Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 286

ª º

2

¼

[8.23]

q (x, v) = C exp 4İȦ/S v /2 U x

0

¬

being a constant such that ³³ q dx dv = 1.

8.4.5. Extrema statistics in a stationary process

Second order representation leads to simple formulation for linear systems at
least, but it is not enough as far as seismic analysis is concerned. As a matter of fact,
the values handled are the maxima reached instead of the average quadratic values.

This section will present the main formulations at our disposal, while keeping in
mind the fact that a definitely more complex problem is involved.

8.4.5.1. Wide-band processes and narrow-band processes
Extrema statistics depend greatly on the bandwidth of the process being
investigated.

Typically we distinguish between the following:
- wide-band processes, the PSD of which covers a wide frequency range. The
soil's seismic motions without any site effect belong to this category;
- narrow-band processes, the PSD of which corresponds to a narrow peak (the
whole energy is located around a middle frequency f 0 ). The response of a weakly
dampened harmonic oscillator with wide-band noise belongs to this category.
- intermediate processes consisting of a low number of well-separated narrow
peaks. The response of a structure on its first natural modes belongs to this category.

Each category corresponds to a quite characteristic rate of the signal. In
particular, wide-band noise presents isolated extremas, whereas narrow-band noise
consists of “extrema packages” (Figure 8.5).

The bandwidth is characterized by the parameter:

1/ 2
02

į = 1 m /m m

[8.24]

2

1

where m 0 , m 1 and m 2 are order 0, 1 and 2 spectral moments.