A second-order third-moment method for calculating the reliability of fatigue
Section snippets
Basic formula of fatigue crack propagation
Several models based on the principles of fracture mechanics for the prediction of fatigue crack growth in components and structures under dynamic loads have been proposed, one of the best known is the Paris–Erdogan law [1]. From fracture mechanics, it is known that the fatigue crack propagation follows the Paris–Erdogan lawwhere da/dN is the crack growth rate, , the range of the stress intensity factor, S, the stress range, β, a constant that depends on the type of
The second-order third-moment method
The first-order second-moment method only considers the means and variances of random variables. But probabilistic fracture mechanics problems generally involve non-normal distributions such as the lognormal, the exponential or the Weibull distribution. The skewness of a probability density function is sometimes used to measure the asymmetry of a probability density function about the mean. The second-order third-moment method considers not only the mean and variance of a probability density
The Weibull distribution
The Weibull distribution is one of the most widely used distributions in reliability calculations. The great versatility of the Weibull distribution stems from the possibility to adjust to fit many cases where the hazard rate either increases or decreases. Further, of all statistical distributions that are available the Weibull distribution can be regarded as the most valuable because through the appropriate choice of parameters (the location parameter, the shape parameter and the scale
Other distributions
If a variable x is normally distributed, the third moment of the random variable x is γx=0 and its density function isThen the mean and variance of the random variable x are given by
If a variable x is exponentially distributed and its density function is given bythen the mean, the variance and the third moment of the variable is given by
If a variable x is lognormally distributed and its density function
Numerical calculation
If a0, C, β and S in , may be all random variables with prescribed probability density functions such as the normal, the lognormal, the exponential and the Weibull distributions, in which the means, the variances and the third moments can be calculated from , , , , , , , , , , (the Weibull distribution calculated from , , ). Then the mean, the variance and the third moment of the crack length a at any given N cycles of stress can be calculated from , , . Then the parameters, m, α and x0 of
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