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

“Over a warranty period of 8 years or 100,000 miles, I require greater than 95% reliability (i.e. fewer than 5% failures), with no less than 90% confidence in my estimation.”

The statistical reliability target comprises three parts:

1. Period : a minimum ‘age’ over which the component is expected to function properly and reliably. This can be expressed in di ff erent units depending on the application, for example: time, distance, number of ground-air ground cycles, or number start-up / shut-down cycles. 2. Reliability : a minimum reliability target, or conversely, the maximum permissible number of failures that are acceptable over the specified period. This will vary depending on the criticality of the failure in terms of both safety and economic risk. 3. Confidence : the required accuracy of the reliability estimation. This is a measure of how confident we are in the estimation of the statistical parameters, for example, the mean and standard deviation. It is dependent on sample size. The larger the sample, the higher the confidence. For small sample sizes, a high additional safety margin is required to achieve the desired confidence target. This paper considers how digital simulation, physical component testing, uncertainty analysis, and statistical re liability analysis, are used to support the fatigue design requirement, and thereby reduce a company’s exposure to unacceptable safety and warranty risks. Fatigue design of mechanical systems has historically followed a deterministic process. That means, for a given set of input loads and component strength parameters, it will return a consistent fatigue life estimate with no variation. In reality, the input loads and component strengths are statistically variable and uncertain. They have a mean expected value, a statistical variability, and uncertainty associated with them. Deterministic design methods take no implicit account of variability or uncertainty. In practice, the designer applies a safety factor greater than one to each input load, and a factor less than one to the strength, in order to account for these uncertainties. Often an additional safety factor is then applied to the final result to allow for ‘modelling errors’. In most cases, the engineer is fairly certain that the qualification is conservative, but cannot state with any confidence what the expected safety margin, reliability, or failure rate will be. Furthermore, uncertainty in the safety margin makes it almost impossible to validate the simulation against a physical durability or reliability test. In comparison, a stochastic design propagates the e ff ects of variability and uncertainty throughout the analysis. That means input loads and strength parameters are expressed statistically by their probability distributions as illus trated in Fig. 1. A stochastic simulation model is therefore verified on its ability to accurately estimate the mean expected fatigue life, and the uncertainty and variability in that life. Such verification yields considerable additional confidence in the simulation. A brief introduction to stochastic design and uncertainty quantification is given by NAFEMS (2018). The x axis in Fig. 1 is represented by a parameter called ‘Damage’. This generic term accounts for damage weighted aging of the component. For example, in a simple constant-amplitude pressure test, the age may be expressed in terms of the number of pressure cycles to failure. However, if the test were to use variable-amplitude pressure cycles, then the damage term must consider the relative e ff ect of the di ff erent pressure amplitudes. It is normal to select a damage parameter which scales linearly with respect to the cumulative exposure to damaging events of di ff erent amplitudes. A common example of this is seen in the Plamgren-Miner linear damage accumulation rule Palmgren (1924), Miner (1945). This linear assumption permits the subsequent use of linear statistical analysis and linear algebra methods in the design process. In reality this assumption is not strictly correct. Damage is found to accumulate non-linearly and is therefore dependent on the sequence of loading events. However, when applied over long periods, to a large population of users with statistically di ff erent usage characteristics, the principal of the central limit theorem ensures the damage parameter is su ffi ciently accurate for design purposes. This is especially true of fatigue because fatigue cracks initiate as a microstructural phenomenon and propagate at a rate that is exponentially proportional to the applied loads. This unavoidable source of uncertainty usually outweighs the error associated with 1.1. Deterministic and Stochastic design