Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 320

pulsation process is linked to the X max (t) process in a deterministic way using the Z 0
function:

: t Ȧ Xt [8.64]

0 ax

Xt X tcos 4 t < t

[8.65]

max

dX/dt t X t Ȧ Xsint 4 < t

[8.66]

max 0 max

d/dtt Ȧ X 4

[8.67]

0 ax

As a consequence, after having specified the deterministic relationships Z 0 (x max )
and S(Z 0 ) (and therefore S(x max )), the problem will consist of characterizing the
random processes X max (t) and \(t) in terms of probability density. Then it is easy to
obtain the PSD of X(t).

It is possible to show that from a theoretical point of view (see [FOG 96] and
[BEL 99]), each of the probability densities confirms a Fokker-Planck differential
stochastic equation that can be solved.

If, in our example, the equation gives an analytical solution, it is not the same
with more complex cases. Thus, for several-dof systems, the differential problem is
large, and its solution is far beyond the scope of engineering calculations. Therefore,
from a practical point of view, in order to evaluate the probability density rate we
will merely use simple physical approximations that will reproduce the frequency
dispersion phenomenon.

In short, the different steps of the method are:
- determination of the amplitude/pulsation relationship. In our example, we will
take the relationship corresponding to the full-line curve in Figure 8.22;
- determination of the amplitude/source intensity relationship. In our example,
we adjust S(x max ), to preserve the average intensity of the oscillator (actually we can
simply explain the average energy of the non-linear oscillator as well as that of the
equivalent linear oscillator);