Civil Engineering Reference
In-Depth Information
where
()
()
2
(
)
⎡
g
=
BG
x
G
x
4
C C
I
ρ
Δ
x
MM
M
1
M
2
⎣
L M u uu
xy
x
y
(7.38)
CCCI
⎛
⎞
⎤
2
(
)
x
++
′
′
ρΔ
⎜
⎟
⎥
L
D
M
w
ww
B
⎝
⎠
⎦
D
⎡
()
()
2
(
)
g
BG
x
G
x
4
C C
I
x
=
ρ
Δ
⎢
⎣
MM
M
1
M
2
D M u uu
xz
x
z
B
D
CCCI
(7.39)
⎤
⎛
⎞
2
(
)
+
′
−
′
ρΔ
x
⎜
⎟
⎥
DLMw
w
B
⎝
⎠
⎦
D
⎡
()
()
2
(
)
g
G
x G
x
4
CCI
x
=
ρ
Δ
⎢
⎣
MM
M
1
M
2
D Lu uu
yz
y
z
B
(7.40)
D
D
⎤
⎛
⎞⎛
⎞
2
(
)
+
CCC
′
−
′
+
CI
ρΔ
x
⎜
⎟⎜
⎟
⎥
DL L
Dw
w
B
B
⎝
⎠⎝
⎠
⎦
Example 7.1
Let us set out to calculate the variances and covariance of the torsion and bending moments
M
,
x
M
M
and
at midspan of the simply supported beam type of bridge illustrated in Fig. 7.6. Let
y
z
C
′
C
C
us for simplicity assume that it has a typical bridge type of cross section where
,
and
D
M
CDB C
′
are negligible and
⋅
. Then
D
L
⎡
⎤
⎢
⎥
2
(
)
()
()
( )
⎡
2
⎤
BC
′
I
G
x
G
x
x
⋅
⋅
⋅
ρ
Δ
σ
⎢
Mw
M
1
M
2
ww
⎥
MM
x
x
⎢
⎥
⎛
xx
2
LL
2
⎞
⎢
ρ
VB
⎥
⎢
⎥
2
2
(
)
(
)
(
)
(
)
∫∫
CI
GxGx
x x x
σ
=
⋅
′
⋅
⋅
⋅
ρ
Δ
⎜
⎟
⎢
⎥
⎢
MM
⎥
⎝
⎜
⎟
Lw
M
1
M
2
ww
1
2
yy
y
y
2
⎠
⎢
⎥
00
⎢
⎥
2
2
σ
⎢
D
CI
⎥
⎛
⎞
⎢
⎥
⎣
M
zz
⎦
()
()
( )
2
G
x G
x
x
⋅
⋅
⋅
ρ
Δ
⎢
⎥
⎜
⎟
Du
M
1
M
2
uu
B
z
z
⎝
⎠
⎣
⎦
()
()
( )
⎡
⎤
⎡
2
⎤
Cov
BC C
′′
I
G
x
G
x
x
⋅
⋅
⋅
ρ
Δ
MM
L M w
M
1
M
2
ww
xy
2
x
y
⎢
⎥
⎛
⎢
⎥
LL
2
⎞
VB
ρ
⎢
⎥
⎢
⎥
Cov
∫∫
0
dx dx
=
⎜
⎟
⋅
M
xz
M
yz
⎜
⎟
12
2
⎢
⎥
⎝
⎢
⎥
⎠
00
0
⎢
Cov
⎥
⎢
⎥
⎣
⎦
⎣
⎦
where it has been taken for granted that
L
L
=
.
exp
