Civil Engineering Reference
In-Depth Information
(
)
Cov
M
where
ρ
=
σ
σ
is the covariance coefficient between
and
MM
MM
M
M
y
y
z
y
z
y
z
M
fluctuations. Similarly, the variance of the shear stress is given by
2
⎡
⎤
⎛
Q
⎞
QM
y
2
2
()
⎢
z
x
⎥
Es t
⎡
⎤
E
σ
=
=
⎜
+
+
⎟
s
⎣
yz
⎦
⎜
⎟
yz
⎢
AA At
2
⎥
⎝
⎠
y
z
m
0
⎣
⎦
2
⎡
2
2
⎤
⎛ ⎞
Q
⎛
Q
⎞ ⎛
Q
⎞
⎛ ⎞ ⎛
Q
M
⎞
Q
M Q
⎛
M
⎞
y
y
y
⎢
⎥
z
x
z
x
z
x
=
E
+
+
+
2
+
2
+
2
⎜ ⎟
⎜
⎟ ⎜
⎟
⎜ ⎟ ⎜
⎟
⎜
⎟
⎜ ⎟
⎜
⎟ ⎜
⎟
⎢
A
A
2
At
A A
A
2
At
A
2
At
⎥
⎝ ⎠ ⎝
⎠
⎝
⎠
⎝ ⎠
⎝
⎠ ⎝
⎠
y
z
m
0
y
z
y
m
0
z
m
0
⎣
⎦
2
2
2
σ
Cov
Cov
⎛ ⎞
⎛ ⎞ ⎛
⎜ ⎟
σ
σ
Cov
⎞
Q
QQ
QM
Q
M
QM
y
yz
y x
z
x
zx
2
2
2
=
+
+
+
+
+
⎜
⎜ ⎟
⎜
⎟
⎜
⎟
⎜ ⎟
A
A
2
A t
AA
A At
2
A At
2
⎝ ⎠
⎝
⎠
⎝ ⎠
y
z
m
0
yz
y
m
0
z
m
0
2
2
2
σ
⎛ ⎞
⎛ ⎞ ⎛
σ
σ
⎞
Q
Q
M
y
z
x
⎜ ⎟
=
+
+
⎜ ⎟ ⎜
⎟
⎜ ⎟ ⎜
⎟
⎜ ⎟
A
A
2
A t
⎝ ⎠ ⎝
⎠
⎝ ⎠
y
z
m
0
σσ
σσ
σσ
QQ
Q M
yz
y x
QM
z
x
+
2
ρ
+
2
ρ
+
2
ρ
QQ
QM
QM
AA
yz
A At
2
y x
A At
2
z x
yz
y
m
0
z
m
0
(1.18)
2
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
2
σ
⎡
⎤
Q
Cov
QQ
⎡
⎤
Q
y
y
Q
y
z
y
z
⎢
⎥
⎢
⎥
2
2
⎢
⎥
EQ
M
Cov
E Q M
QM
Where
=
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢
⎣ ⎦
σ
=
⎢
⎥
Q
z
QM
y
x
z
yx
⎢
⎥
⎢
⎥
2
2
⎢
⎥
⎣
Cov
σ
⎦
z
x
x
⎣
Q
zx
⎦
⎢
⎣ ⎦
M
x
(
)
⎡
⎤
Cov
σσ
⎡
⎤
⎢
ρ
QQ
Q
Q
⎥
yz
y
z
Q
yz
⎢
⎥
⎢
(
)
⎥
⎢
⎥
Cov
and
ρ
=
σ
σ
⎢
⎥
QM
QM
Q
M
yx
yx
y
x
⎢
⎥
⎢
⎥
⎢
⎥
⎢
(
)
ρ
⎥
Cov
σσ
⎣
Q
zx
⎦
⎢
QM
Q
M
⎥
zx
z
x
⎣
⎦
Thus, the main focus is in the following on the calculation of the standard deviation and
the covariance to fluctuating response components, whether they contain dynamic
amplification or not. Simple linear static calculations are considered trivial. However,
some mention of the calculation of time invariant mean values has been included for the
sake of completeness.
