Civil Engineering Reference
In-Depth Information
where it has been introduced that the single-sided spectrum of the modal loading is de-
fined by
1
(
)
()
*
S
lim
a
a
ω
=
⋅
⋅
(4.28)
Q
Q
Q
T
i
π
i
i
T
→∞
()
i
x
1
This will render the displacement response at a position where
φ
=
. The response
x
(e.g. where
at an arbitrary position
has its maximum) may simply be obtained by
recognizing that due to linearity the Fourier amplitude at
φ
x
is given by
()
( )
()
a
x
a
ω
=
φ
⋅
ω
(4.29)
r
i
η
i
r
i
and thus, the response spectrum for the displacement response at
x
x
is given by
=
r
2
()
φ
x
2
ˆ
i
r
(
)
()
()
Sx
,
H
S
ω
=
⋅
ω
⋅
ω
(4.30)
r
i
r
i
Q
i
2
K
i
In structural engineering it has been customary to split the response calculations into a
background and a resonant part as illustrated in Fig. 4.4.
Fig. 4.4
Frequency domain spectra and transfer function
The motivation behind this is that static and quasi-static load effects are more accu-
rately determined directly from time invariant equilibrium conditions. This is particu-
larly important for the determination of cross sectional force resultants (or stresses), as
shown in chapter 7. The variance of the displacement response in Eq. 4.30 split into a
background and a resonant part is given by
