# A Combined Electromagnetic and Mechanical Approach for EU-DEMO Toroidal Field Coils
## 5 Structural Calculation Methodology-Theromechanical Analysis
According to this system, the considered material characteristics are: E\({}_{11}\) = 20 GPa; E\({}_{22}\) = 20 GPa and E\({}_{33}\) = 12 GPa [15]. The TF winding pack (WP) is modelled with an equivalent orthotropic material [16; 17]. The proposed design for the WP has a graded layout: twelve layers of different conductors are placed in order to increase the current density on the magnet cross-section, thus producing a higher magnetic field. Additionally, the jacket of the CICC has a graded thickness to increase the steel amount towards the innermost layer (Figure 10). For the homogenization procedure, one representative volume element (RVE) is defined for each layer (Figure 10a) to obtain twelve different equivalent materials. Each RVE must be a small portion of the whole component large enough to contain the needed  
Figure 10: TF winding pack with the indication of an RVE (**a**) and homogenized winding pack considered for the finite element analyses (**b**). Figure 9: Plots of the mechanical contact interfaces implemented in the model: different colors represent different contact surfaces. information and to reproduce the correct macroscopic material behavior. For periodic materials, as the graded WP, the repetitive unit cell is usually chosen as RVE. The orthotropic elasticity constants can be computed considering six load cases applied to the RVE: three tensile tests and three shear tests [12; 18; 19; 20]. For each load case, a suitable arbitrary macroscopic strain field is applied on the boundary of the RVE, and the corresponding reaction forces are used to assemble the stiffness matrix. Once the material stiffness matrix is completed, the engineering constant can be derived. For an orthotropic behavior, the effective constitutive relation reads  
\[\left( \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \tau_{xy}\\ \tau_{yz}\\ \tau_{zx}\end{array} \right)=\left(\begin{array}{cccc}\mathrm{D}_{11}& \mathrm{D}_{12}&\mathrm{D}_{13}\\ \mathrm{D}_{21}&\mathrm{D}_{22}&\mathrm{D}_{23}\\  
By repeating these steps for each load case, all elements of the material stiffness matrix of Equation (1) are evaluated. The matrix \(\mathbf{D}\) is finally inverted, and the engineering constants \(\mathrm{E_{x}}\), \(\mathrm{E_{y}}\), \(\mathrm{E_{z}}\), \(\mathrm{G_{yy}}\), \(\mathrm{G_{yz}}\), \(\mathrm{G_{xz}}\), \(\mathrm{v_{xy}}\), \(\mathrm{v_{yz}}\) and \(\mathrm{v_{xz}}\) are easily computed as  
\[\mathrm{[C]}=\left( \begin{array}{cccc}\frac{1}{\mathrm{E_{y}}}&-\frac{ \mathrm{v_{yy}}}{\mathrm{E_{y}}}&-\frac{\mathrm{v_{xz}}}{\mathrm{E_{z}}}&&\\ -\frac{\mathrm{v_{yy}}}{\mathrm{E_{x}}}&\frac{1}{\mathrm{E_{y}}}&-\frac{ \mathrm{v_{xy}}}{\mathrm{E_{z}}}&&\\ -\frac{\mathrm{v_{xz}}}{\mathrm{E_{x}}}&-\frac{\mathrm{v_{yz}}}{\mathrm{E_{y}} }&\frac{1}{\mathrm{E_{z}}}&&\\ &&&\frac{1}{\mathrm{G_{xy}}}&\\ &&&&\frac{1}{\mathrm{G_{yz}}}&\\ &&&&\frac{1}{\mathrm{G_{xz}}}\\ \end{array} \right) \tag{4}\]  
The structural model presented is used to assess the static reliability of the system. The six time points detailed in Section 3 are considered. The first one considers the dead weight and the preload application by means of bolted joints and pretensioners. In fact, in the tokamak, there are several systems that must be preloaded at room temperature, at the end or during the assembly phase, to guarantee the correct functioning and avoid possible detachment between the magnets and the support structures. It is therefore essential to check with a dedicated step the correct working of the bolted interfaces. The second time point includes the thermal load application due to the cooling from room temperature to the coil operating temperature. Once the cooldown to 4.2 K of the machine is completed, the other four characteristic instants of the scenario are analysed, considering the Lorentz forces obtained from the electromagnetic calculations. The results of the static analyses in terms of displacement components are reported in Figure 11 at the end of the cooldown, in Figure 12 at the TF energization phase and at the end of flattop.