## 2 The Reference Hydraulic Actuator  
Figure 1 displays the geometry of the title problem, and it clarifies the meaning of the main components and the symbols adopted in the following. The material properties of the reference hydraulic actuator are shown in Table 1. The cylinder tube represents \(\sim\)51% of the overall hydraulic actuator weight for the reference configuration, and the components of Table 2 are labeled as in Figure 1.  
The reference hydraulic cylinder tube is 150 mm in bore radius (\(r_{\text{i}}\)), 172.5 mm in external radius (\(r_{\text{e}}\)), and 2300 mm in length (\(L\)). The piston rod has a hollow circular cross-section where the inner radius (\(r_{\text{ir}}\)) and the outer radius (\(r_{\text{er}}\)) are 25 mm and 80 mm, respectively. The piston rod stroke is 1960 mm (\(L_{\text{r}}\)); the axial length of the piston (\(L_{\text{p}}\)) is 160 mm. The flanges at the extremities of the actuator are in aluminium alloy 7075-T6. The nominal operating pressure is 35 MPa. In [8; 9], the cylinder tube was in S275 JR steel, and it was designed for at 30 MPa and 35 MPa working pressure, similarly our reference geometry. In [8; 9], the cylinder bore radius was equal to 25 mm and equal to 70 mm, i.e., one sixth and about a half of our component, respectively. The S275 JR steel has a lower yielding strength (\(\sigma_{\text{y}}=235\) MPa) compared to the S355 steel used in the present study.  
The cylindrical coordinate system (\(r\), \(\theta\), \(a\)) is adopted for the analytical formulation, and it defines the radial, hoop, and axial directions, respectively, for a generic axisymmetric geometry, see Figure 2. The cylindrical section of the component is internally pressurized, and it is represented by multiple \(j\)-th layers whose radial thickness is equal to \(t_{\phi}\). Since the reference actuator is in steel, it is considered as a homogeneous and isotropic material and its cross-section consists of one layer 22.5 mm thick. For composite material, the mechanical properties (e.g., strength and stiffness) are influenced on the reinforcement orientation angle \(\phi\); therefore, this information is mandatory for the definition of the optimal cylinder design in composite.  
The internal yield pressure (\(p_{\text{i}}\)) of the reference steel cylinder tube was calculated using Lame's solution for thick-walled vessels. The circumferential stress (\(\sigma_{\theta}|_{r=\text{ri}}\)) and the radial displacement (\(u_{r}|_{r=\text{ri}}\)) at the inner cylinder radius were then evaluated accordingly. Results are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ID Label** & **Component** & **Material** & **Weight [kg]** & **\% Weight** \\ \hline
1 & Rod & Steel, 42 CrMo4 & 321 & 39.8\% \\
2 & Head & Aluminum, 7057-T6 & 30 & 3.7\% \\
3 & Cylinder Tube & Steel, S355 & 408 & 50.6\% \\
4 & Piston & Aluminum, 7057-T6 & 24 & 2.9\% \\
5 & End cap & Steel, S355 & 24 & 3.0\% \\  & Overall weight & - & 807 & 100\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Weight of the reference hydraulic cylinder parts.  
Figure 2: The cylindrical coordinate system nomenclature.  
The first-yield condition represents a soft failure mode that precedes the catastrophic failure of the component, which is associated with the fully yielded state at burst pressure. The first-yielded state rather than the full yield was chosen for the following comparative safety analyses since the structural steel yielding phase has only a weak counterpart for composite materials. In addition, limiting the analyses below the first-yield threshold allows considerations based on linearity.