Narrow Results

We provide the exact expression of the reliability of a system under a Bayesian approach, using beta distributions as both native and induced priors at the system level, and allowing uncertainties in sampling, expressed under the form of misclassifications, or noises, that can affect the final posterior distribution. Exact 100(1-α)% highest posterior density credible intervals, for system reliability, are computed, and comparisons are made with results from approximate methods proposed in the literature.

The ability to accurately determine the temporal safe region in time-variant reliability analysis is seminal for reliability-based design. When stochastic excitations are present and discrete-time approaches are invoked, the errors can be large when one uses only one past safe event (and one new failure event) at each time-step. Furthermore, when all previous safe events are accumulated and used, the calculations can be time consuming and the accuracy not ensured. In this paper, a minimal, or a so-called extreme limit-state, surface is obtained to identify the system temporal safe region in an economical manner. To do this, the limit-state surface motion for each failure mode is recorded as a parametric polar plot that provides both magnitude and relative angle of the vectors from the origin to the most-likely failure points (MLFPs) in standard normal space. The angle differences provide correlation and the magnitude differences provide importance. At the component-level, a few logical policies that compare correlation and the magnitude ensure that the safe region is sufficiently recognized. At the system-level, the temporal average of correlations and the magnitudes at the component-level, along with series or parallel system designations, foretells which failure modes are needed to form the system extreme limit-state surface. The impact of the work includes an immediate recognition of the important failure modes and reduced computation for methods such as multi-normal integration. Case studies of both series-system reliability and parallel-system reliability are presented using structural beams excited by stochastic loads and plagued with degrading material properties and dimensions. The accuracy of the extreme LSS is demonstrated cogently. The use of the polar plots as a design tool becomes evident.

A Bayesian procedure for estimating the reliability of a complex system of independent series and parallel subsystems is presented. This method accepts either binomial or Poisson test data (perhaps both or neither) as inputs, as well as prior information, at any and all levels in the system (such as the component, subsystem and system levels). Natural conjugate beta and gamma priors are considered. A numerical example concerning the unreliability of the low-pressure coolant injection system of a certain U.S. commercial nuclear-powered boiling water reactor is used to illustrate the procedure. A Mathematica^® notebook is available for implementing the method.