Civil Engineering Reference
In-Depth Information
Example 2.4
Considering the cantilevered beam shown in Fig. 2.15, then the reduced variance of the base
moment fluctuations is given by:
2
⎛
⎞
NN
1
σ
ˆ
ˆ
⎜
⎟
∑∑
()
( )
(
)
M
GxGx
ˆ
ˆ
x
ˆ
=
⋅
⋅
ρ
Δ
Mn
Mm
y
2
2
⎜
⎟
σ
LN
nm
11
q
y
==
⎝
⎠
ˆ
()
GxGLxLx
ˆ
ˆ
x
xx
x
xL
where
=
=
=
,
Δ=
−
and
Δ=Δ
. Assuming that the
M
M
mn
(
)
(
)
z
and setting for simplicity
z
x
exp
xL
L
L
covariance coefficient
ρ
Δ=
−Δ
=
, then
q
u
y
(
)
(
)
x
exp
x
ˆ
x
Δ= and
0.2
ρ
Δ=
−Δ. Choosing a reduced integration increment
q
y
0.9
T
[
]
x
ˆ
0.1
0.3
0.5
0.7
corresponding position vector
=
then the influence function
ˆ
ˆ
()
( )
GxGx
ˆ
ˆ
multiplications
⋅
are given by
Mn
Mm
ˆ
ˆ
ˆ
()
( )
()
GxGx
ˆ
ˆ
:
Gx
Mn
⋅
Mn
Mm
0.1
0.3
0.5
0.7
0.9
0.1
0.01
0.03
0.05
0.07
0.09
(
0.3
0.03
0.09
0.15
0.21
0.27
0.5
0.05
0.15
0.25
0.35
0.45
ˆ
)
Mm
Gx
0.7
0.07
0.21
0.35
0.49
0.63
0.9
0.09
0.27
0.45
0.63
0.81
The covariance coefficient
(
)
x
ˆ
ρ
Δ
is given by:
q
y
(
)
x
ˆ
:
x
ˆ
n
ρ
Δ
q
y
0.1
0.3
0.5
0.7
0.9
0.1
1
0.82
0.67
0.55
0.45
ˆ
m
0.3
0.82
1
0.82
0.67
0.55
0.5
0.67
0.82
1
0.82
0.67
x
0.7
0.55
0.67
0.82
1
0.82
0.9
0.45
0.55
0.67
0.82
1
