# Global Analysis of Adjustable Pallet Rack Down-Aisle Frames Using the Generalised Beam Theory
## 7 Parametric Studies Conducted with the GBT Finite Element Model
### Effect of Upright Sectional and Member Imperfections  
First of all, the influence of initial upright sectional and member imperfections was assessed. The imperfections were created as a combination of the down-aisle out-of-plumb (S: sway, equivalent forces approach), plus one or two of the following imperfections: SD (symmetric distortional), AD (anti-symmetric distortional), T (twist) and B (bow--Figure 11b). Noticeably, with the aim of reducing the number of analyses to be carried out in the parametric study, the camper imperfection mode (Figure 11a) was omitted; it was considered that the twist imperfection already triggers bending member deformations in the down-aisle direction. For the same reason of reducing the number of analyses, distortional buckling was the only contemplated sectional initial deformation. No local buckling imperfection was applied since, typically, the local deformation mode does not govern the behaviour of multi-stiffened rack sections, such as the one used in the example under analysis. On the other hand, bow member imperfection was included because it commonly produces rather detrimental effects in the axial capacity when combined with the symmetric distortional one. A maximum magnitude of L/1000 was defined for the pure member imperfection (bow and torsional modes), while two different magnitudes were applied for pure distortional imperfections: t and 0.25 t. These values are within the range of commonly used imperfection magnitudes [31]. The shapes of these pure sectional and member imperfections (Figure 15) were obtained by means of GBT eigenbuckling analyses involving only member global or distortional deformation modes. Finally, the initial member imperfection was defined as a linear combination of these pure member imperfections. As a result, the 13 different initial geometrical imperfections shown in Table 3 were evaluated. The direction of each pure member imperfection (+ or -) is defined according to the shape of the eigenbuckling mode. It is important to point out that only one out-of-plumb direction is considered (positive) and that all the combinations of positive and negative imperfections will produce different results in the nonlinear analyses since out-of-plumb is asymmetrical. The shape of two different types of combined member imperfections is shown in Figure 16. \begin{table}
\begin{tabular}{|l|l|l|l|l|l|} \hline
**Type of Imperfection** & **T** & **B** & **AD** & **SD** & **Combined \({}^{*}\)** \\ \hline S & 0 & 0 & 0 & 0 & 0 \\ \hline S/T+ & 1.5 & 0 & 0 & 0 & 1.5 \\ \hline S/T\({}^{-}\) & 1.5 & 0 & 0 & 0 & 1.5 \\ \hline S/T+/AD1+ & 1.5 & 0 & 2.5 & 0 & 2.69 \\ \hline S/T+/AD1\({}^{-}\) & 1.5 & 0 & 2.5 & 0 & 3.48 \\ \hline S/T\({}^{-}\)/AD1+ & 1.5 & 0 & 2.5 & 0 & 3.48 \\ \hline S/T\({}^{-}\)/AD1\({}^{-}\) & 1.5 & 0 & 2.5 & 0 & 2.69 \\ \hline S/B+/SD1+ & 0 & 1.5 & 0 & 2.5 & 2.94 \\ \hline S/T+/AD2\({}^{+}\) & 1.5 & 0 & 0.625 & 0 & 1.52 \\ \hline S/T+/AD2\({}^{-}\) & 1.5 & 0 & 0.625 & 0 & 1.65 \\ \hline S/T\({}^{-}\)/AD2\({}^{+}\) & 1.5 & 0 & 0.625 & 0 & 1.65 \\ \hline S/T\({}^{-}\)/AD2\({}^{-}\) & 1.5 & 0 & 0.625 & 0 & 1.52 \\ \hline S/B+/SD2+ & 0 & 1.5 & 0 & 0.625 & 1.5 \\ \hline \end{tabular} \({}^{*}\) It is noted that the maximum amplitude of member and sectional imperfections do not occur at the same cross-section; consequently, the value in the last column is not equal to the sum of the values presented in the previous columns.  
\end{table} 
Table 3: Imperfection magnitudes (mm) of the pure member modes and combinations. The results of all GBT geometrically nonlinear analyses are compared in Table 4. Firstly, similar global lateral displacement values are observed in all cases. Differences in displacements are only significant at the member level (in the segments of the uprights between joints). These differences lead to major changes in other aspects of the rack behaviour. Specifically, the longitudinal membrane stress distribution clearly changes between simulations.