This paper examines the variability of predicted responses when multiple stress-strain curves (reflecting variability from replicate material tests) are propagated through a finite element model of a ductile steel can being slowly crushed. Over 140 response quantities of interest (including displacements, stresses, strains, and calculated measures of material damage) are tracked in the simulations. Each response quantity’s behavior varies according to the particular stress-strain curves used for the materials in the model. We desire to estimate response variability when only a few stress-strain curve samples are available from material testing. Propagation of just a few samples will usually result in significantly underestimated response uncertainty relative to propagation of a much larger population that adequately samples the presiding random-function source. A simple classical statistical method, Tolerance Intervals, is tested for effectively treating sparse stress-strain curve data. The method is found to perform well on the highly nonlinear input-to-output response mappings and non-standard response distributions in the can-crush problem. The results and discussion in this paper support a proposition that the method will apply similarly well for other sparsely sampled random variable or function data, whether from experiments or models. Finally, the simple Tolerance Interval method is also demonstrated to be very economical.
The purpose of this project was to investigate the use of Bayesian methods for the estimation of the reliability of complex systems. The goals were to find methods for dealing with continuous data, rather than simple pass/fail data; to avoid assumptions of specific probability distributions, especially Gaussian, or normal, distributions; to compute not only an estimate of the reliability of the system, but also a measure of the confidence in that estimate; to develop procedures to address time-dependent or aging aspects in such systems, and to use these models and results to derive optimal testing strategies. The system is assumed to be a system of systems, i.e., a system with discrete components that are themselves systems. Furthermore, the system is 'engineered' in the sense that each node is designed to do something and that we have a mathematical description of that process. In the time-dependent case, the assumption is that we have a general, nonlinear, time-dependent function describing the process. The major results of the project are described in this report. In summary, we developed a sophisticated mathematical framework based on modern probability theory and Bayesian analysis. This framework encompasses all aspects of epistemic uncertainty and easily incorporates steady-state and time-dependent systems. Based on Markov chain, Monte Carlo methods, we devised a computational strategy for general probability density estimation in the steady-state case. This enabled us to compute a distribution of the reliability from which many questions, including confidence, could be addressed. We then extended this to the time domain and implemented procedures to estimate the reliability over time, including the use of the method to predict the reliability at a future time. Finally, we used certain aspects of Bayesian decision analysis to create a novel method for determining an optimal testing strategy, e.g., we can estimate the 'best' location to take the next test to minimize the risk of making a wrong decision about the fitness of a system. We conclude this report by proposing additional fruitful areas of research.
