Civil Engineering Reference
In-Depth Information
1
1
(
)
*
T
⎡
(
)
(
)
⎤
()
*
T
S
lim
a
a
lim
Φa Φa
ω
=
⋅
=
⋅
rr
r
r
⎢
η
η
⎥
T
T
π
π
⎣
⎦
T
→∞
T
→∞
1
(
)
*
T
T
T
Φ
lim
aa ΦΦS Φ
=⋅
⋅
⋅
=⋅
⋅
(9.159)
ηη
η
π
T
T
→∞
(
)
⎡
*
T
T
⎤
T
=⋅
Φ H Φ S Φ H Φ
⋅
⋅
⋅
RR
⎣
η
η
⎦
where
NN
(
)
ˆ
ˆ
ˆ
()
∑∑
T
⎡
12
12
T
T
⎤
o
S
ω
=
A
B ψ SC S ψ BA
⋅
⋅
(9.160)
RR
n
Q
n
n
nm
m
m
Q
m
⎣
⎦
n
m
nm
==
11
The response covariance matrix may then be obtained simply by integration throughout
the frequency domain, i.e.
2
1
⎡
⎤
σ
"
Cov
"
Cov
"
Cov
1
i
1
j
1
N
r
⎢
⎥
⎢
# %#
#
#
⎥
⎢
⎥
2
Cov
Cov
"
σ
"
#
⎢
⎥
i
1
i
ij
∞
⎢
⎥
()
∫
Cov
=
=
S
ωω
d
(9.161)
#
# % #
#
rr
⎢
⎥
rr
⎢
⎥
0
2
Cov
Cov
"
"
σ
#
j
1
ji
j
⎢
⎥
⎢
⎥
#
% #
"""""
⎢
⎥
2
Cov
⎢
σ
⎥
⎣
N
1
N
⎦
r
r
This solution strategy will render identical results to that which has been developed in
chapter 6.3. The only difference is that the mode shapes have been arranged in a
different order, i.e. in this chapter in an order which correspond to the chosen degrees of
freedom in a finite element formulation.
Once
()
rr
S
has been established, then the ensuing determination of the statistical
properties of cross sectional forces is identical to that which has been shown in chapter
9.6 above (see Es. 9.127 - 9.134).
9.9 Dynamic response calculations in time domain
In a time domain solution the total displacement response may be obtained as a sum of
the time invariant solution
r
(given in Eq. 9.86) and a purely dynamic part
()
t
r
, i.e.
