Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 312

For a Gaussian stimulation (the Gaussian linearization method), the E[e(t) 2 ]
variance can be expressed according to the co-variance coefficients of Y(t) and
Y/dt(t) which themselves are functions of Z eq and H eq . The [8.60] system consists of
2 equations in 2 unknowns. Its solution will give the characteristics of the equivalent
linear oscillator.

We can also show that:

2

eq Ȧ EhY, dY t/Y

w ª w º

¼

[8.61]

¬

2Ȧİ E w ª h Y, dY/dt / w dY/dt º

¼

[8.62]

¬

eq eq

The equivalent linear stiffness and damping represent the average statistics of the
“tangent” stiffness and damping values associated with the different values of the
response of the equivalent linear oscillator.

More generally, the method can apply in the non-stationary case of stimulation
owing to a separable process. We are going to discuss the equivalent linearization
method in two simple examples.

8.8.4.1. Application to an elasto-plastic oscillator
The formalism described above does not apply directly to elasto-plastic or
adherence-sliding behavior types. As a matter of fact, conventional models (perfect
elasto-plastic models with bilinear strain hardening) imply dependence on the
history of movement that cannot be represented by a mere h (x, dx/dt). Thus, we
have to resort to either simplified models representing the plastic cycles according to
the movement maximum amplitude or to more sophisticated models that introduce
an additional subsidiary variable confirming a differential equation ( Wen-Bouc 's
model [BOU 94], for example). Figure 8.15 shows the aspect of cycles obtained
with Wen-Bouc's model for different values of its adjustable parameters.