PSI - Issue 75

Felix-Christian Reissner et al. / Procedia Structural Integrity 75 (2025) 382–391 Felix-Christian Reissner / Structural Integrity Procedia 00 (2025) 000–000

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tion of S-N data, particularly under constraints such as limited test data and runouts, remains an open topic in research. A critical but often overlooked factor in statistical S-N modeling is the direction in which scatter is introduced. Classi cal fatigue models typically treat fatigue life as the outcome variable, based on a given load amplitude. Consequently, the variability is modeled in fatigue-life direction. However, recent studies, including Meeker et al. (2024), have shown benefits to modeling variability in the load direction. Nevertheless, the implications for parameter estimation and model convergence are not fully understood. This paper investigates four evaluation strategies for the bilinear Basquin model using maximum likelihood estimation (MLE): (1) variation in the fatigue-life direction, (2) variation in the load direction, (3) a piecewise transformation approach, and (4) a novel pointwise transformation approach. All models are embedded in a consistent probabilistic framework based on a log-normal distribution. Their perfor mance is assessed using systematic Monte Carlo simulations under varying data sizes, standard deviations, and ratios of runouts to the total number of S-N data pairs (runout ratio). The results reveal significant di ff erences in accuracy depending on the evaluation direction. Load-based evaluation is strongest at low runout ratios, whereas piecewise and pointwise strategies become superior as censoring increases. The fatigue-life-based strategy shows the largest bias overall.

Nomenclature

f ( x ) Probability density function (PDF) F ( x ) Cumulative distribution function (CDF) g ( x ) Deterministic S-N model k 1 Slope in high-cycle fatigue regime k 2 Slope in long-life fatigue regime k trans Local transformation ratio L ( θ ) Log-likelihood function L ( θ ) Likelihood function N Normal distribution n fail Number of uncensored S-N data pairs N Number of cycles to failure N k Knee point in cycles p Number of estimated parameters S a Load amplitude S a , k Load at knee point T S Scatter in load direction x S a or N y S a or N δ i Censoring indicator ∆ N log Residual in fatigue-life direction ∆ S log Residual in load direction ε White noise θ Estimated parameter vector µ Meanvalue σ Standard deviation σ N , log

Logarithmic standard deviation in N -direction Logarithmic standard deviation in S -direction Logarithmic standard deviation in x -direction

σ S , log σ x , log

Standard normal PDF Standard normal CDF

ϕ

Φ

ˆ ·

Indicates estimated parameters