV
⎛ ⎞
ω
ρ
ˆ
ˆ
(
)
(
)
()
()
w
⎢
⎥
Sx
,
x
CI
′
H J
ω
=
φ
⋅
⋅
⋅
ω
⋅
ω
⋅
⎜ ⎟
r
r
r
Mw
θ
θ
θ
2
θ
2
mB
⎢
ω
⎥
σ
⎝ ⎠
⎣
θ
θ
⎦
w
Integrating across the entire frequency domain, the following response standard deviations are
obtained:
12
2
()
2
⎛ ⎞
⎡
∞
⎤
S
ρ
BD
V
ω
2
ˆ
ˆ
()
()
()
u
2
()
x
x
C I
∫
H
J
d
σ
=
φ
⋅
⋅
⋅
⎜ ⎟
⎢
⋅
ω
⋅
⋅
ω
ω
⎥
r
r
y
r
Du
⎜ ⎟
⎢
y
y
y
2
m
B
ω
σ
⎥
⎝ ⎠ ⎣
⎦
y
y
u
0
12
2
()
3
⎡
∞
⎤
S
ρ
B
⎛ ⎞
V
ω
2
ˆ
ˆ
()
()
()
w
2
()
x
x
C I
∫
H
J
d
σ
=
φ
⋅
⋅
′
⋅
⋅
ω
⋅
⋅
ω
ω
⎜ ⎟
⎢
⎢
⎥
r
r
z
r
Lw
z
z
z
2
2
m
B
ω
σ
⎥
⎝ ⎠
⎣
⎦
z
z
w
0
12
2
()
4
⎡
∞
⎤
S
ρ
B
⎛ ⎞
V
ω
2
ˆ
ˆ
()
()
()
w
2
()
∫
σ
x
=
φ
x
⋅
⋅
C
′
I
⋅
⋅
H
ω
⋅
⋅
J
ω
d
ω
⎢
⎥
⎜ ⎟
⎢
r
r
θ
r
Mw
θ
θ
2
θ
2
m
B
ω
σ
⎥
⎝ ⎠
⎣
⎦
θ
θ
0
w
Let us focus exclusively on the response in the
y
(drag) direction, and consider a simply supported
horizontal beam type of bridge with span
L
500
m
z
50
m
=
that is elevated at a position
=
. Let
f
()
(
)
y
x
sin
xL
x
L
2
us for simplicity assume that the relevant mode shape
φ
=
π
and that
=
, in
r
()
x
1
L
L
which case
φ
=
. Let us also assume that the entire span is flow exposed, i.e.
=
,
yr
exp
and adopt the following wind field properties:
I
V
0.15
1) the turbulence intensity
=
σ
=
(see Eq. 3.14)
u
u
(
)
0.3
x
f
L
100
z
10
162
m
2) the integral length scale:
=⋅
=
(see Eq. 3.36),
u
f
()
x
f
S
ω
1.08
LV
⋅
u
u
3) the auto spectral density:
=
(see Eq. 3.25)
(
)
2
5 3
σ
x
f
u
11.62
+⋅
ω
⋅
LV
u
ˆ
(
)
(
)
Co
,
x
exp
C
x V
4) the normalised co-spectrum:
ω
Δ=
− ⋅
ω
⋅ Δ
(see Eq. 3.41)
uu
ux
()
CC
9/ 2
1.4
where
=
=
π
≈
.
ux
uy
f
