Being conservative with a limited number of test results

Abstract

In aircraft structural design, failure stresses are obtained from coupon tests and then used to predict failure under combined loads in structural elements. Structural element tests are next used to update the failure envelope for combined loads. It is a common practice to repeat the element tests and then select the lowest test result as a conservative estimate of the mean failure stress. This practice is equivalent to reducing the average test failure stress by a knockdown factor (one that is quite variable). Instead, we propose using the average test result with an explicit knockdown factor obtained from statistical distribution of the test data. We show reductions in the variability of the estimated mean failure stress as well as the likelihood of unconservative estimate. In addition, when the initial distribution or confidence interval of the mean failure stresses is available, we can further decrease the chance of unconservative estimate using Bayesian updating. We demonstrate the gains associated with Bayesian updating when the upper and lower bounds of errors in the analytical predictions are available. Examples with uniform and lognormal distributions of failure stresses compare the lowest-result approach with the two alternatives with the explicit knockdown factor. Both approaches significantly reduce the likelihood of unconservative estimates of the mean failure stress. The average approach reduced this likelihood by about a half and the Bayesian approach by up to an order of magnitude (from 12.5 to 1 % ). We also examine scenarios in which estimates of error and variability are substantially inaccurate. We show that, even then, the likelihood of unconservative estimates reduces significantly. Remarkably, an underestimate of variability also results in about a 2% higher average of the estimated mean failure stress. Thus, we are able to simultaneously use higher average failure stress (leading to lower weight) and reduce the likelihood of unconservative estimates.