## 5 Cylinder Tube Stacking Sequence Definition
### Design for Pressurized Fluid Radial Retention  
The dimensioning of the circumferentially oriented portion (called 90\({}^{\circ}\) plies) of the stacking sequence is driven by the fulfillment of three constraints. The main design constraint for the sizing of the circumferential plies cumulative thickness (\(t_{90^{\circ}}\)) is the thresholding of the circumferential stress value at inner tube radius; such constraint is reported as the first in the summarizing Table 6 below. CFRP pressure vessel and gas bottle walls are mainly dimensioned according to such constraint; steel cylinder actuators are also designed within the respect of an equivalent stress value, where the circumferential component prevails. However, a CFRP axial actuator tube designed according to this first constraint alone would suffer from an excessive radial expansion at the inner surface resulting in both seal leakage [35] and piston misalignment due to increased gap at the bearing bands. The lower stiffness to critical stress ratio of the CFRP material substantially worsens this shortcoming in the title design with respect to a construction steel counterpart, and it raises the thresholding of the inner radial expansion to a major design constraint, which is also reported in Table 6.  
In further contrast with the usual steel construction, the inner surface of the composite tube is tribologically unsuitable to pair seals and bearing bands in sliding contact [35; 36], even when the inner layers are laid in direct contact with the mold. A bonded inner steel liner is therefore employed to endow the sliding seals with a tribologically suitable counterpart. The insertion of a thin steel liner was considered also [8] and sketched in Figure 1 of [24]. In Figure 2, the liner might be represented as the \(i\)-layer located at the inner radius, over which the composite laminae will be stacked. The thickness of the steel liner has been considered equal its minimum technologically allowable value of 2 mm; lower values would create manufacturing difficulties in respecting tolerances and surface quality, whereas higher values would unnecessarily increase the component weight. The thickness of the steel liner is assumed infinitesimal in the pertinent structural calculations, and it inherits the circumferential and axial strain values from the composite layer it is bonded to, i.e., the CFRP tube inner surface; those strains that are cyclic in nature due to the succession of loading \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Constraint** & **Inequality Formulation** & **Prevented Failures** & **Notes** \\ \hline Upper limit on the circumferential stress value & \(\sigma_{\phi}|_{\text{r}-\text{t}_{1}}\leq\sigma_{1}\) & Fracture of the circumferential lamina & \(\sigma_{\phi}|_{\text{r}-\text{t}_{2}}\) from Equation (5) \\  & & & \(\sigma_{1}\) from Table 5 \\ \hline Upper limit on the inner radial displacement & u\({}_{\text{r}-\text{t}_{1}}\leq\text{u}_{\text{lim}}\) & Excessive leakage or piston misalignment & u\({}_{\text{lim}}\) from seal manufacturer requirements \\ \hline Upper limit on the inner circumferential strain & u\({}_{\text{r}-\text{t}_{1}}\leq\text{c}_{\text{lim}}\text{r}_{1}\) & Low-cycle fatigue fractures of the steel layer in the longitudinal direction & u\({}_{\text{r}-\text{t}_{1}}\) from Equation (6) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The three design constraints discussed in the present paragraph, for summarizing purposes. cycles may damage the steel liner and they have to be bounded within allowable values. A third design constraint in terms of admissible strains at the tube inner surface is hence to be considered, see Table 6. The admissible strain value for the steel liner is quantified in the following. Being the liner a non-load-bearing element, its yielding can be tolerated as long as strain induced fractures do not appear.