A second-order third-moment method for calculating the reliability of fatigue

https://doi.org/10.1016/S0308-0161(99)00013-7Get rights and content

Abstract

Probabilistic assessments are a useful supplement to deterministic analyses and can also be used as an aid to decision making in areas such as safety analyses, design studies and deployment of resources on maintenance, inspection and repair. Because the initial crack size, the stress, the material properties and other factors that may affect the fatigue crack growth are statistically distributed, the first-order second-moment method is often adopted to calculate the reliability of fatigue of industrial structures. In this paper, a second-order third-moment method is presented and a three-parameter Weibull distribution is adopted to reflect the influences of skewness of the probability density function. The proposed method that has more characteristics of those random variables that are concerned in reliability analysis is obviously more accurate than the traditional first-order second-moment method with a case study.

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 lawdadN=C(ΔK)n=CβnSnan/2,where da/dN is the crack growth rate, ΔK=βSa, 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 isf(x)=1σ2πexp[−(x−μ)2/(2σ2)].Then the mean and variance of the random variable x are given byμx=μ,σx22.

If a variable x is exponentially distributed and its density function is given byf(x)=λexp[−λ(x−x0)]then the mean, the variance and the third moment of the variable is given byμx=x0+(1/λ),σx2=(1/λ2),γx=(2/λ3).

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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