PSI - Issue 38

Hendrik Bissing et al. / Procedia Structural Integrity 38 (2022) 372–381 Hendrik Bissing, Markus Knobloch, Marion Rauch / Structural Integrity Procedia 00 (2021) 000 – 000

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considering the quantile values of the parameters’ probability distributions. Due to the iterative compu tation in this approach, the calculation stops at an abortion criterion which represents a run out calculation. From that point the curve is represented by a horizontal dashed line with m = 3, for the generally applied slope of the Paris law. However, that does not allow the conclusion that an endurance limit under fatigue crack propagation conditions is reached.

a) Parameter C

b) Parameters ΔK 0 and a 0

Fig. 5. SN p curves of various input parameters with 2.5 % and 97.5 % quantile values of the probabilistic density functions

The Paris law parameter C shows parallel SN p curves with almost one order of magnitude shift in the number of applicable cycles, when comparing the 2.5 % quantile with the 97.5 % quantile. The quantile curves of ΔK 0 run almost congruent for large stress amplitudes, illustrating that the influence is small. The fast turn off point for the 97.5 % quantile slightly below 400 N/mm² emphasises the strong sensitivity of this concept to this parameter. The same sensitivity can be seen for the initial crack depth a 0 . For a very small starting crack depth, represented by the a 0,2.5% curve, the computation stops, and crack propagation is not computable within reasonable boundaries. However, the effect of a 0 is more pronounced than the effect of the lower boundary condition of the crack propagation law, ΔK 0 . 2.2. Proof-of-concept study with data from literature In this part, a Monte-Carlo simulation, including 99 simulation runs for each pre-defined stress range from 40 N/mm² to 400 N/mm² in steps of 10 N/mm², provides a proof-of-concept study with all input parameters from Table 1 and Table 2, randomly considered within the entire TSM. For the CPM, the application cases 3 and 4 are used to apply most realistic 3D cases, Fig. 4c and d, according to Newman et al. (1981). To verify the entire TSM, the Monte-Carlo simulation results are plotted together with a large compilation of experimental results for butt welds that originate from several sources, Fig. 6, (Oliver et al., 1979, Sonsino et al., 2005, Schaumann et al., 2013). Fig. 6a illustrates the results distinguished for each concept, whereas Fig. 6b shows the entire TSM with the upstream SLA and the downstream CPM. Fig. 6b confirms a great compliance between the calculated number of cycles and experimental results in the range of 100 N/mm² to 400 N/mm². It also shows that, for stress ranges exceeding approximately 250 N/mm², the crack propagation law is the governing concept, apparent by the fact that no or just a few points exceed the abortion criterion of the CPM. Below this value, the influence of the crack initiation becomes decisive and a lot more calculative results exceed the applied limit of the CPM. For smaller stress ranges below 100 N/mm², the numerical results differ significantly from the experimental results. This effect also reflects the rare experimental data with failure of the specimen in this stress range and thus the difficulty of a concept verification for stress ranges below 100 N/mm². The small amount of run out tests do neither confirm, nor contradict the numerical results with overall proof. Therefore, a lower limit of the application of the TSM is yet to be determined. Fig. 6 also shows the characteristic SN curve of the notch case 90 according to DIBt (2012), which is used for the design of wind energy tower structures in engineering practice (SN 90,WE ) with only one turn- off point at 5∙10 6 load