## 2 Overall Analysis Approach and Implemented Methods
### Structural Sizing
\[tc=max\bigg{(}\frac{\Delta p\cdot Rc_{int}}{\sigma_{all}\cdot\eta_{w}-0.6\cdot \Delta p};\ \frac{\Delta p\cdot Rc_{int}}{2\cdot\sigma_{all}\cdot\eta_{w}+0.4\cdot \Delta p};\ 1.6\ \text{mm}\bigg{)}\] (19)  
The internal radius is updated with the value resulting from \(Rc_{int}=R_{str}-tc\) and Equation (19) is applied iteratively until \(tc\) converges within an assigned threshold. * Similarly, the thickness of vessel endcaps (\(te\)) is calculated according to [38] using the formula in Equation (20), in which the function \(k(f_{end})\) is defined in Equation (21) and the second term is introduced to guarantee the applicability of the first expression. \[te=max\bigg{[}k(f_{end})\cdot\frac{\Delta p\cdot Rc_{int}}{\sigma_{all}\cdot \eta_{w}-0.1\cdot\Delta p};\ 5\cdot 10^{-4}\cdot L\ ;1.6\ \text{mm}\bigg{]}\] (20) \[k(f_{end})=\bigg{\{} \begin{array}{ll}\frac{2+f_{end}-^{2}}{6}&0.0005 \leq te/L<0.002\\ 1&te/L\geq 0.002\end{array} \] (21)  
At the first step it is assumed that \(Re_{int}=Rc_{int}\), then the internal radius is updated with the value resulting from \(Re_{int}=R_{str}-te\) and Equation (20) is iterated until the convergence of \(te\);  
* The obtained thickness values allow to calculate the longitudinal mass distributions as well as the total mass of the tank, taking the structure, the insulation layer, and assuming the whole internal volume can be filled with LH\({}_{2}\);
* Given the mass distribution, it is possible to calculate both the maximum (tensile) and minimum (compressive) axial local stress and perform the verifications given in Equation (22). In this latter, the first line is the comparison between the limit longitudinal tensile stress, resulting from both the internal pressure (first term in brackets) and the bending moment, and the strength of the welded material, whereas the second line is the local buckling verification for the cylindrical part of the vessel at ultimate load factors. For buckling design, a null value was assigned to \(\Delta p\) since a positive value reduces the compressive stress in tanks' shells. This assumption is conservative, since a minimum \(\Delta p\) of about 20% of external atmospheric pressure is constantly kept by means of check valves that prevent the inversion of mass flow in the outlets. Column buckling verification was implemented too, but it is not reported here for conciseness, since local buckling is more critical in most of the observed cases. \[\left\{ \begin{array}{ll}\frac{\left(\Delta p\cdot Rc_{int}}{2\cdot tc}-0.2 \cdot\Delta p\right)+\eta_{r}\frac{B_{max}\cdot Rc_{int}}{l}\leq\sigma_{all} \cdot\eta_{w}\\ max\bigg{[}1.5\cdot\eta_{r}\frac{B_{max}\cdot R_{str}}{l_{c}};1.5\cdot\eta_{x} \frac{N_{max}}{\pi\left(R_{str}\cdot^{2}-Rc_{int}\right)}\bigg{]}\leq\Delta_{ b}\Big{(}\frac{L}{D_{int}},\frac{d_{exp}}{L},\ \text{$te$}\Big{)}\cdot\frac{0.125}{f^{5} }\cdot E\cdot\frac{tc}{R_{str}}\end{array} \right.\] (22)  
The variables in Equation (22) not yet introduced are:  
* \(B_{max}\), the maximum bending moment, calculated considering both the longitudinal mass distribution varying along the tank and the distance between the supporting frames (\(d_{supp}\));
* \(N_{max}\), the maximum compressive axial load;
\(I_{c}\), the inertia moment of the central tube; \(FS\), a design factor defined depending on the ratio between the predicted buckling stress (\(\sigma_{b}\)) and the yielding stress as in Equation (23); \[FS= \begin{cases}2&\sigma_{b}\leq 0.55\text{-}\sigma_{y}\\ 2.407-0.741\frac{\sigma_{b}}{\sigma_{y}}&\text{for }0.55\text{-}\sigma_{y}< \sigma_{b}<\sigma_{y}\\ 1.667&\sigma_{b}\geq\sigma_{y}\end{cases} \] (23) \(E\), the modulus of elasticity of the structural material; \(\Delta_{b}\), a correction function, depending on the length-to-diameter ratio (\(L/D_{str}\)), the nondimensional distance between supports (\(d_{supp}/L\)), and endcap wall thickness (\(te\)), derived from FEM analyses as described in Section 2.4.1.