PSI - Issue 79

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

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likelihood estimation (MLE) frameworks, see Pascual (1999); Meeker et al. (2024); Sto¨rzel and Baumgartner (2021). However, while the influence of runouts on point estimates of S-N parameters is qualitatively understood, their quan titative e ff ect on the accuracy and reliability of confidence intervals has not been systematically assessed. Confidence intervals (CIs) provide a measure of statistical uncertainty and are valuable for evaluating the robustness and reliability of estimated S-N parameters. In particular, they allow engineers to assess the precision of the knee point and slope parameters, which directly influence fatigue design margins. Despite their importance, the computation and interpretation of CIs in fatigue analysis remain inconsistent across standards and guidelines like the FKM Guideline, DIN 50100, ASTM E739-10 or ISO 12107, where either confidence intervals are not provided or nonlinear models are not considered. These limitations motivate a probabilistic framework capable of handling nonlinearity, censored data, and small sam ple sizes typically encountered in fatigue testing. Likelihood-based confidence intervals, and in particular those derived from the profile likelihood ratio, provide a robust and conceptually sound approach to quantify uncertainty under these conditions. They are assumed to provide reliable results under censoring and with small sample sizes Meeker et al. (2017), but systematic validation in the context of S-N curves, especially for bilinear models with runouts, remains scarce. The objective of this work is therefore to quantify the influence of runouts on the coverage behavior of profile likelihood confidence intervals for bilinear S-N models. The bilinear Basquin model is chosen as it represents a practical and widely used formulation. Using a Monte Carlo framework, the coverage probability, i.e., the propor tion of intervals containing the true parameter value, is evaluated for di ff erent proportions of runouts. This allows a quantitative assessment of whether profile-likelihood CIs achieve their nominal confidence levels under realistic test conditions.

2. Background and Theory

The characterization of fatigue behavior is commonly based on S-N data, where S a , i denotes the applied stress amplitude and N i the corresponding number of cycles to failure. In experimental practice, a series of specimens is tested under di ff erent constant load amplitudes, and the resulting pairs ( S a , i , N i ) describe the relation between load and fatigue life. Since the data typically span several orders of magnitude in both stress and lifetime, logarithmic transformations S a , log , i = log 10 ( S a , i ) and N log , i = log 10 ( N i ) are applied, yielding a more tractable representation that reveals approximately linear trends. These trends form the basis of S-N models.

2.1. Bilinear Basquin S-N model

The linear Basquin model Basquin (1910) is one of the best-known S-N models and is widely used. Here, the bilinear Basquin model is employed, as it allows for di ff erent slopes in the long-life and high-cycle regimes, thereby capturing the characteristic change in fatigue behavior near the knee point. Usually, the fatigue life N i is modeled as the dependent variable while the stress amplitude S a , i is the independent variable. This follows the physical relationship between stress amplitude S a , i and fatigue life N i . However, in this work the relationship is reversed. Meeker et al. (2024) show that taking the fatigue life N i as the independent variable simplifies the scatter. Using this approach, the scatter can be modeled as constant over the entire S-N curve, which is a good approximation for most engineering materials. Finally, the bilinear S-N model is given by: S a , log , i =   S a , k , log + 1 k 1 ( N log , i − N k , log ) , if N i ≤ N k , S a , k , log + 1 k 2 ( N log , i − N k , log ) , if N i > N k . (1) Here, N k , log denotes the knee point, S a , k , log the corresponding load amplitude, k 1 the slope before the knee point, and k 2 the slope after the knee point. The scatter of the fatigue data is modeled assuming a normal distribution of the logarithmic load amplitude S a , log , i ,