Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 305

only a little dissipation, like great movement effects or moderate shocks that can be
deemed elastic.
It is also necessary to distinguish the weak non-linearities which often appear at
the level of imperfect linking and which do not question the modal structure of the
system from the strong non-linearities that radically modify the mode structure and
the behavior.
The former are taken into account through a slight modification of the resonance
frequencies and mostly through the introduction of a relatively small damping term
compared with the stiffness and inertia terms. In most such cases we give up an
actual physical modeling to merely globally adjust a viscous dampening coefficient
H n for each mode, which corresponds to linear modeling.
The latter can only be dealt with using specific non-linear models. In current
practice, it does not prevent engineers from deriving equivalent linear models from
these models, within restricted parametric ranges.
This last aspect will be the focus of our attention in the rest of the chapter.
We will not systematically study the different non-linear models used in seismic
analysis, but we will rely on two characteristic examples:
- the elasto-plastic behavior model, which is the most often used, or the
“according to Coulomb 's law adherence-sliding-type” behavior, which has the same
nature;
- the “elastic shock-type” behavior model.
8.8.3. Notion of “inelastic spectra”
This notion is the most used method in seismic analysis practice for taking the
elasto-plastic behavior of the structure into account.
Consider a perfect elasto-plastic m-mass 1-dof oscillator loaded by an J(t)
imposed acceleration of its support. It is characterized by its resonance pulsation Z
in its elastic phase, its limit elastic displacement x e , its limit force mZ 2 x e and its
ductility coefficient P (Figure 8.11, where D = 0).
Suppose that J(t) consists of two opposed I intensity impulses in the shape of 'W
time slots and separated by a time period W, with 'W << W (Figure 8.12).