5. The NAFEMS Stochastics Challange Problem

This section introduces a practical uncertainty quantification problem from the NAFEMS Stochastics Working Group. The original challenge was also published in Benchmark Magazine, April 2022.

The problem is a simple engineering reliability problem. Consider a mechanical part subjected to uncertain loading. The resistance or strength of the part is represented by R, and the applied stress or load effect is represented by S.

In the course material, R can be interpreted as the fatigue strength of a mechanical part, estimated from a limited number of tested specimens. Similarly, S represents the load acting on the part in operation, estimated from field measurements.

The available data

Only a small number of measurements are available. This is important because the uncertainty is not only caused by natural variability in strength and load, but also by limited knowledge of the true distribution parameters.

The question

The limit state is defined as:

g = R - S

If g > 0, the resistance is greater than the stress and the part survives. If g < 0, the stress exceeds the resistance and the part fails. The probability of failure is therefore:

p_f = P(g < 0)

The challenge is not only to calculate a single estimate of p_f. We also want to understand how uncertain that estimate is, because the distributions of R and S have been inferred from small datasets.

This example separates two important types of uncertainty:

• Aleatory uncertainty: the natural variability of strength and load from one case to another.
• Epistemic uncertainty: the uncertainty caused by limited data and incomplete knowledge of the true distribution parameters.

Additional testing or field measurement could reduce the epistemic uncertainty. However, the natural variability of strength and load would still remain part of the reliability problem.

The key engineering questions are:

• How large is the probability of failure?
• How uncertain is that probability of failure estimate?
• Which input, R or S, contributes most to the uncertainty?
• What value should be used to support an engineering decision?

This sets up the next section, where the problem is written as a probabilistic model and solved numerically using Stan.