## 2 Overall Analysis Approach and Implemented Methods
### Structural Sizing  
The structural sizing of a generic non-integral unstiffened tank is performed combining the approach described by [26], based on Federal Aviation Regulation (FAR) Part-25 ([40]), and the ASME Boiler and Pressure Vessel Code [38]. According to this latter, the allowable tensile stress of the material (\(\sigma_{all}\)) can be calculated as in Equation (17), in which \(\sigma_{y}\) is the yielding strength and \(\sigma_{u}\) is the ultimate strength of the material, both depending on the operating temperature \(T\). \[\sigma_{all}=min\Big{[}\frac{\sigma_{y}(T)}{1.5};\ \frac{\sigma_{u}(T)}{3.5} \Big{]} \tag{17}\]  
Since Equation (17) frames the structural design in the elastic stress-strain interval, only limit conditions prescribed by FAR Part-25 were considered. They are:  
* Difference between internal pressure and external pressure at limit condition (\(\Delta p\)) calculated as in Equation (18): \[\Delta p=(k_{vent}\cdot p_{int}-p_{cr})\cdot RF\] (18) where: \[\begin{array}{ll}\circ&p_{int}\ \text{is the vessel internal pressure, whose reference value is set to 144.8 kPa (21 psi) according to the general conclusions drawn in [26];}\\ \circ&k_{vent}\ \text{is the ratio between the venting pressure ($p_{vent}$), i.e., the maximum value that internal pressure reaches during normal operations, and \(p_{int}\). According to [26], a typical value of \(k_{vent}\) is 1.1;}\\ \circ&p_{cr}\ \text{is the atmospheric pressure at cruise, calculated using the ICAO International Standard Air model for the desired cruise altitude;}\\ \circ&RF\ \text{is the relief valve tolerance, set to 1.1 according to FAR Part-25 and also adopted in [26]. * Inertial loads due to flight manoeuvres or gusts at the following (not combined) limit load factors, derived dividing by 1.5 the ultimate values prescribed by FAR Part-25: \[ \begin{array}{ll}\circ&\text{Equal to 3 in the radial direction ($n_{r}$), referred to the most critical case for loads acting in the cross-section plane, i.e., downward loads (ultimate \(n_{z}=+4.5\));}\\ \circ&\text{Equal to 6 in the axial direction ($n_{x}$).}\end{array} \] Thermal loads were not considered, since it is assumed the whole metallic structure of the tank, being thermally isolated from the external atmosphere by the insulating layer, is at a constant temperature, hence there are no gradients causing additional stress. In addition, it is assumed that the supporting frames are designed to allow the thermal contraction of the tank structure, which avoids the local introduction of additional loads. The structural design is then implemented through the following procedure:
* Given the insulation material thickness (\(t_{ins}\)) as an input, the internal radius of the central part of the vessel (\(Rc_{int}\)) is initialised with the external radius value \(R_{str}=R-t_{ins}\) (see Figure 7);* The thickness of the vessel cylindrical part (\(tc\)) is calculated considering welded joints, longitudinal or circumferential, as the most critical areas in which the static strength needs to be verified; therefore, both longitudinal and hoop stresses are calculated and compared to admissible values of the given material and type of joint. The adopted formulas are given in Equation (19), where the first and second expressions provide the thickness resulting from a design for circumferential and longitudinal stress, respectively, whereas the third term is the minimum thickness imposed by [39]. In Equation (19), \(\eta_{w}\) is the welding efficiency as defined in [38], which for butt joints is set to \(1\) if full radiographic examinations are carried out, 0.85 in the case of spot examinations, and 0.7 otherwise. For the present study, a prudential value of 0.85 was chosen.