Civil Engineering Reference
In-Depth Information
9.7 Dynamic response calculations in frequency domain
If the lowest eigen frequency of the structure is in a region where dynamic effects can
not be neglected, say below about 4 Hz, then a quasi-static solution shown in Chapter 9.6
above will no longer suffice and a full dynamic analysis will be required. However, the
time invariant static solution shown in Chapter 9.5 is still valid, and thus,
t
he 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
()
()
()
r r r
. However, in the
following a stochastic solution in frequency domain will be shown, and from this only
the statistical properties of the response will emerge, i.e. the result of the response
calculation is the covariance matrix
r
t
, i.e.
tot
t
=+
t
⎡
2
1
⎤
Cov
Cov
Cov
σ
"
"
"
1
i
1
j
1
N
r
⎢
⎥
⎢
# %#
#
#
⎥
⎢
⎥
2
Cov
"
"
Cov
#
σ
⎢
⎥
i
1
i
ij
⎢
⎥
T
E
⎡ ⎤
Cov
=
r r
⋅
=
⎢
#
# % #
#
(9.101)
⎥
rr
⎣ ⎦
⎢
⎥
2
Cov
"
Cov
"
#
σ
⎢
j
1
ji
j
⎥
⎢
⎥
#
% #
"""""
⎢
⎥
2
Cov
⎢
σ
⎥
⎣
N
1
N
⎦
r
r
where
N
is the number of degrees of freedom in the system. Thus, extreme values of
displacement events are given by
r
=+
r
r
=+
r σ
k
(9.102)
tot
max
p
r
max
where
r
is given in Eq. 9.86,
k
is a peak factor defined in Chapter 2.4 and
σ
is a
vector containing all the standard deviations of the chosen set of displacement degrees of
freedom in the system.
σ
may be extracted from the square root of the vector contained
on the diagonal of the covariance matrix, see Eq. 9.101.
In a frequency domain approach it is a necessary requirement that all load and
response quantities are stationary such that a Fourier transform will render predictable
coefficients throughout the entire time window of the process. I.e. motion induced load
contributions which evolve from the previous history of the process (indicial functions)
can not be included in such a solution strategy. Thus, response calculations in frequency
domain must be based on (see Eq. 9.80)
() (
) () (
) ()
()
t
t
t
t
(9.103)
Mr
+−
C C
r
+−
K K
r
=
R
ae
ae
