Deterministic Fracture Mechanics (DFM) assessments of structural components (e.g. pressure vessels and piping used in the nuclear industry) containing defects can usually be carried out using the R6 procedure. The aim of such an assessment is to demonstrate that there are sufficient safety margins on the applied loads, defect size and fracture toughness for the safe continual operation of the component. To ensure a conservative assessment is made, a lower-bound fracture toughness, and upper-bound defect sizes and applied loads are used. In some cases, this approach will be too conservative and will provide insufficient safety margins. Probabilistic Fracture Mechanics (PFM) allow a way forward in such cases by allowing for the inherent scatter in material properties, defect size and applied loads explicitly. Basic Monte Carlo Methods (MCM) allow an estimate of the probability of failure to be calculated by carrying out a large number of fracture mechanics assessments, each using a random sample of the different random variables (loads, defect size, fracture toughness etc). The probability of failure is obtained by counting the proportion of simulations which lead to assessment points that lie outside the R6 failure assessment curve. This approach can give good results for probabilities greater than 10−5. However, for smaller probabilities, the calculation may be inefficient and a very large number of assessments may be necessary to obtain an accurate result, which may be prohibitive. Engineering Reliability Methods (ERM), such as the First Order Reliability method (FORM) and the Second Order Reliability Method (SORM), can be used to estimate the probability of failure in such cases, but these methods can be difficult to implement, do not always give the correct result, and are not always robust enough for general use. Advanced Monte Carlo Methods (AMCM) combine the two approaches to provide an accurate and efficient calculation of probability of failure in all cases. These methods aim to carry out Importance Sampling so that only assessment points that lie close to or outside the failure assessment curve are calculated. Two methods are described in this paper: (1) orthogonal sampling, and (2) spherical sampling. The power behind these methods is demonstrated by carrying out calculations of probability of failure for semi-elliptical, surface breaking, circumferential cracks in the inside of a pressure vessel. The results are compared with the results of Basic Monte Carlo and Engineering Reliability calculations. The calculations use the R6 assessment procedure.
Stability analysis of underground mine hard rock pillars via combination of finite difference methods, neural networks, and Monte Carlo simulation techniques
