Civil Engineering Reference
In-Depth Information
Performing the multiplication
TT
T ββ T
and then the following is obtained:
⋅
⋅
⋅
i
i
TT
T ββ T
⋅
⋅
⋅
=
i
i
{
(
)
⎡
⎤
−
φ
′′′
EI
⋅
2
(
)
(
)(
)
yz
(
)(
)
(
)(
)
⎢
⎥
′′′
EI
′′′
EI
′′′
EI
′′′
EI
′′
EI
′′′
EI
′′
EI
φ
φ
φ
φ
φ
−
φ
φ
yz
yz z y
yz z y
yz yz
⎢
)
}
⎥
(
φ
′
GI
−
φ
′′′
EI
θ
t
θ
w
⎢
⎥
⎢
{
⎥
(
)
′′′
EI
−
φ
⋅
2
⎢
(
)
z
y
(
)(
)
(
)(
)
⎥
′′′
EI
′′′
EI
′′
EI
′′′
EI
′′
EI
φ
φ
φ
−
φ
φ
⎢
z
y
z
y z
y
z
y yz
⎥
)
}
(
′
GI
′′′
EI
φ
−
φ
⎢
⎥
t
w
θ
θ
⎢
⎥
{
{
(
)
(
)
φ
′
GI
−
φ
′′′
EI
φ
′
GI
−
φ
′′′
EI
⎢
⎥
θ
t
θ
w
θ
t
w
2
(
)
GI
EI
φ
′
−
φ
′′′
⎢
⎥
)
}
)
}
θ
t
θ
w
(
(
⋅
φ
′′
EI
⋅
φ
′′
EI
⎢
⎥
z
y
yz
⎢
⎥
2
(
)
(
)(
)
⎢
⎥
Sym
.
φ
′′
EI
−
φ
′′
EI
φ
′′
EI
z
y
z
y yz
⎢
⎥
⎢
⎥
2
(
)
′′
EI
φ
⎢
⎥
⎣
yz
⎦
i
(7.87)
It is readily seen from Eq. 7.87 that the covariance matrix associated with an arbitrary
mode
i
has the properties
()
x
Cov
=
FF
r
R
i
2
⎡
⎤
σ
σ
⋅
σ
−
σ
⋅
σ
σ
⋅
σ
−
σ
⋅
σ
VV
VV
VV
VV
M M
VV
M M
VV
M M
yy
yy
zz
yy
x x
yy
y y
yy
z z
⎢
⎥
⎢
⎥
2
σ
−
σ
⋅
σ
σ
⋅
σ
−
σ
⋅
σ
VV
VV
M M
VV
M M
VV
M M
⎢
⎥
zz
zz
x x
zz
y y
zz
z z
⎢
⎥
2
σ
σ
⋅
σ
σ
⋅
σ
⎢
⎥
MM
MM
MM
MM
MM
xx
xx
yy
xx
zz
⎢
⎥
2
Sym
.
σ
−
σ
⋅
σ
⎢
⎥
MM
MM
MM
yy
yy
zz
⎢
⎥
2
⎢
σ
⎥
⎣
M
zz
⎦
i
(7.88)
I.e., for an arbitrary mode
i
Cov
1 for
mn
VV VM
,
,
VM
,
MM
,
MM
⎧
⎪
+
=
y
z
y
y
z
y
x
y
x
z
mn
i
ρ
=
=
(7.89)
⎨
mn
i
−
1 for
mn
=
VM
,
VM
,
VM
,
VM
,
M M
σσ
⋅
⎪
⎩
mm
nn
y
x
y
z
z
x
z
z
y
z
i
i
which could be expected, because within a single mode all coupling between cross
sectional force components are caused by the structural properties already contained in
the relevant mode shape, and thus, all covariance coefficients will either be plus or
minus unity (depending on the chosen sign conventions). Thus, the problem is reduced
to the calculation of variance contributions from each of the modes that have be
