PSI - Issue 57

Andrew Halfpenny et al. / Procedia Structural Integrity 57 (2024) 718–730 Andrew Halfpenny / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 4. Weibull regression and contour plots

3.3. Statistical Reliability Analysis

Monte Carlo simulation produces a sample of simulated fatigue life results. Statistical Reliability Analysis is then performed for each failure mode. As fatigue damage is exponentially proportional to stress, it is common to use an A Priori assumption that the PDF (Probability Density Function) of life follows either a log-normal or Weibull distribution. The Weibull distribution is given in eq. 4.

β η

η

β − 1

η

e −

x − γ

β

x − γ

(4)

p ( x ) =

Where p ( x ) is the Weibull PDF of life x in the range ( x ≥ γ ). η is known as the characteristic life. β is the shape parameter (or slope). γ is the location parameter (or, where γ > 0, the failure-free life ). (This is commonly referred to as a 3-parameter Weibull curve. In the case of a 2-parameter Weibull curve, the location parameter γ = 0 ) A typical Weibull plot for 5 experimental data points is shown in Fig. 4. Fig. 4 a) shows the life data plotted on Weibull probability paper. The solid line represents the regression line and the dotted lines the 2-sided 90% confidence bounds. This implies, with a confidence of 90%, that the true regression line will lie somewhere between these two extremities. Fig. 4 b) shows an alternative view of the Weibull parameters. The contour plot describes the possible range of parameters β and η within the 90% confidence bounds. Knowledge of the confidence bounds are especially important for small sample sizes. The cumulative density function Q ( x ) is given as eq. 5. This is also known as the Weibull unreliability function. Q ( x ) = x γ p ( t ) dt = 1 − e − x − γ η β (5) The Weibull reliability function R ( x ), is simply one minus the cumulative density function and is given as eq. 6. R ( x ) = 1 − Q ( x ) = e − x − γ η β (6) The Weibull failure rate function λ ( x ) is given as eq. 7.

β η

η

β − 1

x − γ

p ( x ) R ( x )

(7)

λ ( x ) =

=

Thee ff ectsof β , η and γ are illustrated in Fig. 5. In the case of a 2-parameter Weibull distribution, the characteristic life is taken as η , whereas, in the case of a 3-parameter Weibull distribution, it is taken as θ = η + γ . The shape parameter β describes the type of failure. For a 2-parameter Weibull distribution, these are assumed as: