Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 289

8.4.5.3.2. Determination of the reliability function and the peak factor
Reliability is defined as the probability for the process not to cross the threshold
in a given T time period, which is usually chosen to be non-dimensional.

N = f (0) T represents the average number of semi-cycles performed by the
process during the T time period. The reliability function is noted: W (r, N).

For wide-band processes and for relatively high thresholds (r t 2.5 to 3), the
“threshold-crossing” occurrence is both rare and isolated.

Therefore, we can represent its occurrence owing to a Poisson distribution:

Probability (n occurrences in the T time period) =

^ `

n

ª f r T º / n! exp ª f r T º

¬ ¼ ¬ ¼

Reliability involves a zero occurrence probability. Hence, by replacing f(r) with
its [8.29] expression:

ª º

2

W r, N exp N exp r / 2

¼

[8.30]

¬

we can recognize Gumbel 's asymptotic law.

From reliability, we can define the absolute maximum reached by the process
during the time period T. This is a stochastic variable the probability density of
which is:

w w [8.31]

p r, N W/

max r

The most frequently used average value is the average maximum , which is also
called peak factor P (N) when it is referred to the mean square deviation.

f
ww

0

³

ȝ Nr / r dr

[8.32]