Civil Engineering Reference
In-Depth Information
is then defined by
{
}
{
}
T
⎡
⎤
(
)
(
)
T
⎡
−
1
⎤
⎡
−
1
⎤
E
⎡
⎤
E
C v
=
FF kAKR kAKR
⋅
=
⋅
⎢
⎥
FF
⎣
mm
⎦
mm
mm
⎣
⎦
⎣
⎦
mm
⎣
⎦
⎧
T
⎫
⎛
(
)
⎞
−
1
T
−
1
T
T
⎡
⎤
=⋅
kAK RRK Ak
⋅
⋅
E
⋅
⋅
⋅
⋅
(7.53)
⎨
⎜
⎟
⎬
mm
⎣
⎦
mm
⎝
⎠
⎩
⎭
⎧
⎡
(
)
T
⎤
⎫
1
1
TT
−
−
=
kAKC K Ak
⋅
⋅
⋅
ov
⋅
⋅
⋅
⎨
⎬
mm
⎢
R
⎥
mm
⎣
⎦
⎩
⎭
where the
6
N
by
6
N
nodal load covariance matrix
⎡
⎤
%#
$
⎢
⎥
p
⎫
T
⎡
⎤
Cov
=
E
R R
⋅
=
"
C
ov
"
where
=
1,2,3,
,
N
(7.54)
…
⎢
⎥
⎬
⎭
RR
⎣
⎦
⎢
R
pk
k
⎥
$#
%
⎢
⎥
⎣
⎦
Its content is
N
nubers of 6 by 6 covariance matrices between force components
associated with nodes
p
and
k
, each is given by
T
⎡
⎤
2
2
⎛
⎞ ⎛
⎞
V
V
ρ
ρ
ˆ
ˆ
⎡
T
⎤
⎢
ˆ
ˆ
⎥
E
E
Cov
=
R R
⋅
=
⋅
Q v
⋅
⋅
⋅
Q v
⋅
⎜
⎟ ⎜
⎟
RR
⎣
p
k
⎦
⎜
⎟ ⎜
⎟
pk
2
2
⎢
⎥
⎝
⎠ ⎝
⎠
⎣
p
k
⎦
2
2
⎛ ⎞ ⎛ ⎞
ρ
V
ρ
V
ˆ
ˆ
T
T
⎡
ˆ
ˆ
⎤
E
=
⋅
⋅
Qv
⋅
⋅
v Q
⋅
(7.55)
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞
p
⎣
p
k
⎦
k
2
2
p
k
2
2
V
V
ρ
ρ
ˆ
ˆ
ˆ
T
ov
=
⋅
⋅
QC Q
⋅
⋅
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
p
v v
k
2
2
pk
p
k
where
⎡
uu
uv
uw
⎤
p
k
p
k
p
k
1
⎢
⎥
ˆ
T
⎡
ˆ
ˆ
⎤
Cov
E
v
v
E vu
vv
vw
=
⋅
=
⋅
(7.56)
⎢
⎥
⎣
⎦
vv
p
k
p k
p k
p k
pk
2
V
⎢
⎥
wu
wv
ww
⎢
⎥
⎣
⎦
p
k
p
k
p
k
As previously mentioned (see Eq. 7.19), it is a usual assumption in wind engineering that
cross-covariance between different velocity components is insignificant, i.e. that all off
diagonal terms in Eq. 7.56 may be neglected, in which case
ˆ
Cov
I
I ρ
(7.57)
pk
≈⋅
⋅
vv
p
k
pk
