## 5 Cylinder Tube Stacking Sequence Definition
### The Effect of the Fluid Weight on the Misalignment Actuator Condition  
Before applying any loading conditions, one of the main input factors acting in a hydraulic cylinder is the deflection of the cylinder due to its own weight and the fluid weight, [15]. This tiny deflection leads to an initial misalignment on the cylinder tube and piston-rod assembly. To analyze the influence of the axial pushing load under this misaligned condition, a FE analysis was performed. This simulates the actuator loaded by its own weight and the fluid present in the pressurized chamber, in a pin mounted horizontal configuration, at the pushing load exerted at the testing pressure (42 MPa). The model setup presented in the previous Section 6.2 has been used. The weight of the fluid was therefore evaluated by referring to the mineral base oil Tellus S2 (ISO VG46), where its density is assumed to be 872 kg/m\({}^{3}\). The overall weight of the fluid within the cylinder at the maximum extension of the piston rod thus becomes approximately 132 kg. Since the density of the oil is known, its weight has been modeled by the evaluating the equivalent nodal forces, acting along the Y-direction, of the cylinder oil chamber discretized by the course mesh, see Figure 7.  
The actuator own weight was mimicked by using a gravity and distributed load applied to the elements of the entire model, where the gravity acceleration is imposed equal to 9.81 m/s\({}^{2}\) and it acts along the Y-direction. The buckling FE analysis adopts the boundary conditions previously presented and summarized in Figure 6.  
The results obtained for the reference and the composite cylinder Case D are presented below. The stress field, occurring at the piston rod, due to the cylinder's own weight and the fluid weight, was evaluated by considering the geometric nonlinearity due to the large rotation hypothesis. In other words, the equilibrium equations are determined by updating the deformed configuration of the structure uploading the stiffness matrix proper of the present FE problem. The pushing load has been gradually increased (100 uniformly spaced steps), and for each load level the Newton-Raphson  
Figure 8: Deformed configuration (Scale Factor \(\times\) 50) of the cylinder tube, while pushing under its own weight: (**a**) reference and (**b**) composite Case D configuration considering small strain assumption, (**c**) reference and (**d**) composite Case D configuration considering large rotation assumption. iteration has been performed up until the relative errors on both the incremental displacements and the residual nodal equilibrium forces fell below the 1%o threshold. The bending stress (\(\sigma_{\text{Mf}}\)) and axial stress (\(\sigma_{\text{N}}\)) values of the rod were calculated by collecting the bending moment and the axial force on the steel rod, for both the reference steel cylinder and the multi-material composite cylinder Case D. The compressive axial stress at the rod is constant on the two configurations and is equal to \(-163.64\) MPa. Under the assumption of small strain deformation, the hydraulic actuator with the composite cylinder presents a lower bending stress with respect to the reference, approximately \(-50\%\). In fact, the decrease in bending stiffness of Case D--raised by the buckling analysis above--is compensated by the reduced weight (approximately \(-80\%\)). However, due to misaligned axial load, the increase in stress also appears to be relatively small between the actual and the composite solution. On the other hand, considering the non-linear effect due to the large rotation assumption, the bending moment at the piston-rod increases from 12 MPa up to 37 MPa for the reference configuration, and from 8 MPa up to 45 MPa for the multi-material solution. The bending stress achieves 22.6% and 26.0% of the axial force (\(\sigma_{\text{N}}\)) for the reference and the composite Case D solution, respectively.