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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3.2.1. Lifetime-Based Evaluation (N-direction) In the first strategy, the statistical variation is assumed to lie entirely in the fatigue-life direction . For a given load amplitude S a , the S-N model estimates the logarithmic fatigue life log 10 ( ˆ N ) = g (log 10 ( S a )). The residuals that are minimized are defined by:

∆ N log = log 10 ( N ) − log 10 ( ˆ N )

(7)

where ˆ N is estimated and N the observed fatigue life. The logarithmic standard deviation σ N , log is estimated from the residuals ∆ ˆ N log . Based on these explanations and S-N data, the likelihood function (see Eq. 6) can be constructed and the MLE can be conducted. 3.2.2. Load-Based Evaluation (S -direction) In the second strategy, the stochastic variation is assumed to occur in the load direction. In this case, the fatigue life N is treated as the input variable, and the S-N model yields an estimated load amplitude ˆ S a . This strategy is identical to the lifetime-based evaluation, with the only di ff erence that the load model is used instead of the fatigue life model. 3.2.3. Piecewise Transformed Evaluation according to Sto¨rzel and Baumgartner (2021)(N ( S ) -direction) The third strategy introduces a hybrid approach in which the residuals are modeled in load direction (see Eq. 7, but using the load amplitude instead of the fatigue-life). A transformation is then applied to project this variability into the lifetime direction based on the slopes k 1 and k 2 of the bilinear Basquin model. To ensure a constant standard deviation in the load direction, the logarithmic standard deviation is first calculated from the load residuals. These residuals, along with the standard deviation, are then transformed into the lifetime domain using the slope parameters of the bilinear model: σ N , log =   σ S , log k 1 , if N ≤ N k σ S , log k 2 , if N > N k , ∆ ˆ N log =   ∆ S log k 1 , if N ≤ N k ∆ S log k 2 , if N > N k (8) The likelihood is then computed in the same way as in the lifetime-based strategy. This strategy preserves the physically correct interpretation of uncertainty in the fatigue-life direction while deriv ing the distribution shape from the load direction, thereby enabling a piecewise constant standard deviation in the transformed fatigue-life direction. 3.2.4. Pointwise Transformed Evaluation (N ( S ) -direction) The fourth strategy introduces the second hybrid approach. Although this approach is conceptually based on Sto¨rzel and Baumgartner (2021), it is presented in this form for the first time in this paper. It extends the previous approach by calculating the residuals ∆ N log and ∆ S log in both the load and fatigue-life directions. A pointwise transformation based on the local transformation ratio k trans = ∆ N log / ∆ S log is conducted. This local ratio is then used to convert the standard deviation from the load domain into the fatigue life domain according to

σ N , log = σ S , log k trans .

(9)

As in the previous approach, the likelihood is subsequently computed based on the transformed residuals. This strategy retains the interpretation of uncertainty in the fatigue-lifetime direction but derives the shape of the distribution from the load space. The pointwise transformation ensures a continuous transition at the knee point.