Civil Engineering Reference
In-Depth Information
(
)
(
)
Eq x t q x t
⎡
,
,
⎤
⋅
=
⎣
⎦
y
1
y
2
2
2
⎧
2
⎫
(7.22)
2
⎛
⎞
ρ
VB
D
⎡
D
⎤
⎪
⎛
⎞
⎛
⎞
⎪
(
)
(
)
2
CI
x
C
′
C I
x
⋅
⋅
ρΔ
+
−
⋅
ρ Δ
⎜
⎟ ⎨
⎬
⎜
⎟
⎢
⎜
⎟
⎥
⎜
⎟
Du
uu
D
L
w
ww
2
B
B
⎝
⎠
⎝
⎠
⎣
⎦
⎝
⎠
⎩
⎪
⎪
⎭
I
/
V
I
/
ww
V
where
σ= are the
u
- and
w
-component turbulence
intensities. Thus, the variance of the background part is given by (see Eq. 7.17)
=
σ
and
u
u
2
2
⎛
⎞
VB
ρ
2
()
()
()
x
∫∫
G
x
G
x
σ
=
⋅
⋅
⋅
⎜
⎟
Mr
⎜
⎟
M
1
M
2
z
2
z
z
B
⎝
⎠
L
exp
(7.23)
2
⎧
2
⎫
D
⎡
D
⎤
⎪
⎛
⎞
⎛
⎞
⎪
(
)
(
)
2
CI
x
C
′
C I
x dxdx
⋅
ρΔ
+
−
⋅
ρ Δ
⎨
⎬
⎜
⎟
⎢
⎜
⎟
⎥
Du
uu
D
L
w
ww
12
B
B
⎝
⎠
⎝
⎠
⎣
⎦
⎪
⎪
⎩
⎭
The volume integral in Eq. 7.23 represents a spatial averaging of the fluctuating load
effect with respect to the bending component
M
at a certain spanwise position
x
.
This is identical to that which has previously been dealt with in Chapter 2.10 (see
Example 2.4).
While Eq. 7.23 provides the calculation procedure for the background part of the
cross sectional force component
M
at
x
alone, it is convenient to establish more
general procedures comprising the background response of several components, e.g. the
bending moments
M
as well as the torsion moment
M
and
M
. These force
y
x
components are in general given by
⎡
()
( )
⎤
Gxqxt
⋅
,
⎡ ⎤
M
M
x
θ
⎢
⎥
x
⎢ ⎥
(
)
()
( )
x
,
t
M
∫
⎢
Gx
qx t
,
⎥
x
M
=
=
⋅
(7.24)
⎢ ⎥
Br
y
M
y
y
⎢
⎥
⎢ ⎥
L
M
exp
()
( )
⎢
Gxqxt
,
⎥
⋅
⎣ ⎦
z
B
⎣
M
z
⎦
z
G
nxyz
,,
where
,
=
, are the static influence functions for cross sectional force
M
n
M
at
x
. By adopting the definition
components
M
,
M
and
x
y
⎡
⎤
0
0
G
M
x
⎢
⎥
()
⎢
⎥
G
x
=
⎢
0
G
0
(7.25)
M
M
y
⎥
⎢
⎥
G
0
0
⎣
M
z
⎦
