Civil Engineering Reference
In-Depth Information
2
T
T
L
⎡
⎤
1
1
2
2
()
()
( )
lim
∫
M t
dt
lim
∫ ∫
G
x
q
x t
,
dx
dt
σ
=
⎡
⎤
=
⎢
⋅
⋅
⎥
(2.100)
⎣
⎦
M
M
y
T
T
T
→∞
T
→∞
⎢
⎥
⎣
⎦
0
0
0
(
)
y
qxt
, as this is the only time domain
variable on the right hand side of the equation. This may be obtained by splitting the
squared integral into a product of two identical integrals, only made distinguishable by
letting them contain different space variables, one labelled
,
It is desirable to perform statistics only on
x
and the other
x
, Thus,
the following is obtained
TL
L
⎡
⎤ ⎡
⎤
1
2
() ( )
() ( )
σ
=
lim
∫∫
G
x
⋅
q
x
,
t
⋅
dx
⋅
∫
G
x
⋅
q
x
,
t
⋅
dx
dt
⎢
⎥ ⎢
⎥
M
M
1
y
1
1
M
2
y
2
2
T
T
→∞
⎢
⎥ ⎢
⎥
⎣
⎦ ⎣
⎦
00
0
(2.101)
LL
T
⎡
⎤
1
()
()
(
)
(
)
∫∫
G
x
G
x
lim
∫
q
x
,
t
q
x
,
t dt dx dx
=
⋅
⋅
⋅
⎢
⎥
M
1
M
2
y
1
y
2
1
2
T
T
→∞
⎢
⎥
⎣
⎦
00
0
(
)
y
qxt
is given by
,
Recalling that the cross covariance function of
T
1
(
)
(
)
(
)
Cov
x
,
0
lim
∫
q
x t
,
q
x
x t dt
,
Δ==
τ
⋅
+Δ
qq
y
y
yy
T
T
→∞
0
(2.102)
T
1
(
)
(
)
(
)
∫
=
lim
qxtqxtdtCov
,
⋅
,
=
Δ
x
y
1
y
2
q
y
T
T
→∞
0
where the separation
Δ=
x
xx
−
, and introducing the covariance coefficient
21
(
)
(
)
2
ρ
Δ=
x
v x
Δ
/
σ
, it is seen that Eq. 2.101 simplifies into
q
q
q
y
y
y
LL
2
2
()
()
( )
∫∫
GxGx
x x x
σσ
=⋅
⋅
⋅
ρ
Δ
(2.103)
Mq
M
1
M
2
q
1
2
y
y
00
The square root of the double integral
1/2
LL
⎡
⎤
()
()
( )
J
∫∫
G
x
G
x
x dx dx
=
⋅
⋅
ρ
Δ
(2.104)
⎢
⎥
M
M
1
M
2
q
y
1
2
⎢
⎥
⎣
⎦
00
is in wind engineering often called the joint acceptance function, because it contains the
necessary statistical (i.e. variance) averaging in space. Thus,
