A survey of probabilistic methods used in reliability, risk and uncertainty analysis: Analytical techniques 1

Abstract

This report provides an introduction to the various probabilistic methods developed roughly between 1956-1985 for performing reliability or probabilistic uncertainty analysis on complex systems. This exposition does not include the traditional reliability methods (e.g. parallel-series systems, etc.) that might be found in the many reliability texts and reference materials (e g Kapur and Lamberson, 1977). Rather, the report centers on the relatively new, and certainly less well known across the engineering community, analytical techniques. Discussion of the analytical methods has been broken into two reports. This particular report is limited to those methods developed between 1956-1985. While a bit dated, methods described in the later portions of this report still dominate the literature and provide a necessary technical foundation for more current research. A second report (Analytical Techniques n) addresses methods developed since 1985. The flow of this report roughly follows the historical development of the various methods so each new technique builds on the discussion of strengths and weaknesses of previous techniques. To facilitate the understanding of the various methods discussed, a simple 2-dimensional problem is used throughout the report. The problem is used for discussion purposes only; conclusions regarding the applicability and efficiency of particular methods are based on secondary analyses and a number of years of experience by the author. This document should be considered a 'living document' in the sense that as new methods or variations of existing methods are developed, the document and references will be updated to reflect the current state of the literature as much as possible. For those scientists and engineers already familiar with these methods, the discussion will at times become rather obvious. However, the goal of this effort is to provide a common basis for future discussions and, as such, will hopefully be useful to those more intimate with probabilistic analysis and design techniques. There are clearly alternative methods of dealing with uncertainty (e g fuzzy set theory, possibility theory), but this discussion will be limited to those methods based on probability theory.

Possibility and Evidence-Based Reliability Analysis and Design Optimization

“…In different phases of modeling and simulation, uncertainty arises from the following (Robinson, 1998;Agarwal et al, 2004;Huang et al, 2004;2006c;2012b):…”

Probabilistic risk analysis of corrosion associated failures in cast iron water mains

“…Type II uncertainty can be reduced by collecting more information and data (Cullen and Frey 1999). Melching (1995), Madsen et al (1986), Ditlevsen and Madsen (1996), Melchers (1999), and Robinson (1998) Estimation, and Harr's Point Estimation. Monte Carlo simulation is widely used for replicating real world phenomena involving random parameters with known or assumed probability distributions.…”

SPUF - a simple polyurethane foam mass loss and response model.

“…The mean value method [10,11] was used to calculate the mean and standard deviation of the solid fraction as a function of the heating conditions by assuming that the input parameters are independent random variables and that the response is linear. The mean solid fraction, µ Sf , and the standard deviation of the solid fraction, σ Sf , was determined using a simple Taylor series expansion of solid fraction, S f (ξ i ), about the mean of the individual random variables or input parameters, µ i , by neglecting higher order terms as follows:…”