Civil Engineering Reference
In-Depth Information
The second term of [8.45] represents the mathematical expectancy of the f
parameter random variable:
TT
2
³³
ª
º
,
,
,
* **
f
t
t
exp
2iʌf
t
t
dt dt
[8.46]
¬
¼
00
By using [8.46] we obtain, if we suppose a slow a(t) variation:
T
2
1
³
2
ª
S , 0
º |
Sfat t
[8.47]
¬
¼
pv
2
0
We take a standardized envelope with a given (half-sine) shape. The integral
given by [8.47] only depends on the unknown T. In order to obtain T from the ORS,
for example, for an f frequency centered within the seismic range and for two given
damping values, we can consider the ratio:
R
f,İ,İ'
S
f, İ' / S
f, İ
[8.48]
pv
pv
This relationship characterizes the duration of the stimulation. Actually, if we
consider the case when the duration of the transient phase of the response of the f
frequency and the H damping frequency oscillator where H' is small with regard to T,
for instance, R is given by:
1/ 2
o
[8.49]
R f,İ,İ' İ/İ' ȝ N, į'
/ȝ N,į
From Figure 8.7 abacuses, the latter relationship allows us to obtain N and
therefore T. Then relation [8.47] gives S(f).
8.6.3.
Determination of the floor spectra
As we now know the characteristics of the source separable process, it is
possible to determine those of the structure response. To do so, all we have to do is
apply equation [8.13] which gives the PSD of the response at point r and time W
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