PSI - Issue 57

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

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Andrew Halfpenny / Structural Integrity Procedia 00 (2023) 000–000

Fig. 5. The e ff ects of Weibull β , η and γ on the life of a component

• β < 1 : infant mortality – the failure rate, represented by the green curve in Fig. 5 c), shows significant early failures diminishing rapidly with respect to time. • β = 1 : constant failure rate – the failure rate, represented by the blue curve in Fig. 5 c), shows a constant failure rate with respect to time. (Note: as β → 1, the Weibull distribution tends to an exponential distribution when γ = 0.) • β > 1 : fatigue / wear out failure – the failure rate, represented by the red curve in Fig. 5 c), shows failure rates increasing with respect to time.

Note: in the case of a 3-parameter Weibull curve, the values of β can be smaller on account of the failure free life given by γ .

Estimation of the statistical parameters is based on curve fitting. There are two general methods available as de scribed by Nelson (1990) and ReliaSoft (2015b). The performance characteristics of both methods make them suitable to the following situations: 1. Rank Regression Estimation (RRE) : is preferred for small sample sizes such as those found in physical quali fication tests. Special measures may be taken for tests stopped prior to failure (known as run-out, suspended, or right-censored tests), these are discussed in Halfpenny et al (2019) and ReliaSoft (2015a). 2. Maximum Likelihood Estimation (MLE) : is preferred for larger sample sizes such as those found in simulation tests. These methods take implicit account of suspensions. Halfpenny et al (2019) further recommends that fa tigue simulations that return ‘life beyond endurance’, (where the maximum stress is less than the fatigue strength of the material), be modelled as suspensions. Conversely, in cases where Monte Carlo simulation returns a result of ‘static failure’, (where the maximum stress exceeds the static strength limitations of the material), Halfpenny et al (2019) recommends running additional simulations using the ‘Design for Robustness’ approach to deter mine if this is a statistically likely result. Such failures are often indicative of failure through anticipated abusive loading, or raise safety concerns, and therefore warrant further consideration.

4. Case study: A statistical comparison of simulation with physical test

A typical comparison of simulated fatigue results with qualification test data was presented by Halfpenny et al (2019) Five qualification tests were performed to failure and are represented as black diamonds in Fig. 6. A factor of two was observed between the minimum and maximum values. A number of simulated runs were performed to model the variability due to fatigue parameter uncertainty (an aleatoric uncertainty), and FE modelling error (an epistemic uncertainty). The statistical reliability of the simulated runs are shown as red points in Fig. 6.