# 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 Perforations  
The influence of perforations has also been assessed. Additional GBT geometrically nonlinear analyses, including upright perforations, have been carried out. The characteristics of the applied initial geometrical imperfections are described in Table 9. In order to reduce the number of analyses, only the negative twist member imperfection (T-) is considered when combined with sectional imperfections because it results in higher lateral displacements. The same procedure is used to obtain the shape and magnitude of the combined imperfections. However, some differences are found, especially when sectional deformations are included because slightly different eigenshapes are obtained for perforated members. Finally, it is noted that the warping degree of freedom at the floor column cross-section is completely restrained (as in Section 7.1).  
The main results are shown in Tables 10 and 11. The increase in lateral displacement due to perforations is small. Contrarily, the maximum longitudinal membrane stress is larger. This is an expected result since perforations decrease the section area and modulus of the column. Furthermore, differences between longitudinal membrane stress distribution are still found for different types of initial imperfection, especially in the cross-section lips,  
\begin{table}
\begin{tabular}{|p{113.8 pt}|p{113.8 pt}|p{113.8 pt}|} \hline
**Type of Imperfection** & **Max. Lateral Disp (mm) (Ratio Warp. Free/Warp. Fixed)** & **Max. Long Stress (MPa) (Ratio Warp. Free/Warp. Fixed)** \\ \hline S & 6.59 (1.00) & \(-126.77\) (1.01) \\ \hline S/T+ & 6.42 (0.98) & \(-128.16\) (0.98) \\ \hline S/T- & 6.76 (1.02) & \(-132.29\) (1.00) \\ \hline S/T+/AD1+ & 6.33 (0.96) & \(-173.98\) (1.07) \\ \hline S/T+/AD1\(-\) & 6.53 (1.00) & \(-177.39\) (1.06) \\ \hline S/T\(-\)/AD1+ & 6.66 (1.00) & \(-159.25\) (1.01) \\ \hline S/T\(-\)/AD1\(-\) & 6.68 (1.01) & \(-170.3\) (1.06) \\ \hline S/B+/SD1+ & 6.61 (1.00) & \(-166.2\) (0.99) \\ \hline S/T+/AD2+ & 6.39 (0.98) & \(-136.71\) (1.04) \\ \hline S/T+/AD2\(-\) & 6.44 (0.99) & \(-140.57\) (1.07) \\ \hline S/T\(-\)/AD2\(-\) & 6.74 (1.01) & \(-133.51\) (1.01) \\ \hline S/T\(-\)/AD2\(-\) & 6.79 (1.02) & \(-134.71\) (0.99) \\ \hline S/B+/SD2+ & 6.60 (1.00) & \(-132.3\) (1.00) \\ \hline \end{tabular}
\end{table}
Table 7: Maximum lateral displacement and maximum compressive longitudinal membrane for the case with free warping at the floor column cross-section.  
\begin{table}
\begin{tabular}{|p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}|} \hline  & **Axial (M1)** & **Chamber (M2)** & **Bow (M3)** & **Twist (M4)** & **Distortional Modes (M6–10)** & **TOTAL (M1–10)** \\ \hline S & 1 & 0.132 & 0.016 & 0.010 & 0.007 & 1.135 \\ \hline S/T\(-\) & 1 & 0.125 & 0.018 & 0.112 & 0.035 & 1.167 \\ \hline S/T\(-\)/AD1\(+\) & 1 & 0.123 & 0.063 & 0.169 & 0.503 & 1.377 \\ \hline S/T\(-\)/AD1\(-\) & 1 & 0.138 & 0.065 & 0.137 & 0.492 & 1.461 \\ \hline S/B+/DS1\(+\) & 1 & 0.128 & 0.137 & 0.019 & 0.398 & 1.446 \\ \hline S/T\(-\)/AD2\(+\) & 1 & 0.125 & 0.016 & 0.128 & 0.131 & 1.187 \\ \hline S/T\(-\)/AD2\(-\) & 1 & 0.125 & 0.027 & 0.102 & 0.135 & 1.192 \\ \hline \end{tabular}
\end{table}
Table 8: Stress ratios (_SRi_ in Equation (14)) values for free warping degree of freedom at the floor column cross-section.  
On the other hand, in Table 11 (and Figure 24), it can also be observed that the influence of distortional modes on the longitudinal membrane stress distribution seems to be slightly more relevant for perforated members. This can be explained by the fact that, in the particular rack column under study, the inclusion of perforations reduces the stiffness of those distortional deformation modes that play a relevant role in the deformed shape.