# Global Analysis of Adjustable Pallet Rack Down-Aisle Frames Using the Generalised Beam Theory
## 6 Validation of the GBT Finite Element Model  
This section is aimed at validating the GBT model (Section 5) by comparing it with the shell model (Section 4). Perforated and non-perforated uprights are considered (Figures 13a and 13b, respectively), with holes explicitly modelled with shells and accounted for in GBT throughout the formulation presented in [19]. Geometrically linear and nonlinear static analyses are applied. The following results are compared: (i) The lateral displacements in X direction at each loading level (Figure 13a); (ii) the longitudinal membrane stress distribution in an upright section found at 400 mm from the floor (section \(S\) in Figure 13a). The level displacements in X were selected because they were expected to be significantly influenced by the study parameters (presence of holes and geometric nonlinearity). On the other hand, section \(S\) was selected for being significantly demanded and for being far enough from the connections, whose modelling with GBT and shells is different, leading to possible localised differences. Tables 1 and 2 display the lateral displacements in X (mm) of each loading level (Figure 13a) for models with non-perforated and perforated columns, respectively. Results  
Figure 11: Flexural and torsional global modes for the upright ((**a**) mode 2, (**b**) mode 3 and (**c**) mode 4) and the beam ((**d**) mode 2, (**e**) mode 3 and (**f**) mode 9) cross-section, respectively. Figure 12: GBT beam elemental basis for the upright cross-section. Schematics of the eccentricities determined between the reference point and centroid (\(\overset{\rightarrow}{C}\)) and the reference point and the shear centre (\(\overset{\rightarrow}{S}\)). are displayed for geometrically linear and nonlinear analyses, as well as for the shell and GBT models. The column Diff. (%) presents the differences in percentage between shell and GBT results. Table 1 shows small differences between the two models in geometrically linear analyses, except at the first level. A similar trend can be observed in the nonlinear analysis but with more significant differences. Remarkably, the absolute error remains rather constant (around 0.05 mm and 0.1 mm for linear and nonlinear analysis, respectively), resulting in the highest relative error at the lower level, where the displacement is small. The results obtained are reasonably accurate given the different modelling approaches applied for the joints and the restraining of the cross-ais \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{**Level (Figure 13a)**} & \multicolumn{6}{c|}{**Displacements in \(x\) [mm]**} \\ \cline{2-7}  & \multicolumn{3}{c|}{**Linear**} & \multicolumn{3}{c|}{**Nonlinear**} \\ \cline{2-7}  & **Shell** & **GBT** & **Diff (\%)** & **Shell** & **GBT** & **Diff. (\%)** \\ \hline
1 & 0.97 & 0.93 & \(-4.68\) & 1.34 & 1.25 & \(-7.01\) \\ \hline
2 & 2.28 & 2.23 & \(-2.24\) & 3.19 & 3.05 & \(-4.39\) \\ \hline
3 & 3.44 & 3.39 & \(-1.41\) & 4.79 & 4.62 & \(-3.50\) \\ \hline
4 & 4.29 & 4.25 & \(-1.05\) & 5.92 & 5.74 & \(-3.05\) \\ \hline
5 & 4.84 & 4.80 & \(-0.84\) & 6.62 & 6.43 & \(-2.80\) \\ \hline \end{tabular}
\end{table} 
Table 1: Displacement comparison between shell and GBT models with non-perforated uprights. \begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{**Level (Figure 13a)**} & \multicolumn{6}{c|}{**Displacements in \(x\) [mm]**} \\ \cline{2-7}  & \multicolumn{3}{c|}{**Linear**} & \multicolumn{3}{c|}{**Nonlinear**} \\ \cline{2-7}  & **Shell** & **GBT** & **Diff (\%)** & **Shell** & **GBT** & **Diff (\%)** \\ \hline
1 & 1.01 & 0.95 & \(-5.71\) & 1.40 & 1.29 & \(-8.09\) \\ \hline
2 & 2.33 & 2.28 & \(-1.87\) & 3.28 & 3.15 & \(-4.22\) \\ \hline
3 & 3.48 & 3.46 & \(-0.46\) & 4.89 & 4.76 & \(-2.67\) \\ \hline
4 & 4.32 & 4.32 & 0.03 & 6.01 & 5.90 & \(-1.85\) \\ \hline
5 & 4.84 & 4.88 & 0.62 & 6.68 & 6.59 & \(-1.35\) \\ \hline \end{tabular}
\end{table} 
Table 2: Displacement comparison between shell and GBT models with perforated uprights. Figure 13: Models for validation.