Civil Engineering Reference
In-Depth Information
then the following is obtained:
Cov
=
vv
ˆˆ
(
)
(
)
s
0
0
s
0
0
⎡
ρ
Δ
ρ
Δ
⎤
uu
11
n m
uu
1 2
n m
⎢
⎥
(
)
(
)
0
ρ
Δ
s
0
0
ρ
Δ
s
0
⎢
⎥
vv
11
n m
vv
12
n m
⎢
(
)
(
)
⎥
0
0
s
0
0
s
ρ
Δ
ρ
Δ
ww
11
n m
ww
1 2
n m
⎢
⎥
(
)
(
)
ρ
Δ
s
0
0
ρ
Δ
s
0
0
⎢
⎥
uu
21
n m
uu
2 2
n m
⎢
⎥
(
)
(
)
0
s
0
0
s
0
ρ
Δ
ρ
Δ
vv
21
n m
vv
2 2
n m
⎢
⎥
(
)
(
)
⎢
0
0
ρ
Δ
s
0
0
ρ
Δ
s
⎥
⎣
⎦
ww
21
n m
ww
2 2
n m
(9.97)
where
is the separation in the global coordinate system between points
Δ
ij
(
)
i
=
1 or 2
n n
and
j
=
1 or 2
mm
, and where
ρ
Δ
s
,
=
uv
, or
w
, are the
p
pp
ij
covariance coefficients of the flow components, see Eq. 3.33, i.e.:
1
⎧
⎫
2
2
⎡
⎤
2
⎪
⎪
⎛
⎞ ⎛
⎞
XX
ZZ
−
−
⎪
⎪
(
)
⎢
⎥
i
j
i
j
s
exp
⎜
⎟ ⎜
⎟
ρ
Δ=
−
+
(9.98)
⎨
⎬
pp
ij
⎢
⎥
⎜
x
⎟ ⎜
z
⎟
L
L
⎪
⎪
⎝
⎠ ⎝
⎠
p
p
⎢
⎥
⎣
⎦
⎪
⎪
⎩
⎭
where
x
L
and
z
L
are the integral length scales of
uv
, or
w
in
X
or
Z
=
p
directions (see Eq. 3.35). Thus (see Eq. 9.92 above),
Cov
=
rr
⎧
NN
⎫
(
)
(9.99)
T
⎪
(
)
⎪
1
1
(
)
−
(
)
−
∑∑
T
T
T
KK
−
⋅
A
⋅
Bψ Cov ψ BA
⋅
⋅
⋅
⋅
⋅
⋅
KK
−
⎨
⎬
ˆˆ
ae
n
n
n
vv
m
m
m
ae
⎪
⎪
⎩
⎭
nm
==
11
The extreme value of the response
tot
r
may then readily be obtained from Eq. 9.91.
Since this solution is quasi-static, the corresponding cross sectional element forces may
be calculated from
max
(
)
k
F
=
k d
=
k A
r σ kAr
+
=
(9.100)
tot
n
tot
n
n
p
r
n
n tot
n
n
max
max
max
