Civil Engineering Reference
In-Depth Information
00 0
10 0
01 0
00 1
00 0
00 0
⎡
⎤
D
C
B
⎢
⎥
⎡
⎤
2
0
⎢
⎥
⎢
⎥
D
⎢
⎥
100
001
⎢
⎥
⎡
⎤
ˆ
where
B
=
0
0
C
BC
′
,
θ
=
and
ψ
=
⎢
⎥
⎢
⎥
⎢
⎥
q
L
m
m
m
−
⎣
⎦
⎢
⎥
⎢
⎥
′
M
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎢
⎥
⎢
⎥
⎣
⎦
Fig. 7.11
Simply supported beam type of bridge
The wind load covariance matrix associated with the cross product between external load
components in arbitrary nodes
p
and
k
is given by (see Eqs. 7.55 -7.59)
2
⎛
2
⎞
VB
ρ
ˆ
ˆ
(
)
T
Cov
=
⋅
Q C
⋅
ov
⋅
Q
where
x
⎜
⎟
Cov
=⋅
I
I ρ
⋅
Δ
R R
⎜
⎟
p
v v
k
v v
p
k
pk
pk
pk
2
pk
pk
⎝
⎠
(
)
[
]
[
]
and
I
==
I
diag I
I
I
,
ρ
Δ=
x
diag
ρ
ρ
ρ
, and where
Δ
is
p
k
u
v
w
pk
pk
uu
vv
w
pk
the absolute value of the distance between nodes
p
and
k
. Thus,
2
2
⎛
VBL
N
⎞
ρ
(
)
ˆ
ˆ
TT
T
Cov
=
⋅
θ B ψ IIρθB ψ
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⎜
⎟
R R
⎜
⎟
m
q
m
p
k
pk
m
q
m
pk
2
m
m
⎝
⎠
2
2
⎛
VBL
N
⎞
ρ
ˆ
⇒
Cov
=
⋅
Cov
⎜
⎟
R R
⎜
⎟
R R
p
k
2
p
k
⎝
⎠
where the non-dimensional cross covariance matrix
ˆ
Cov
is given by
R R
p
k
ˆ
Cov
=
R
pk
0
0
0
0
0
0
⎡
⎤
⎢
⎥
2
D
CI
⎢
⎛
⎞
⎥
(
)
02
x
0
0
00
⋅
ρ
Δ
⎢
⎜
⎟
⎥
Du
uu
pk
B
⎝
⎠
⎢
⎥
⎢
2
(
)
(
)
⎥
(
)
2
0
0
CI
′
x
CBC I
′
′
x
0
0
⋅
ρ
Δ
−
⋅
ρ
Δ
Lw
ww
pk
L
M w
ww
pk
⎢
⎥
⎢
⎥
(
)
2
(
)
2
(
)
0
0
CBCI
′
′
x
BCI
′
x
0
0
−
⋅
ρ
Δ
⋅
ρ
Δ
⎢
⎥
LMw
w k
Mw
w k
⎢
⎥
0
0
0
0
0
0
⎢
⎥
⎢
0
0
0
0
0
0
⎥
⎣
⎦
