PSI - Issue 75

Fritz Wegener et al. / Procedia Structural Integrity 75 (2025) 363–374 Wegener et al. / Structural Integrity Procedia 00 (2025) 000–000

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4. Analytical fatigue assessment

4.1. Implementation of the notch-strain approach

As discussed in section 2.2, the determination of cyclic material behavior is required in the calculation of the crack initiation life for the local loading conditions as cyclically stabilized stress-strain curve and as fatigue resistance via the strain-life curve. Since no cyclic material data are available for the tested bolting assemblies, the behavior is approximated based on the UML acc. to Ba¨umel and Seeger (1990) and the results from quasi-static tensile tests, shown in Figure 4 (a).

Fig. 4: (a) Stress-strain curves for quasi-static tensile tests (mean curves); (b) Stress-strain curves for cyclic loading determined using UML; (c) Strain-life curves determined using UML; (d) P J -life curves.

Averaged test data of the tensile tests and resulting cyclic material parameters are given in Table 2. Based on these data, the cyclic stress-strain as well as the strain-life-relationship are derived using the Ramberg-Osgood equation and Co ffi n-Manson relation, see Figure 4 (b) and (c). Table 2: Quasi-static (experimental) and cyclic (UML) material data of the investigated bolting assemblies. Series E R m K ′ n ′ σ ′ f b ε ′ f c no. [N / mm²] [N / mm²] [N / mm²] [-] [N / mm²] [-] [-] [-]

M1-M12 202,490 1,173 2,032 0.15 1,760 -0.087 0.384 -0.580 M2-M12 196,150 1,116 1,927 0.15 1,674 -0.087 0.392 -0.580 M2-M56 216,000 1,107 1,882 0.15 1,661 -0.087 0.433 -0.580

Local stresses and strains in the first load-bearing thread are calculated using an axially symmetric 2D FE model in combination with the approximated material data and the Ramberg-Osgood law for the cyclic stress-strain relationship. For all investigated bolting assemblies, a thread tolerance combination 6g-6AZ is implemented. As shown in Figure 5 (a), the actual load path of the experimental fatigue tests is divided for this purpose in two FE calculations. In the first calculation step (top row), the upper test load S o = S m + S a is applied as it is the case at the beginning of the fatigue tests. This yields the upper local stress and strain at the root of the first load bearing thread. In a second calculation step (bottom row), only the nominal stress amplitude is applied and local stress and strain at the thread root are derived. Based on these results, the actual stress-strain hysteresis (right) is constructed taking into account the Masing behavior of the material, Masing (1926). The calculated local stress-strain hysteresis is characterized by upper and lower local stress and strain and is further processed by means of a damage parameter in order to compensate for mean stress e ff ects. Based on the findings of Eichsta¨dt (2019) the damage parameter P J acc. to Vormwald (1989) is used to rate the calculated stress-strain hystereses. The e ff ective stress range ∆ σ ef f is calculated based on the local stress at crack opening σ op . As shown in Figure 5 (d), for the given example the stress-strain hysteresis lies completely above the crack opening stress σ op . Therefore, the whole hysteresis is considered for damaging at the thread root and the e ff ective stress range results to ∆ σ ef f = σ o − σ u . This is the case for all investigated series and load levels, due to the high nominal mean stresses.