# 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 Floor Connection Boundary Conditions  
The characteristics of the upright-to-floor connection play a relevant role in the structural behaviour of pallet rack structures in the down-aisle direction. Commonly, this connection presents a semi-rigid flexural behaviour, and it is modelled with a rotational  
Figure 21: Longitudinal membrane stress distribution (S/T\(-\)/AD1\(-\) member imperfection) of the (a) axial, (b) camber, (c) bow, (d) twist, (e) distortional modes, and (f) with all deformation modes involved in the analysis. \begin{table}
\begin{tabular}{|l|l|l|l|l|l|l|} \hline  & \multicolumn{1}{c|}{**Axial (M1)**} & \multicolumn{1}{c|}{**Camber (M2)**} & \multicolumn{1}{c|}{**Bow (M3)**} & \multicolumn{1}{c|}{**Twist (M4)**} & \multicolumn{1}{c|}{**Distortional Modes (M6-10)**} & \multicolumn{1}{c|}{**TOTAL**} \\ \hline S & 1 & 0.132 & 0.016 & 0.009 & 0.012 & 1.129 \\ \hline S/T\(-\) & 1 & 0.128 & 0.018 & 0.071 & 0.043 & 1.183 \\ \hline S/T\(-\)/AD1\(+\) & 1 & 0.122 & 0.085 & 0.151 & 0.468 & 1.364 \\ \hline S/T\(-\)/AD1\(-\) & 1 & 0.126 & 0.074 & 0.105 & 0.417 & 1.396 \\ \hline S/B\(+\)/DS1\(+\) & 1 & 0.127 & 0.138 & 0.019 & 0.420 & 1.454 \\ \hline S/T\(-\)/AD2\(+\) & 1 & 0.127 & 0.030 & 0.090 & 0.137 & 1.167 \\ \hline S/T\(-\)/AD2\(-\) & 1 & 0.128 & 0.015 & 0.069 & 0.116 & 1.219 \\ \hline \end{tabular}
\end{table} 
Table 5: Stress ratios _SRi_ (Equation (14)) of the most meaningful sectional modes. spring (linear or nonlinear). Furthermore, its stiffness (moment-flexural rotation) depends on the upright axial internal force. Thus, several experimental tests are carried out at different axial load levels. On the other hand, although the sectional deformation of the cross-section is usually restrained at the upright-to-floor connection, the warping degree of freedom can present diverse levels of constrain (from free to completely restrained), depending on its design characteristics. Consequently, additional analyses with free warping displacement (for torsional and distortional modes) of the column cross-section at the floor connection were carried out to study the sensitivity of the model results. In these additional analyses, the sectional and member imperfections of the rack column were obtained following the same procedure described in Section 7.1, but with released warping at the upright bottom section. Notice that the shape of these imperfections can be different than those presented in Figure 15 due to the change in the boundary conditions. The shape and magnitude of the imperfections can be observed in Figure 22a,b and Table 6; Figure 22c displays the resulting warping torsional moment diagrams. The main results of these analyses are shown in Tables 7 and 8, similarly to Tables 4 and 5. The change in the boundary conditions seems to have a marginal effect on the maximum global sway displacement. Contrarily, higher maximum membrane longitudinal \begin{table}
\begin{tabular}{|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 & 3.13 \\ \hline S/T\(-\)/AD1\(-\) & 1.5 & 0 & 2.5 & 0 & 3.46 \\ \hline S/T\(-\)/AD1\(-\) & 1.5 & 0 & 2.5 & 0 & 3.13 \\ \hline S/B+/SD1\(+\) & 0 & 1.5 & 0 & 2.5 & 2.94 \\ \hline S/T+/AD2\(+\) & 1.5 & 0 & 0.625 & 0 & 1.63 \\ \hline S/T+/AD2\(-\) & 1.5 & 0 & 0.625 & 0 & 1.96 \\ \hline S/T\(-\)/AD2\(+\) & 1.5 & 0 & 0.625 & 0 & 1.96 \\ \hline S/T\(-\)/AD2\(-\) & 1.5 & 0 & 0.625 & 0 & 1.63 \\ \hline S/B+/SD2\(+\) & 0 & 1.5 & 0 & 0.625 & 1.52 \\ \hline \end{tabular}
\end{table} 
Table 6: Imperfection magnitudes (mm) of the pure member modes and combinations. Load case with a free warping degree of freedom at the floor column cross-section. Figure 22: The shape of (**a**) S/T-/AD1- and (**b**)S/B+/SD1+ member imperfection with the free warping degree of freedom at the upright-to-floor connection.