Civil Engineering Reference
In-Depth Information
from which the response covariance matrix may readily be obtained by frequency
domain integration, i.e.
2
1
⎡
⎤
σ
"
Cov
"
Cov
"
Cov
1
i
1
j
1
N
r
⎢
⎥
⎢
# %#
#
#
⎥
⎢
⎥
2
Cov
Cov
"
σ
"
#
⎢
⎥
i
1
i
ij
∞
⎢
⎥
T
()
⎡ ⎤
∫
Cov
=
E
r r
⋅
=
=
S
ωω
d
#
# % #
#
rr
⎣ ⎦
⎢
⎥
rr
⎢
⎥
0
2
Cov
Cov
"
"
σ
#
j
1
ji
j
⎢
⎥
⎢
⎥
#
% #
"""""
⎢
⎥
2
Cov
⎢
σ
⎥
⎣
N
1
N
⎦
r
r
(9.126)
For the ensuing calculations of the statistical properties of cross sectional response forces
at element end points it is necessary also to determine the covariance between the
displacement response and its derivatives. The general solution to the problem of
determining the covariance between a stationary process
()
()
x t
and its derivatives
x t
()
x t
has been shown in Chapter 2.9 (see Eq. 2.94). Recalling that for a stationary
and
[
]
[
]
process
Ex x
⋅
=
Ex x
, then
⋅
=
0
⎡
2
⎤
10
−
ω
⎡
Cov
Cov
Cov
⎤
rr
rr
rr
∞
⎢
⎥
⎢
⎥
2
∫
0
0
d
Cov
Cov
Cov
=
⎢
ω
⎥
⋅
S
ω
(9.127)
⎢
⎥
rr
rr
rr
rr
⎢
⎥
⎢
⎥
0
Cov
Cov
Cov
2
4
0
−
ω
ω
⎣
⎦
⎢
⎥
rr
rr
rr
⎣
⎦
Since the displacement response vector associated with element number
n
is given by
dAr
it is seen that
=⋅
n
Cov
T
⎡
⎤
⎡
(
) (
)
⎤
T
T
⎡
⎤
Ar
⋅
Ar
⎡ ⎤
d
nn
dd
⋅
rr
⋅
0
0
0
0
n
n
⎡⎤
⎢
⎢⎥
⎢
⎢
⎣⎦
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ⎥
Cov
T
⎢
⎥
⎢
(
) (
)
⎥
T
T
Ar
⋅
Ar
⎢
dd
⋅
⎥
⎢ ⎥
rr
⋅
dd
nn
n
n
T
E
E
⎢
⎥
E
⎢
⎥
=
=
=
A
⋅
⋅
A
=
(9.128)
⎢
⎥
⎢ ⎥
n
n
T
Cov
T
(
) (
)
T
⎢
⎥
⎢
⎥
dd
⋅
rr
⋅
Ar
⋅
Ar
⎢
⎥
⎢ ⎥
dd
nn
n
n
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ⎥
T
T
T
Cov
dd
⋅
(
) (
)
rr
⋅
⎢
⎥
⎢
⎥
⎣
⎦
Ar
⋅
Ar
⎣ ⎦
⎣
dd
⎦
n
⎣
⎦
n
n
nn
while
