## 3 Analysis and Results
### RSM Results for Single, Tandem, and Tridem Axles
#### 3.1.3 Tridem Axle with Dual Tires  
The stress of a tridem axle with dual tires was analyzed using the PITRA PAVE software. As shown in Table 7, with increasing depth, the stresses generated by the loads decrease significantly. The distance between the tires in both the Y and X directions plays a significant role in the stress levels. The smaller the distance, the greater the stress. To identify this relationship, a statistical analysis was conducted using response surface methodology (RSM). This analysis aims to develop a predictive relationship for stress based on loads, pavement depth, and tire spacing. A base 10 logarithmic transformation was applied to enhance model accuracy by stabilizing variance. This is because, without the transformation, the performance of the models was poor. The fit summary in Table 8 shows that the quadratic model is suggested, while the cubic model is rejected. The quadratic model's significant \(p\)-value and high adjusted R\({}^{2}\) indicate a strong correlation between tire spacing, depth, and stress response. The ANOVA results listed in Table 9 describe the significant influence of tire spacing and measurement depth on stress development, with \(p\)-values consistently below 0.05. Additionally, the fit statistics in Table 7 show a minimal difference between the adjusted and predicted R\({}^{2}\) values. The adequate precision value, exceeding 4, indicates the reliability of the model. Equation (3) shows the quadratic model. \[\sigma_{z}=10^{(2.40058-0.0642388-A-0.026072-B-0.005852-C+0.0002068-A-B-0.000149 -A+C-0.000013-B+C+0.000725-A^{2}+0.000386-B^{2}+0.00007-C^{2})} \tag{3}\]  
\[\sigma_{z}=\text{Vertical stress};\]  
A = thickness of the pavement;  
B = spacing between tires in the Y direction;  
C = spacing between tires in the X direction. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & **F-Value** & \(p\)**-Value** & **Performance** \\ \hline Model & 30.07 & 9 & 3.34 & 951.31 & \textless{}0.0001 & significant \\ A-A & 2.41 & 1 & 2.41 & 684.99 & \textless{}0.0001 & \\ B-B & 0.0547 & 1 & 0.0547 & 15.58 & 0.0002 & \\ C-C & 0.0583 & 1 & 0.0583 & 16.6 & 0.0001 & \\ AB & 0.0726 & 1 & 0.0726 & 20.66 & \textless{}0.0001 & \\ AC & 0.0574 & 1 & 0.0574 & 16.36 & 0.0001 & \\ BC & \(7.8\times 10^{-6}\) & 1 & \(7.8\times 10^{-6}\) & 0.0022 & 0.9626 & \\ A\({}^{2}\) & 2.4 & 1 & 2.4 & 684.32 & \textless{}0.0001 & \\ B\({}^{2}\) & 0.0135 & 1 & 0.0135 & 3.85 & 0.053 & \\ C\({}^{2}\) & 0.0076 & 1 & 0.0076 & 2.16 & 0.1451 & \\ Residual & 0.3126 & 89 & 0.0035 & & & \\ Cor Total & 30.38 & 98 & & & & \\  & & & & Fit statistics & & \\ Std. Dev. & 0.0593 & & & R\({}^{2}\) & 0.9897 & \\ Mean & 0.9337 & & & Adjusted R\({}^{2}\) & 0.9887 & \\ C.V. \% & 6.35 & & & Predicted R\({}^{2}\) & 0.9869 & \\  & & & & Adeq Precision & 100.3383 & \\ \hline \end{tabular}
\end{table} 
Table 6: ANOVA for quadratic model for tandem axle model. \begin{table}
\begin{tabular}{l l l l l l} \hline
**Source** & **Sequential \(p\)-Value** & **Adjusted R\({}^{2}\)** & **Predicted R\({}^{2}\)** & **Performance** \\ \hline Linear & \textless{}0.0001 & 0.9027 & 0.8962 & \\
2 FI & 0.2316 & 0.904 & 0.8904 & \\ Quadratic & \textless{}0.0001 & 0.9887 & 0.9869 & Suggested \\ Cubic & \textless{}0.0001 & 0.9971 & 0.9962 & Aliased \\ \hline \end{tabular}
\end{table} 
Table 5: Fit summary of the various models for tandem axle with dual tires. Figure 5 shows the residual plots of the quadratic model. These plots shown in Figure 5 include the normal probability plot, residuals versus predicted values, residuals versus run number, and actual versus predicted values. However, these plots describe how well the model fits the data and predicts the stress accurately. The normal probability plot is essential for assessing whether the residuals, or errors, follow a normal distribution. The data points on this plot should form a straight line, indicating that the residuals are normally distributed.