Civil Engineering Reference
In-Depth Information
ˆ
ˆ
()
( )
(
)
GxGx
ˆ
ˆ
x
ˆ
The inner product
⋅
⋅
ρ
Δ
is then:
Mn
Mm
y
n
1
2
3
4
5
m
ˆ
ˆ
()
( )
( )
GxGx
ˆ
ˆ
x
ˆ
⋅
⋅
ρ
Δ
Mn
Mm
y
1
0.01
0.025
0.034
0.039
0.041
2
0.025
0.09
0.123
0.141
0.149
3
0.034
0.123
0.25
0.287
0.302
4
0.039
0.141
0.287
0.49
0.517
5
0.041
0.149
0.302
0.517
0.81
ˆ
ˆ
()
( )
( )
ˆ
ˆ
ˆ
is symmetric about the
diagonal
m
= and increasing with increasing distance from the base of the beam. The reduced
variance of the base moment fluctuations is given by:
As can be seen, the inner product
GxGx
⋅
⋅
ρ
Δ
x
Mn
Mm
y
2
⎛
⎞
1
σ
[
⎜
M
q
y
L
⎟
(
0.01
0.09
0.25
0.49
0.81
2
0.025
0.034
0.039
0.041
=
+
+
+
+
+
⋅
+
+
+
⎜
2
⎟
2
5
σ
⎝
⎠
)
0.123
0.141
0.149
0.287
0.302
0.517
0.2
+
+
+
+
+
+
⎤
≈
⎦
Thus, the standard deviation of the base moment is given by
2
0.45
L
⇒
σ
≈
⋅
σ
⋅
M
q
y
(See further calculations in Chapter 7.3.)
