Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 314

For strong non-linearities with slope ratios lower than 0.5, the non-linear
calculation reveals the existence of two phenomena:
- a phenomenon at the elasto-plastic cycle frequency;
- a low-frequency phenomenon that corresponds to the cumulative effects of
plastic displacements regarding the inelastic spectrum method referred to above.
A 1-dof linear oscillator cannot represent this “two-frequency” behavior, thus, in
that case we can observe discrepancies as far as the variance estimate is concerned,
as well as poor PSD results (Figure 8.16).
8.8.4.2. Application to a 1-dof elastic shock oscillator
Figure 8.17 shows the diagram of a 1-dof- shock oscillator and the aspect of the
behavior law. K c is the stop stiffness, which is typically high relative to that of the
oscillator. This law is the f(x) type and it fits quite well with the stochastic
linearization technique.
Figure 8.17. Diagram and law of a 1-dof-shock oscillator
However, applying this technique here gives very disappointing results:
- The graphs in Figure 8.18 extracted from [GUI 90] derive from non-linear
digital simulations. They show the physical behavior of the shock oscillator. They
represent the mass displacement PSDs for different stationary wide-band noise type
stimulation levels (the K no-dimension associated parameter being the ratio between
the play and the standard deviation of the response of the without shock oscillator).
For high K values (very important play and therefore no shocks or few shocks) or
small K values (play nearly equal to zero, hence an almost perfect connection with
the stop), we have the behavior of a weakly dampened linear oscillator (we took H =
0.01). On the other hand, when K is in the order of 1, the PSDs are quite flat. This