Civil Engineering Reference
In-Depth Information
it follows from Eqs. 7.14 and 7.24 that
VB
ρ
{
}
ˆ
(
)
( )
(
)
()
( )
∫
∫
M
x
,
t
=
G
x
⋅
q
x t dx
,
=
⋅
G
x
⋅
B v
⋅
x t
,
dx
(7.26)
Br
M
M
q
2
L
L
exp
exp
where
ˆ
B
is the load coefficient matrix defined in Eq. 5.12, i.e.
(
)
⎡
(
)
(
)
⎤
2/
DBC
DBC
/
C
′ −
D
D
L
⎢
⎥
(
)
ˆ
⎢
⎥
()
(
)
B
x
=
2
C
C
′
+
D B C
/
(7.27)
q
L
L
D
⎢
⎥
⎢
⎥
2
BC
BC
′
M
M
⎢
⎥
⎣
⎦
T
(
)
(
)
(
)
and where
=
⎣
v
in the case of a horizontal bridge type of structure
(see Eq. 5.9). The background covariance matrix
x t
,
⎡
uxt
,
wxt
,
⎤
⎡
2
⎤
σ
Cov
Cov
MM
MM
MM
⎢
xx
xy
xz
⎥
⎢
⎥
()
2
Cov
x
=
⎢
ov
σ
ov
(7.28)
MM
r
M
M
M
M
M
M
⎥
B
y
x
y
y
y
z
⎢
⎥
2
Cov
Cov
σ
⎢
⎥
MM
MM
MM
⎣
⎦
zx
zy
zz
B
is then obtained from
(
)
⎡
(
)
T
(
)
⎤
x
E
xt
,
xt
,
Cov
=
M
⋅
M
MM
r
⎣
B
r
B
r
⎦
B
T
⎡
⎤
⎧
⎫ ⎧
⎫
2
VB
⎛
ρ
⎞
⎢
⎪
(
)
⎪ ⎪
(
)
⎪
⎥
ˆ
ˆ
E
∫
dx
∫
dx
=
GBv GBv
⋅
⋅
⋅
⋅
⋅
(7.29)
⎨
⎬ ⎨
⎬
⎜
⎟
⎢
⎥
Mq
Mq
2
⎝
⎠
⎪
⎪ ⎪
⎪
⎢
⎥
L
L
⎩
exp
⎭ ⎩
exp
⎭
⎣
⎦
2
VB
ρ
{
}
⎛
⎞
ˆ
ˆ
()
( )
T
( )
T
T
()
⎡
⎤
=
∫∫
G
x
⋅
B
⋅
E
v
x
,
t
⋅
v
x
,
t
⋅
B
⋅
G
x
dx dx
⎜
⎟
M
1
q
1
2
q
M
2
1
2
⎣
⎦
2
⎝
⎠
L
exp
Introducing Eq. 7.21 and adopting the assumptions in Eq. 7.19, then
2
(
)
⎡
x
0
⎤
σρ
⋅
Δ
{
}
u u
⎡
(
)
T
(
)
⎤
22
(
)
E
xt
,
xt
,
V
x
v
⋅
v
≈
⎢
⎥
=
⋅
I ρ
⋅
Δ
(7.30)
⎣
1
2
⎦
v
2
(
)
0
σρ
⋅
Δ
x
⎢
⎥
⎣
⎦
w
w
