# A Combined Electromagnetic and Mechanical Approach for EU-DEMO Toroidal Field Coils
## 6 Fatigue Life Assessment--(S-N) Approach  
For the fatigue assessment of the toroidal field coils and support structures (IIS, OIS and GS), the procedure adopted is based on the prescriptions of the ITER design criteria [21; 22; 23; 24]. Considering the complete set of static analyses performed previously, it is possible to identify two time points corresponding to the minimum and maximum of the function describing the Tresca equivalent stress over time. Only the time interval involving cyclic loads must be considered, i.e., from the TF energization to the end of flattop. The radial and the vertical forces remain constant over time, while the OOP actions work as a superimposed cyclic load. The minimum is recorded at the energization phase (Figure 15), which is the early stage of the plasma scenario. The maximum is reached at the end of flattop (Figure 16) where the combination of in-plane and out-of-plane loads produces the highest stress peaks. As mentioned, to obtain a conservative estimate of the fatigue damage on the structure, it is assumed to have 20,000 constant cycles. Figure 14: The most stressed regions are recorded in the TF Case. In the last image, the internal bending effect of the case on the side facing the plasma at end of flattop is shown: the winding pack detaches almost completely from the case lid. Figure 13: Plots of the equivalent Tresca stress at cooldown (**a**), TF energization (**b**) and end of flattop point (**c**). The procedure for the extraction of the equivalent alternate fatigue stress is made of the following steps:  
1. The six components (\(\sigma_{\rm x}\), \(\sigma_{\rm y}\), \(\sigma_{\rm z}\), \(\tau_{\rm xy}\), \(\tau_{\rm yz}\), \(\tau_{\rm xz}\)) of the stress tensor for each finite element are stored in two matrices, collecting the extremes of the stress cycle (Figures 15 and 16). To this aim, a proper local coordinate system is defined for the elements considered. The x and z components are oriented, respectively, radially and toroidally along the TF curvilinear abscissa, while the y component is tangent to the curvilinear abscissa. 2. The stress range matrix (\(\sigma_{\rm Rx}\), \(\sigma_{\rm Ry}\), \(\sigma_{\rm Rz}\), \(\tau_{\rm Rgy}\), \(\tau_{\rm Ryz}\), \(\tau_{\rm Rxx}\)) is computed by subtracting the minimum stress matrix from the maximum stress one. 3. The three principal stresses (\(\sigma_{\rm R1}\), \(\sigma_{\rm R2}\), \(\sigma_{\rm R3}\)) of the range matrix and the three principal direction cosines ((l,m,m)\({}_{\rm I}\), (l,m,n)\({}_{\rm I}\), (l,m,n)\({}_{\rm JII}\)) are computed. 4. The \"equivalent alternating stress\" (S\({}_{\rm alt}\)) is defined as half the largest principal stress range difference: S\({}_{\rm alt}\) = 0.5 max l (\(\sigma_{\rm R}\)\({}^  
* The \"equivalent mean stress\" (S\({}_{\text{mean}}\)) is equal to the sum of the two principal mean stress components having the same direction cosines as the two principal stress range components determining the S\({}_{\text{alt}}\) value. * The equivalent mean stress is finally corrected by adding the residual stress related to the base material, conventionally equal to 50 MPa, or to the weld regions, conventionally equal to 250 MPa (Figure 18). Figure 17: Plot of the equivalent alternating stress considering as stress cycle the interval from the energization phase to the end of flattop. Figure 18: Plots of the equivalent mean stress for each element of the metallic components considering a residual stress of 50 MPa (base metal) or 250 MPa (welds). The equivalent mean stresses so obtained must be compared with the (S-N) experimental curve available for the 316 LN at cryogenic temperature (4.2 K). Fatigue experimental tests on material samples are usually run for \"constant amplitude-fully reversing cycles\", i.e., the same stress magnitude is applied in tension and in compression (which corresponds to a load ratio R = \(\sigma_{\text{min}}/\sigma_{\text{max}}=-1\)).