Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 298

Then we obtain either a PSD that is a function of two frequencies f 1 and f 2 , or a PSD
that is a function of frequency f and time t.

Obviously, adjusting a PSD to two variables during a series of observations or of
experiments is far more problematic and often illusive. A way to simplify involves
supposing that the U (t, W) function evolves more slowly according to the t variable
than according to the W variable (which often proves to be true). It is then easier to
carry out adjustments.

We can also assume that the t evolution occurs in a deterministic way, and thus
adjust a frequency content varying in time according to some given law. We can be
still more restrictive by considering an average, and therefore constant with time
frequency content, and by making only the overall amplitude time-dependent. The
latter assumption leads us to the divisible model of seismic stimulation:

* t at Ft

[8.44]

where a(t) is a deterministic envelope and F(t) is a S(f) PSD stationary random
process.

As we will see, the latter representation is fairly easy to adjust to conventional
seismic data. Generally a simple envelope (half-sine) is taken. Even a constant level
slot gives satisfactory results because the most important effect to take into account
is the start of the stimulation for t = 0 time and its stopping for time t = T.

8.6.2. Adjusting a separable process from the ORS data

We have seen that the seismic stimulation is given to an engineer in the shape of
a set of ORS drawn up for several damping values. Thus, the problem involves
adjusting an a(t) envelope and a S(f) PSD from S pv (f,H) or S pa (f,H) functions.

As we remember that the ORS have the meaning of maximum statistic averages,
we can show that the relationship:

2

2

ª º *

pv S , 0

E f

[8.45]

¬ ¼

is quite well confirmed.