## 2 Overall Analysis Approach and Implemented Methods
### Structural Sizing
#### 2.4.1 Validation of the Structural Model with FEM Analyses  
The results of the analytical structural model presented in Section 2.4 were compared with FEM analyses considering the two test cases presented in Table 5, where the last row indicates the value of the correction function \(\Delta_{b}\) introduced in Equation (22). This was obtained by means of FEM analyses aiming to compare the critical load obtained for buckling eigenvalue equal to unity with the estimation provided by the right-hand term in the 2 nd line of Equation (22) without any correction. The FEM simulation campaign was carried out using the two test cases reported in Table 5 as baseline configurations,and varying the parameters that it depends on, in order to cover a wide range of cases of interest. The two test cases were created in order to underline the differences between structural sizing dominated by tensile or buckling strength. In fact, the 1 st case is a tank with a shorter cylindrical part and a higher internal pressure, whereas the 2 nd tank is longer and has a larger unbraced length; hence, the design of the 1 st cylinder is driven by the circumferential stress acting on welded joints, whereas the 2 nd cylinder is designed to withstand buckling loads. The finite element mesh of each tank is made up of shell elements only; non-structural elements, such as hydrogen and the insulation layer, were modelled as non-structural masses. Different loading conditions were considered as combinations of inertial and pressurisation loads for static sizing verifications. For the bending-induced buckling analyses, only the ultimate downward inertial load was considered; thus, ignoring the beneficial effect of pressurisation on critical load evaluations; the latter was predicted by means of an eigenvalue analysis, according to Equation (24), where \(P_{dend}\) is a pre-existing applied load (preload or dead load) and \(\Delta P\) is the perturbation load introduced in the buckling analysis step. For these analyses, a null preload and a perturbed load equal to the ultimate downward inertial load were considered. Normally, the lowest positive eigenvalue is of interest. \[P_{cr}=P_{dead}+\lambda_{i}\cdot\Delta P \tag{24}\]  
For all the analyses, the structural connections between the tank and the fuselage were modelled as multi-point constraints. In details, at the two sections where the structural supports are located, the points on the longitudinal axis were designated as \"master nodes\" and multi-point constraint relationships were created with the \"slave nodes\" located on the cylindrical surface. For one of the master nodes the degrees of freedom associated to radial, \begin{table}
\begin{tabular}{l c c} \hline \hline
**Input** & **Test Case \#1** & **Test Case \#2** \\ \hline \(D_{Str}\) [m] & 3.00 & 3.00 \\ \hline \(L/D_{Str}\) [\(-\)] & 1.00 & 2.00 \\ \hline \(f_{end}\) [\(-\)] & 0.60 & 0.60 \\ \hline \(d_{supp}/L\) [\(-\)] & 0.70 & 0.90 \\ \hline \(P_{int}/p_{ref}\) \(\left(p_{ref}=144.8\) kPa) & 1.40 & 1.00 \\ \hline \(t_{ins}\) [mm] & 100 & 100 \\ \hline Structural material and characteristics & AA 2219 (Table 4) & AA 2219 (Table 4) \\ \hline Insulation material and characteristics & Polystyrene foam (Table 3) & Polystyrene foam (Table 3) \\ \hline
**Output** & **Test Case \#1** & **Test Case \#2** \\ \hline \(tc\) [mm] & 2.1 & 2.4 \\ \hline \(te\) [mm] & 1.7 & 3.0 \\ \hline Tube structural mass [kg] & 166 & 383 \\ \hline Endcaps (2) structural mass [kg] & 97 & 175 \\ \hline Total insulant mass [kg] & 132 & 207 \\ \hline Total tank empty mass [kg] & 395 & 765 \\ \hline Internal volume [m\({}^{3}\)] & 29.6 & 50.7 \\ \hline Allowable tensile stress on welded joints [MPa] & 159 & 159 \\ \hline Nominal buckling strength (long cylinders) [MPa] & 6.6 & 7.6 \\ \hline FEM derived correction factor \(\Delta_{b}\) [\(-\)] & 0.50 & 0.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Input and output of the analytical model for two test cases.