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Single Degree-of-Freedom Linear Oscillator Read chapter Read first chapter

2013 | OriginalPaper | Chapter

9. Peak Factor and Random Fatigue

Author : André Preumont

Published in: Twelve Lectures on Structural Dynamics

Publisher: Springer Netherlands

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Abstract

This second chapter on random vibration is devoted to the analysis of the statistics of the random response in order to assess the reliability of the structure; two important failure mechanisms are examined: the failure by threshold crossing and fatigue. The first part of this chapter is devoted to the peak factor, that is the statistics of the extreme value of the random process during a given observation period. Approximate formulae are derived. The relationship between the peak factor and the so-called Response Spectrum traditionally used in earthquake engineering is elucidated. The second part of this chapter is devoted to random fatigue according to the linear damage theory (Miner). Classical results of uniaxial Gaussian stresses with zero mean are reviewed first; the case of biaxial loading is also addressed, by the definition of a special uniaxial equivalent von Mises stress; the finite element formulation of the method is outlined. A set of problems is proposed at the end of the chapter.

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previous chapter Random Vibration
next chapter Rotor Dynamics
Footnotes
1
The probability distribution function \(F(x)\) of a random variable is related to its probability density function \(p(x)\) by
$$\begin{aligned} F(x)=\int ^{\ x}_{-\infty }p(a)\ da \qquad \mathrm{or } \qquad p(x)=\frac{dF(x)}{dx} \end{aligned}$$
 
2
\(1-h(n)\) is the probability that the \(n\)th extremum is also below the threshold \(\eta \sigma \) if all the previous extrema are below the threshold.
 
3
In the large earthquake that struck Japan in March 2011 (Magnitude 9 on the Richter scale), maximum ground accelerations above 1\(g\) were observed at more than 100 km from the epicenter, and the duration of the strong ground motion was about 2 minutes.
 
4
Taking only the first contribution of (9.13), one gets the explicit formula:
$$\begin{aligned}\varPhi _0(\omega _n)=\frac{2\xi \omega _n^3}{\pi }.\frac{S_d(\omega _n,\xi )^2}{2\ln (\omega _nT/\pi )}\end{aligned}$$
 
5
This relationship ignores the endurance limit and the statistical scatter in the material behavior; the S–N curve may also be defined in a probabilistic way.
 
6
The Gamma function is an extension of the factorial function \(\varGamma (n)=(n-1)!\) for non integer numbers. It is defined by
$$\begin{aligned}\varGamma (x)=\int _0^{\infty }t^{x-1}e^{-t}dt\end{aligned}$$
 
7
The frequency content of a time-history of \(s_c(t)\) is different from that of the stress components; if the stress components are harmonic at a frequency \(\omega \), \(s_c(t)\) has its frequency content centered on \(2\omega \).
 
Metadata
Title
Peak Factor and Random Fatigue
Author
André Preumont
Copyright Year
2013
Publisher
Springer Netherlands
Book
Twelve Lectures on Structural Dynamics

Print ISBN: 978-94-007-6382-1

Electronic ISBN: 978-94-007-6383-8

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-94-007-6383-8_9

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