Civil Engineering Reference
In-Depth Information
(
)
ˆ
Δ
is the distance between points
i
and
j
, and where
Ss
,
where
Δ
ω
,
ij
pp
ij
uv
, or
w
, is the reduced cross spectral density between flow component
p
at point
i
and itself at point
j
(i.e. the cross spectral density between
=
p
uv
, or
w
at ends 1 or 2 of
w
at ends 1 or 2 of element
m
). As shown in Eq. 2.88,
ˆ
pp
element
n
and
uv
, or
S
may be expressed by the product of the reduced single point spectra at points
i
and
j
and the reduced co spectrum between the same points, i.e.
ˆ
(
)
ˆ
ˆ
ˆ
(
)
()
()
Ss
,
S
S
o
s
,
Δ
ω
=
ω
⋅
ω
⋅
Δ
ω
(9.118)
pp
ij
p
p
pp
ij
i
j
E.g., adopting a Kaimal type of auto spectrum and simple exponential co-spectrum
decay, then (see Eqs. 3.25 and 3.41)
x
f
()
()
⎫
S
ω
ALVZ
⋅
ˆ
p
u
()
p
p
i
i
⎪
⎧
⎪
S
ω
=
=
p
i
2
5 3
⎪
σ
x
f
⎡
()
⎤
p
v
w
=
11.5
ALVZ
p
i
+
ω
⎨
⎪
⎩
⎧
⎪
⎪
⎪
⎢
p
p
i
⎥
⎣
⎦
⎪
⎪
()
()
()
x
f
S
ω
ALVZ
⋅
p
p
p
j
ˆ
j
1
2
1
2
1
2
1
2
()
S
ω
=
=
⎪
n
p
j
2
5 3
σ
⎪
⎡
x
f
⎤
p
j
11.5
+
ALVZ
ω
⎪
n
⎢
p
p
j
⎥
i
(9.119)
⎣
⎦
=
⎬
⎪
⎨
⎪
⎪
⎩
⎧
⎪
⎪
m
⎧
⎫
⎪
2
2
(
)
(
)
⎡
cXX
⎤ ⎡
cZZ
⎤
ω
−
+
−
⎪
⎪
⎪
⎣
px
i
j
⎦ ⎣
pz
i
j
⎦
⎪
m
(
)
ˆ
⎪
Co
Δ
s
,
ω
=
exp
−
⎨
⎬
⎪
pp
ij
V
⎪
⎪
⎪
n
ij
⎪
⎪
⎩
⎭
⎪
⎪
n
j
=
⎨
⎪
⎪
⎩
1
2
()
()
⎡
⎤
V
=
V Z
+
V Z
m
⎪
ij
⎣
i
j
⎦
⎭
m
and where indices
n
or
m
refers to element numbers and 1 or 2 refers to element end
numbers. By defining the reduced auto spectral density matrices associated with
elements
n
and
m
ˆ
⎡
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
⎤
⎫
()
i g SSS SSS
S
ω
=
n
⎢
u
v
w
u
v
w
⎥ ⎪
⎣
1
1
1
2
2
2
⎦
⎬
n
n
n
n
n
n
(9.120)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
()
⎡
⎤
⎪
diag S
S
S
S
S
S
S
ω
=
m
⎢
u
v w u v w
mm m m m m
⎥
⎣
1
1
1
2
2
2
⎦⎭
and the reduced covariance matrix between corresponding element ends
