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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This continuous transition can be beneficial for optimization compared to a piecewise transformation. However, the resulting distribution in the transition zone is not normally distributed.

4. Methodology

This study aims to compare di ff erent strategies for evaluating S-N models by analyzing their ability to recover model parameters under controlled conditions. To generate data sets, a log-normal distribution in load direction is assumed, with a constant logarithmic standard deviation σ S , log . Fatigue life N is sampled from a uniform distribution on a logarithmic scale between 10 4 and an upper limit. The default limit is approximately 1 . 16 · 10 7 cycles, resulting in about 5 % runouts. Corresponding load amplitudes S a are sampled from a normal distribution in load direction, truncated such that values exceeding a defined upper load limit (three standard deviations below the load amplitude at N = 10 4 ) are resampled. In real-world experiments, tests are often stopped after a maximum number of cycles (e.g., 10 7 ). To account for this, S-N pairs with N > 10 7 are treated as right-censored observations (runouts) in the likelihood function. The default number of S-N data pairs is 15. For each evaluation strategy, 10 000 Monte Carlo simulations are performed and convergence is checked. In each simulation, a dataset is generated with varying configurations, and the model parameters are estimated using MLE. The following parameters are systematically varied: (1) number of S-N data pairs, (2) logarithmic standard deviation σ S , log , and (3) slope parameter k 1 . The base model used for data generation is summarized in Table 1.

Load amp. at knee point Knee point Slope HCF Slope LLF Log. std. deviation

N k

k 1

k 2

Symbol

S a , k 125

σ S , log 0.03

1 · 10 6

Value

5

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Table 1. Parameters of the base model

5. Results

In the following, the results of the simulations are presented. The simulations are conducted using Python 3.12.4, primarily utilizing the main packages SciPy 1.14.1 (see Virtanen et al. (2020)) and NumPy 1.26.4 (see Harris et al. (2020)). Because the evaluation in fatigue-life direction does not provide the logarithmic standard deviation in load direction it is estimated separately after the MLE. The results are visualized using box plots, which summarize the distribution of the estimated parameters over 10 000 simulation repetitions. Each boxplot displays the median (central line), the interquartile range (the box), and the whiskers (interquartile range multiplied by 1 . 5).

5.1. Number of S-N Data Pairs

The first investigation focuses on the influence of the number of S-N data pairs, as illustrated in Fig. 1. The number of data points is varied from 10 to 30 S-N pairs. The evaluation in fatigue-life direction consistently shows the poorest performance overall, yielding the least accurate estimates for all model parameters, including the load amplitude at the knee point S a , k , the corresponding fatigue life N k , the slope k 1 and the logarithmic standard deviation σ S , log .