## 3 Analysis and Results
### RSM Results for Single, Tandem, and Tridem Axles
#### 3.1.1 Single Axle with Dual Tires  
Table 1 provides a comprehensive summary of the outputs generated by the PITRA PAVE v1.0.0 software. The data indicate that, as the depth of the assessment points increases, the values of the stresses exhibit a significant decline. This trend highlights the relationship between depth and stress distribution within the pavement structure, emphasizing the importance of considering depth in the design and analysis of pavement systems. Moreover, the analysis reveals that reduced spacing between tires correlates with increased stress levels. This phenomenon can be attributed to stress overlap, where closely spaced tires distribute their load over a smaller area, resulting in higher localized stress concentrations. Understanding this relationship is crucial for optimizing tire configurations to minimize pavement damage. Further, to investigate the interactions between tire spacing (in the Y direction), pavement thickness, and the resultant stress response, a statistical evaluation was conducted using response surface methodology (RSM). This approach allowed for the development of a predictive model that assesses stress at various depths within the pavement layer. Importantly, a base 10 logarithmic transformation was applied to the data to enhance the model's performance, as preliminary analyses indicated that the untransformed model did not yield satisfactory results. Table 2 presents the fit summary of the various models generated during this analysis. Notably, the quadratic model emerged as the most effective predictor of stress, supported by its significant \(p\)-value and superior adjusted and predicted R\({}^{2}\) values when compared to alternative models. In contrast, the cubic model was found to be aliased, suggesting that it may not provide reliable predictions within the studied parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Run**} & \multirow{2}{*}{ \begin{tabular}{c} **Y Spacing (A)** \\ **(in)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Thickness (B)** \\ **(in)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Stress** \\ **(psi)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Run** \\ **(in)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Y Spacing (A)** \\ **(in)** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **Thickness (B)** \\ **(in)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Stress** \\ **(psi)** \\ \end{tabular}  } \\ \hline
1 & 14 & 0 & 100.03 & 18 & 7 & 30 & 4.5288 \\
2 & 14 & 5 & 49.749 & 19 & 7 & 35 & 3.3682 \\
3 & 14 & 10 & 19.733 &  
Table 3 presents the ANOVA results for the quadratic model. The analysis reveals that the impact of tire spacing alone on the model is insignificant, as indicated by a \(p\)-value greater than 0.05. Conversely, the effect of pavement thickness is significant, with a \(p\)-value below 0.05, showing its importance in stress response. Additionally, the interaction between pavement thickness and tire spacing was found to be significant. This is because, at depths close to the surface, there is no overlap between stresses but, as the location of analysis goes deeper, there will be an overlap between the developed stresses from the tires. Table 3 also summarizes the model's fit statistics. Notably, the difference between the adjusted and predicted R\({}^{2}\) values is less than 2, indicating strong agreement between the predicted and observed values. The adequate precision value of 76, which far exceeds the threshold of 4, further suggests that the model is not only effective but also capable of reliably navigating the design space, making it a valuable tool for engineers in pavement design. Figure 3 illustrates several important diagnostic plots for the quadratic model: the normal probability plot, residuals versus predicted values, residuals versus run, and predicted versus actual values.