# Global Analysis of Adjustable Pallet Rack Down-Aisle Frames Using the Generalised Beam Theory
## 4 Shell Finite Element Model of the Structure  
The structure was analysed with Ansys Mechanical APDL [27]. A single-column modelling approach, similar to that proposed by [PERSON] et al. [28], was applied (see also [29]). The model consists of one single upright made with Mindlin-Reissner shell elements, with half-span beams connected at both sides, modelled with 6-DOF/ node Euler-Bernoulli beam elements. Two models for the upright were analysed, with and without perforations; in both cases, the cross-section corners were considered sharp. A cyclic symmetry condition was imposed between the two beam ends at each level, consisting of coupling all their degrees of freedom (Figure 6). This model, with slight modifications, was presented in previous research [26]. The model is well suited for vertical and horizontal actions (Figures 5(a) and 5(b), respectively). On the other hand, overturning or the behaviour of the lateral upright frames (those that only present beams on one side) cannot be captured; typically, additional flexural and torsional effects arise in those frames [6]. The use of such a model is aimed at simulating the behaviour of internal bays, and its use is endorsed by the high number of bays (Figure 5). The steel of all the components was modelled as being linear elastic, with E = 210 GPa and \(\upsilon\) = 0.3. The behaviour of floor and beam-to-upright joints is semi-rigid, presenting  
Figure 5: Sketch of the rack. Figure 6: Single-column model. linear relationships between bending moment and rotation. Geometric nonlinearity was accounted for by co-rotational formulations in both beam and shell elements. The load corresponding to the weight of the stored goods is uniformly distributed on the beams, resulting in 17/1.8 kN/m (Figures 5 and 6a). Additionally, lateral forces of 47.6 N, related to the sway imperfections according to the standard EN15512 (\(\phi=1/350\)) [1], are applied at one half-span beam end, corresponding to the midspan point (Figure 6b). The self-weight of the steel is not considered, as it is negligible compared to the load of the stored goods. The beam-to-upright joint was modelled by a sufficiently rigid assembly of bar elements (Figure 7) attached to the beam at point B (Figure 7b). Several couplings are placed between such an element and the upright, aiming to clamp the rotations with respect to X-X and Z-Z. Two uniaxial linear springs are placed at the top and bottom of such an assembly to model the rotational stiffness of the semi-rigid connection (Figure 7b). The springs are calibrated to have the same joint stiffness as that obtained from the code-type test [1], 160 \(\times\) 10\({}^{6}\) Nmm/rad. The details of the modelling of this joint can be found in [26]. With respect to the floor connection, the nodes of the upright bottom section are linked to the rigid rotation of one node (O in Figure 8) with respect to the Y-Y axis, which simultaneously presents a rotational spring linked to a completely restrained node (OO in Figure 8); this rotational spring exhibits the stiffness obtained from the floor connection test [1], 139 \(\times\) 10\({}^{6}\) Nmm/rad. In this modelling, X-X and Z-Z rotations at the floor are constrained, as well as warping. The modelling of the floor (Figure 8) and beam-to-upright (Figure 7) connections presented in this section is only valid for moderately small rotations [26]. Diagonals of the upright frame were not modelled, but their constraining effect on the connecting upright sections is approximately accounted for. This is carried out by coupling the \(X\) displacement of the two points of the cross-section that connect with the bracing diagonal (Figure 9a). Moreover, the shear stiffness of the upright frame is approximately represented with uniaxial springs in the Y direction, placed between two consecutive diagonals (Figure 9b); the approximate (and rather imprecise) character of this modelling is acceptable when studying the down-aisle stability [4].