## 5 Cylinder Tube Stacking Sequence Definition
### The Bucking Analysis  
Analytical and experimental investigations have been developed to study the critical buckling load of hydraulic cylinders. In the present paper, the hydraulic cylinder was modeled in analogy with the approach proposed in ISO standards [38] and based on [PERSON]'s method [39]. However, a FE model was used rather than the analytical formula in order to consider the relevant shear deflection of the composite tube, due to the orthotropic behavior of this class of materials. Although the ISO norm assumes that the axial load is transmitted through the tube during the pushing phase, whilst in the actual working operation, such load is supported by the constrained fluid, and the composite barrel is relieved from the compressive load, retarding buckling incipience. Comparative analyses were conducted following the ISO approach, which can be pragmatically described _as a fixed-end pipe of length, connecting two rigid and rigidly supported tanks_ ([40], p. 37). The mounting case method was modeled by FE as a pin-mounted hydraulic cylinder, where the cylinder cross sections at the piston location and at the further head cap extremity are assumed to be rigid, and the relative boundary conditions are presented in Figure 6.  
Figure 5: Relative radial deformation versus distance from the sealing point. It is worth to be noted that the actuator rod is laterally supported by the cylinder at the head cap bearing and at the piston mid-point. The axial load is transmitted from the rod end to the cylinder end and a frictionless contact between the cylinder and the piston is considered. The FE analysis of the cylinder tube was performed by adopting bilinear, four-noded shell elements (element type 75, see [37]) whereas beam elements (element type 98, see [37]) were used for the piston rod. The formulation of both the shell and beam elements included transverse shear effects. In line with the cylinder axial direction, a fine mesh of the cylinder itself at the piston location is used; in fact, the average shell elements size is 10 mm. A coarse mesh of 40 mm was used to discretize the remaining part of the cylinder. In the circumferential direction, for the overall cylinder length, the component was meshed by 25 elements. In addition, the beam elements, applied for the piston rod model, exhibit a uniform average size equal to 25 mm. A lateral view of actuator is illustrated in Figure 5, where both the fine and the course mesh of the cylinder is shown. The cylinder tube stacking sequence in composite was defined by homogeneously spreading the composite plies along its thickness. In fact, the circumferential plies are alternated by the axial plies, similarly to the model presented in the previous Section 6.1. For instance, the composite stacking sequence of Case D was modeled by considering 17 iso-stacking sequence regions. From the inner radius of the cylinder, the plies are stacked as follows: 1 st stack) the inner metallic liner, 2 nd-16 th stack) the quasi-isotropic stacking sequence [0, -45, 90, +45]4 and 17 th stack) 0 ply. The steel layer was inserted at the cylinder inner radius for Cases B-D and was omitted in the test Case A for comparison, see Table 7. It appears evident from Figure 7 that in the case of the bare 0\({}^{\circ}\)/90\({}^{\circ}\) stacking sequence, Case B, a local and critical shear deformation occurs between the two rod support points, significantly lowering the buckling critical load with respect to the reference steel assembly (Table 8). For the reference actuator in steel, the pressure at which the buckling occurs is lower than the burst pressure; therefore, for the further FE analyses (e.g., the dynamic forecasts), the models will be loaded at the testing pressure prescribed for this component (42 MPa).