Why assessing low probabilities of failure by extrapolation could be required

Why assessing low probabilities of failure by extrapolation could be required June 2019 - “Calculating the Probability of Failure (PoF) may fail when this value is very low”, explains Jack Reijmers, Specialist Engineer at Nevesbu. The Probability of Failure is the probability that the loading exceeds the load carrying capability or resistance. A safe structure implies a low PoF. However, sometimes the PoF is too low to calculate. The analysis produces a distribution of probabilities, but failure lies outside the range of calculated values. Especially in those cases where reliability analysis is of utmost importance, it is detrimental that the calculation of the PoF fails. The PoF can be estimated with extrapolation from available results. However, the distribution of probabilities shows a tail with a fast decreasing slope. Therefore, extrapolation from this tail is inaccurate. An alternative is to use the reliability index. Because the relationship between failure and the reliability index is much more gradual, extrapolation from this function is much more accurate.

How to estimate the Probability of Failure by extrapolation

The extrapolation method based on the reliability index is applied to a simple beam model, and the extrapolation is based on a linear fit. Because the extrapolation is realised by a linear function, the resulting PoF is very accurate. Furthermore, the lineair fit can be transformed back to the distribution tail. The extrapolation function follows the fast decreasing slope accurately. Since this method offers the possibility to estimate an accurate PoF, it establishes a sound approach of reliability analysis when low probabilities are involved. Hence, the extrapolation based on the reliability index offers a valuable method to estimate the PoF.

High reliability --> low PoF --> inaccurate probabilities (or even not available)

Loading and load carrying capability of a structure are subject to uncertainty. This uncertainty can be translated to a mean value and variance, and this leads to statistical distributions of the loading and the load carrying capability or resistance. An extremely safe structure shows distributions that are widely apart, with a very small overlap. In that case the limit-state function, defined by G (limit-state) = R (resistance) – S (distribution of load), shows a very small probability of being less than zero (the loading exceeds the resistance). Analyses may fail to produce these low probabilities. The tail of the distribution shows a fast decreasing slope, hence extrapolation from available probabilities of the limit-state function is inaccurate. The figure below shows the cumulative probability for the limit-state. This figure indicates failure for G < 0.



Figure 1 - Distribution of load (line S), resistance (line R) and cumulative probability of the limit-state (line G)

Extrapolation on reliability index

The PoF can be translated to a reliability index. This is the number of standard deviations from the mean value, based on a normal distribution. The limit-state function (G) as a function of this reliability index shows a much better basis for extrapolation. Depending on the deviation from a normal distribution it is even close to a straight line. Therefore the reliability index, giving the probability that loading exceeds the resistance, can accurately be calculated.

Resulting Probability of Failure

Extrapolation based on the reliability index results in the desired PoF. The accuracy of the extrapolation is demonstrated by the perfect fit when the reliability index is translated back to PoF. The values based on the extrapolation function follow the tail of the distribution nicely, and this gives confidence in the calculated PoF.


Figure 2 - Cumulative distribution of the limit-state, with and without extrapolation

This procedure is described in a paper that will be presented by Jack Reijmers at the NAFEMS World Congress in Qu├ębec, 17 – 20 June 2019. If you would like to know more about this procedure, you can send an e-mail to info@nevesbu.com or call +31 88 943 3400.

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