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Recent advance of anomalous statistical methods in engineering reliability analysis

Failure probability, a quantity to assess the reliability of engineering structures or components, is generally very small and sensitive to the distributions chosen for the random factors, such as stress range, water level fluctuations, and fatigue accumulated damage. To capture the heavy tail, skewness, and peakness existing in the random variables, we introduce the Lévy stable distributions and Mittag-Leffler distribution, and give the distribution selection criterion in conjunction with entropy methods.
The Gaussian and Cauchy distributions are two special cases of the Lévy stable distributions, and the Mittag-Leffler distribution is generalization of the exponential distribution. Both of the two class distributions have complicated mathematical forms, and their second moment orders do not exist expect for some special cases. To easily adopt them in engineering experimental data processing, we have developed Matlab toolbox with friendly graphical user interface to numerically calculate the quantities of the distributions.
The anomalous statistical methods we proposed extend the applications of the two class distributions in engineering reliability analysis, which improve or generalize the existing statistical methods, such as Lévy Monte-Carlo strategy for stability analysis of hydraulic structures, dynamic Miner’s rule with Mittag-Leffler distribution for bridge fatigue life prediction, and cumulative entropy method with Lévy stable distributions for circular concrete-filled steel tubular stub columns capacity design. Compared with the traditional statistical methods, the anomalous statistical methods provide alternative and powerful tool to evaluate the engineering reliability.

Author(s):

Yingjie Liang    
Hohai University
China

Wen Chen    
Hohai University
China