# Modeling Palletized Products: The Case of Semi-Filled Bottles under Top-Load Conditions
## 4 Numerical Models  
### Shell (2D) Model  
Two-dimensional shell elements were used in this first finite elements (FE) study to evaluate the behavior of PET bottles under compression in the case of top-load tests. In particular, both a non-linear static and a buckling analysis were done by ANSYS Workbench v.18.1. The geometry was simplified as displayed in (Figure 9a): all secondary details like fittings, chamfers, and notches were removed, since they marginally affect the system; the cap was merged to exclude contact problems; the thickness was simply defined as uniform and equal to 0.3 mm. Four-node shell elements, 13,258 in total, were used to create the FE grid. This mesh, optimized for buckling analysis, can be observed in the area of cap and neck in Figure 9b. In the base of the bottle all the translations and rotations were constrained (Figure 9b). The effect of water and air inside the bottle was replaced by an internal pressure, increasing over time roughly in accordance with the equation of state of perfect gases. The contact between bottle and plate was defined as a contact with friction, with a coefficient of friction equal to 0.2 representing the friction between PET and steel. Material properties were defined in line with Table 1 with a bilinear isotropic hardening model with a plastic zone starting from a yield value of 75 MPa (Figure 4). Figure 8: (**a**) Experimental stress–strain curve; (**b**) calculation of elastic modulus on the linear zone. With the scope to perform the quasi-static analysis, the bottle was loaded by a rigid plate moved by a displacement of 10 mm in the vertical (\(\gamma\)) direction (Figure 9c). This value was chosen by considering the maximum displacement in the elastic region as measured by top-load tests (Figure 6). In some ways, it represents the experimental evidence of the maximum load that this specific bottle can resist without moving its status to the plasticity. The simulation of maximum load is widely used in the PET bottle industry and the vertical load test determines the weak points inside the bottles. Thus, the numerical analysis limited to this value can be extremely useful, for example, in detecting problems in the blow molding line and to check if the component failures are repeated in the same points. Also, in this case, a non-linear structural simulation was performed with the maximum load for bottle verification in order to obtain a reliable model of an empty PET bottle. The simulation was divided into two steps: load in terms of displacement up to 10 mm, and pressure increase up to 1 MPa. Non-linearity due to geometry was taken into account through large displacements. The stress versus strain status following a 10 mm displacement is shown in Figure 10 in terms of (equivalent) von-Mises stress (Figure 10a) and total deformation (Figure 10b). In particular, the maximum stress was 164 MPa. This stress state exceeds the yield point, showing the bottle in plastic conditions. Figure 10: Quasi-static analysis in top-load conditions for a 10 mm vertical displacement: diagram of stress (**a**) and strain (**b**). Figure 9: (**a**) Bottle geometry before and after simplification; (**b**) detail of geometry and mesh by shell elements; (**c**) application of forces and boundary conditions (case of non-linear static analysis). The buckling analysis was performed using the same conditions of simulation. It made it possible to detect the geometrical configuration of instability together with the related buckling load factor (BLF). This factor acted as a load multiplier with the scope to evaluate the buckling load for each condition of instability. When the BLF is greater than one, the structure is safe apart from when the external load is increased. It also means that when the buckling factor is less than one, the structure could collapse under a given load.