# 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
The results indicate that adding a distortional member imperfection produces significantly higher maximum longitudinal membrane stress values, especially when the highest distortional member imperfection magnitude is used (t). Similarly, and as a consequence, the internal forces are also found to depend on the imperfection shape and magnitude. The sensitivity of the flexural (camber) and warping torsional (bi-moment) internal moments diagrams to the imperfections is shown in Figures 17 and 18 and Figures 19 and 20, respectively. Remarkably, the effect of the imperfections on the warping torsional moment can be significant (see Figure 19). Figure 16: Shape of (a) S/T\(-\)/AD1\(-\) and (b) S/B+/SD1+ member imperfection (the magnitude of the imperfection modes has been magnified for easier observation). Figure 15: Shape of a pure twist/torsion (a), bow (b), anti-symmetric distortional (c), and symmetric distortional member imperfection (d). The magnitude of the imperfection modes has been magnified for easier observation. Figure 17: Flexural moment internal force diagram along the column length according to the type of initial geometrical imperfection with a sectional member amplitude of t.  
Figure 18: Flexural moment internal force diagram along the column length according to the type of initial geometrical imperfection with a sectional member amplitude of 0.25 \(\times\) t. \begin{table}
\begin{tabular}{|l|l|l|} \hline
**Type of Imperfection** & **Max Lateral Disp (mm)** & **Max Long Stress (MPa)** \\ \hline S & 6.59 * & \(-\)126.07 \\ \hline S/T\(+\) & 6.53 & \(-\)131.11 \\ \hline S/T\(-\) & 6.64 & \(-\)132.94 \\ \hline S/T\(+\)/AD1\(+\) & 6.57 & \(-\)161.94 \\ \hline S/T\(+\)/AD1\(-\) & 6.51 & \(-\)167.80 \\ \hline S/T\(-\)/AD1\(+\) & 6.67 & \(-\)158.27 \\ \hline S/T\(-\)/AD1\(-\) & 6.61 & \(-\)161.12 \\ \hline S/B\(+\)/SD1\(+\) & 6.60 & \(-\)167.51 \\ \hline S/T\(+\)/AD2\(+\) & 6.54 & \(-\)131.60 \\ \hline S/T\(+\)/AD2\(-\) & 6.52 & \(-\)131.74 \\ \hline S/T\(-\)/AD2\(+\) & 6.65 & \(-\)131.63 \\ \hline S/T\(-\)/AD2\(-\) & 6.64 & \(-\)136.71 \\ \hline S/B\(+\)/SD2\(+\) & 6.59 & \(-\)131.67 \\ \hline \end{tabular}  
* Note: The values presented in Tables 1 and 2 represent the lateral displacements at each level. It is important to note that these values may not coincide with the maximum displacement of the column, which is provided in the present table.  
\end{table} 
Table 4: Comparison of maximum lateral displacement and maximum compressive longitudinal membrane stress of the rack column. It is noted that the analyses were carried out considering the most relevant member and sectional GBT deformation modes. It was observed that the most significant changes in the analysis results are not caused by whether a deformation mode is active or not but rather by the inclusion of its corresponding imperfection. For instance, a comparison of the results obtained with AD1 and AD2 imperfection magnitudes in Table 4 illustrates this point. It can be concluded that imperfections are important in activating the distortional modes. This may be a consequence of the particular characteristics of the structure being analysed, GBT model simplifications applied and type and level of loading, as briefly discussed in the last section of the article. One of the main trademarks of GBT is its modal nature, which allows assessing the influence of each deformation mode on the stress distribution of the rack frame column. A stress ratio value (\(SRi\)) (Equation (14)) is determined for each deformation mode, where \(\mid\sigma_{Mi}\mid\) corresponds to the maximum longitudinal membrane stress (in modulus) caused by mode \(i\).