# Global Analysis of Adjustable Pallet Rack Down-Aisle Frames Using the Generalised Beam Theory
## 1 Introduction
Those considered most relevant are included in the following list [1; 2; 3]: (i) columns are made of open thin-walled profiles, thus they generally undergo significant sectional deformations, ultimately leading to local and distortional buckling; (ii) columns are weak and flexible against warping torsion, and their cross-sections are monosymmetric, meaning that the effects of centroid and shear centre eccentricity are relevant; (iii) columns show multiple small perforations, which have a non-negligible impact on the stiffness and strength of the structure [4; 5]; (iv) the slender nature of the profiles and the global structure increases the geometrically nonlinear effects and, consequently, the influence of initial imperfections; (v) beam-to-column and floor joints are semi-rigid, exhibiting nonlinear behaviour and important load transfer through contact between components (bolts and welding are relatively uncommon); (vi) this latter aspect also intensifies the impact of the looseness in both down- and cross-aisle joints; and (vii) the eccentricities at the joints may have an important effect. Most of these factors are routinely considered in design, specifically when performing global analyses of rack structures. Concerning recent research on global analysis, it is worth mentioning the work by [PERSON] et al., who showed the key role of the warping degree of freedom in pallet racks [6; 7; 8; 9]. By comparing analyses conducted using 3D beam elements with 6 and 7 degrees of freedom  
Figure 1: Typical adjustable pallet racking systems. (where the 7 th degree is warping), they proved that warping needs to be considered in linear buckling analyses to correctly determine the elastic buckling factor of the structure (see also [10]). Furthermore, in second-order geometrically nonlinear analysis, the seventh degree of freedom allows for the determination of the warping moment (bi-moment) and also accounts for the interaction between torsion and bending, which increases horizontal displacements and bending moments. These effects of warping seem to be especially important at the lateral columns of the rack structure (end columns presenting beams at only one side) [6]. Nowadays, most studies using beam elements already incorporate the 7 th degree of freedom in the analysis, as seen in [11; 12; 13]. Another topic of recent research on the analysis of racks is the effect of local and distortional deformations and that of their corresponding imperfections on the overall behaviour of the structure. [PERSON] and [PERSON] compared in [14] the strength predictions of racks derived from beam and shell finite element models. Beam model predictions were actually determined from geometrically nonlinear analysis (GNA) internal forces and checking capacities according to current standards, while the \"advanced\" design approach was applied to directly determine the ultimate strength of the racks from the shell model and geometrically and materially nonlinear analysis with imperfections (GMNIA). They proved that as the slenderness of the column sections increases and, consequently, the sectional deformations become relevant, the predictions from the beam element models worsen with respect to those of the shell elements, overestimating the strength. The reason for the unsafe calculated design loads is attributed to the fact that the GNA analysis-based design does not take into account the additional second-order deformations produced by local and/or distortional buckling of the columns. In another study from the same research team [15], it was demonstrated that, despite the relevant effects of the sectional deformations on the final results of the global analyses, the influence of the corresponding sectional imperfections is not important, especially for unbraced down-aisle rack frames. Notably, part of this study is devoted to discussing the effect of distortional imperfections (see also [16]).