## 3 Results
### Results of the Internal Stresses' Distribution and Trends  
The changes in stress distributions when varying different payloads can provide additional insights into how pallet performance can be affected besides the bending response studied. As previously mentioned, pallets are commonly evaluated by applying a uniformly distributed load on the pallet's top deck. A similar analysis on the pallet analog developed provides a maximum Von Mises equivalent stress of 6.78 MPa on the center span of the structure, when loaded with a uniform pressure of 4.14 kPa. The aforementioned pressure is equivalent to 109 kg of total load for the unit load segment, and the resulting plot is shown in Figure 3. The pallet analog was modelled after PMMA which has a yield strength of 72 MPa [29]. The current internal stresses under uniform loading represent 9.37% of yield. As shown in Figure 4, the Von Mises equivalent stress presents a distribution that closely follows a parabola, with a symmetric distribution along the pallet analog and with the vertex matching the center of the unsupported span. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Cross** \\ **Validation** \\ **Index (k)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Error** \\ **Measurement** \\ \end{tabular} } & \multicolumn{5}{c}{**Model**} \\ \cline{3-7}  & & **3** & **4** & **5** & **6** & **Model** \\  & & **Columns** & **Columns** & **Columns** & **Columns** & **Average** \\ \hline
1 & & 0.26 & 0.06 & 0.10 & 0.05 & 0.12 \\
2 & & 0.22 & 0.17 & 0.07 & 0.04 & 0.12 \\
3 & MAPE & 0.24 & 0.06 & 0.07 & 0.07 & 0.11 \\
4 & & 0.13 & 0.06 & 0.09 & 0.03 & 0.08 \\
5 & & 0.12 & 0.09 & 0.08 & 0.17 & 0.11 \\ \hline
1 & & 0.06 & 0.04 & 0.05 & 0.05 & 0.05 \\
2 & & 0.05 & 0.07 & 0.04 & 0.02 & 0.05 \\
3 & RMSE & 0.08 & 0.03 & 0.04 & 0.06 & 0.06 \\
4 & & 0.05 & 0.04 & 0.06 & 0.03 & 0.05 \\
5 & & 0.10 & 0.05 & 0.06 & 0.09 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Root Means Square Error (RMSE) and Mean Absolute Percent Error (MAPE) for the Gaussian process prediction for each cross validated data set. Figure 3: Plot of the equivalent Von Mises stress (Pa) of the pallet analog under uniformly distributed loading of 4.14 kPa. When discrete loading is applied through the use of stacked corrugated boxes, the stress distribution changes. As shown in Figure 4, the distribution of stresses along the pallet analog changes as the payload characteristics are changed. In this example, each of the payloads have the same characteristics except for the dimensions of the boxes. Unit load height and all coefficients of friction remained constant. Table 8 shows the percent change for the deflection response and for the equivalent Von Mises stresses. Although the change is not completely equivalent, both variables presented a similar trend of change when decreasing the size of the boxes. Figures 5-8 show the equivalent Von Mises stress plots for each of the unit load segments studied. Using the Von Mises stresses for the high-strain, ductile material modelled (PMMA), allows for a reasonable estimation of the yield strength of the pallet analog. The reduction in the peak stresses experienced when comparing the specific payloads versus uniform loading indicate that the pallet analog yield strength can increase from 63.7% (6 columns) to 245% (3 columns). This increase is generated exclusively through the load bridging effect. In analyzing the stress concentrations along the pallet analog, every payload presented high stresses at the ends of the boards, where boxes were contacting the corners of the pallet analog and at the locations of the rigid supports. Since these pressure concentrations were considered a result of the modeling simplification, they can be ignored for the purposes of the load bridging analysis. Excluding these sections, most of the stress distributions followed a clear trend, where the stresses present a change in each of the locations where the boxes' edges contacted the pallet deck.