Probabilistic Life Prediction Isn't as Easy as It Looks

Abstract

Many engineers effect “probabilistic life prediction” by replacing constants with probability distributions and carefully modeling the physical relationships among the parameters. Surprisingly, the statistical relationships among the “constants” are often given short shrift, if not ignored altogether. Few recognize that while this simple substitution of distributions for constants will indeed produce a nondeterministic result, the corresponding “probabilities” are often woefully inaccurate. In fact, even the “trend” can be wrong, so these results can't even be used for sensitivity studies. This paper explores the familiar Paris equation relating crack growth rate and applied stress intensity to illustrate many statistical realities that are often ignored by otherwise careful engineers. Although the examples are Monte Carlo, the lessons also apply to other methods of probabilistic life prediction, including FORM/SORM (First/Second Order Reliability Method) and related “fast probability integration” methods.

An Integrated Prognostics Method Under Time-Varying Operating Conditions

“…The two material parameters and in Paris' law were considered as statistically correlated random variables to account for the randomness of a crack propagation process. Following the literature [11], [38], [45]- [47], in this paper, is assumed to obey a bivariate normal distribution. This assumption is based on fatigue experiments [38], [46], and supported by the central limit theorem [11].…”

“…The variability of the crack propagation process should be reflected in the and statistics. It was reported in [45] that a strong correlation between and must be taken into consideration to achieve acceptable prediction accuracy. In practice, assume a set of experimental trajectories of stochastic crack growth is available, i.e., a set of curves representing crack growth rate vs .…”

“…A standard statistical analysis can be applied to the sample set of to infer the best joint probabilistic distribution of . It is found that the bivariate normal distribution is usually a valid hypothesis for the joint density of [38], [45]- [47]. The density obtained in this way can be considered as prior information; however, the values of and for a specific unit have very narrow distributions, or should be treated deterministically.…”

Nondestructive Evaluation (NDE) Technology Initiatives (NTIP). Delivery Order 0039: Statistical Comparison of Competing Material Models

“…3 Thus the marginal distribution of the data, y, (see Appendix E, figure 8) can be determined by integrating over the model parameter space:…”

“…The use of the Greek letter π π π π is to avoid confusion with the Latin p when both are used in the same equation. 3 In many circumstances the model parameters are constants (e.g. the mean, µ =const.).…”