## 5 Cylinder Tube Stacking Sequence Definition
### Design for Axial Load Retention  
For the present hydraulic component, the axial load (P\({}_{\text{a}}\)) is due to pressure \(p_{\text{i}}\) acting on the head while pulling, or on the pushing piston when retained at full stroke in the absence of position-limiting valves. Such loads can be, respectively, evaluated by referring to the cross where the pressure acts, as follows,  
\[p_{\alpha}=p_{i}\cdot\left(\mathbf{r}_{i}^{2}-r_{er}^{2}\right)\cdot\pi \tag{9}\]  
or as  
\[\text{P}_{\text{a}}=\text{P}_{\text{i}}\cdot\mathbf{r}_{\text{i}}^{2}\cdot\pi \tag{10}\]  
where \(r_{\text{er}}\) is the rod outer radius.  
The 0\({}^{\circ}\) axially oriented ply amount is defined by the minimum cross section area A\({}_{0^{\circ}}\) required in order not to overcome the longitudinal tensile strength of the material. For instance, this value \(\sigma_{1}\) corresponds to the strength of the unidirectional ply parallel to the reinforcement on the principal orthotropic system.  
Unlike the circumferential case, the steel liner could be isolated from the composite tube axial deformation through an antifriction coating; however, this solution was not adopted in the present test case. The composite cylinder tube axial strain (\(\epsilon_{\text{lim}}\)) has to be limited as well to preserve the inner bonded steel liner. Therefore, the required amount of axially oriented unidirectional plies can be calculated as follows.  
\[A_{0^{\circ}}=\text{max}\left(\frac{P_{a}}{\sigma_{1}};\frac{P_{a}}{\epsilon_{ \text{lim}}\cdot E_{1}}\right) \tag{11}\]  
Since circumferential retaining plies become less effective if stacked on an increasing thickness of material, it is advisable to wrap the axial plies over the outer circumferential layers. Unfortunately, due to manufacturing and technological constraints, circumferential and axial UD plies need to be  
Figure 4: Radial displacement at inner radius vs. wall thickness at first-yield pressure.  
alternated, or an unbalanced fabric has to be used. The required axial ply thickness was evaluated as 2.0 mm using the UHM fiber.  
Therefore, by adopting the present methodology, the composite cylinder tube in UMH fiber that satisfies both the radial stress due to pressurization and the axial load detention due to the piston pushing and pulling requires a minimum overall thickness of 7.3 mm, reported in Table 7 as Case A. The optimal solution that introduces the inner steel liner is 13.3 mm thick named as Case B. It is composed of the steel liner (t liner = 2.0 mm) and 7.3 mm of UHM UD plies obtained as the sum of the plies required with the reinforcement aligned to the cylinder axis (t \({}_{0^{\circ}}\) = 2.00 mm, \(\phi\) = 0\({}^{\circ}\) see Figure 2) and to the hoop direction (t\({}_{90^{\circ}}\) = 5.3 mm). Case B is the most promising and compact solution in lieu of 22.5 mm of steel shell as in the reference assembly.  
The composite solution that either omits or considers the liner is referred to as Case A and Case B, respectively.  
Cases C and D are also introduced in Table 7; they derive from Case B with an addition of UD plies oriented at \(\pm 45^{\circ}\), see Figure 2, this aspect will treat below.  
Buckling Loading Conditions, Dynamic Modal Targets and Radial Deformation of the Cylinder at the Piston Locus  
The radial deformation of the cylinder with the piston movement, buckling phenomenon, and dynamic natural response represent three main design targets for a hydraulic cylinder. In this section, these structural requirements are discussed in the light of the application of CFRP orthotropic materials by FE analysis using the commercial, closed source solver MSC.Marc2017.1. The modeling details are given below.