## 3 Analysis and Results
### RSM Results for Single, Tandem, and Tridem Axles
#### 3.1.2 Tandem Axle with Dual Tires  
The stresses induced by a tandem axle with dual tires were evaluated using the PITRA PAVE software. As shown in Table 4, the resulting stress values exhibit a clear decrease as the measurement depth increases. Additionally, the spacing between tires in both the Y and X directions plays a significant role in stress development. Specifically, reduced spacing tends to result in higher stress levels within a certain range. To better understand this relationship, a statistical analysis was performed using response surface methodology (RSM), focusing on the depth at which the stress was measured and the tire spacing. This method enabled the construction of a predictive model that effectively captures the complex dynamics of stress distribution. The use of a base 10 logarithmic transformation further enhanced the model's accuracy by stabilizing variance and improving the overall fit. The fit summary provided in Table 5 suggests the use of a quadratic model, while the cubic model was excluded due to aliasing issues. The quadratic model was chosen because of its significant \(p\)-value and high adjusted R\({}^{2}\) value, both of which indicate a strong correlation between the key influencing factors and the response variable. The detailed ANOVA results for the quadratic model are shown in Table 6, underscoring the significant impact of both tire spacing and stress measurement depth on stress development, as evidenced by consistently low \(p\)-values below 0.05. Moreover, the fit statistics summarized in Table 6 reveal only a minimal difference between the adjusted and predicted R\({}^{2}\) values, which further confirms the model's reliability. The adequate precision value, exceeding 4, reinforces the model's robustness, making it a reliable tool for practical applications in pavement design. \[\sigma_{z}=10^{(2.37052-0.057929-A-0.0038622-B-0.020319-C-0.000087+A-B+0.000267 +A-C-0.000025-B-C+0.000043-B^{2}+0.000059-C^{2})}*\left(\frac{\text{{tire load}}}{4500}\right) \tag{2}\]  
where:  
\(\sigma_{z}=\text{Vertical stress}\);  
A = thickness of the pavement;  
B = spacing between tires in the X direction;  
C = spacing between tires in the Y direction. Figure 4 shows the residual plots for the quadratic model, including the normal probability plot, residuals versus predicted values, residuals versus run, and actual versus predicted values. These plots are crucial for evaluating the model's performance through the assumptions. The normal probability plot is specifically used to assess the distribution of the residuals. As seen in Figure 4, the residuals are close to a straight line, indicating that the errors are normally distributed. This alignment suggests that the quadratic model is well suited for the data. Homoscedasticity, which refers to the assumption of equal variance across the range of predicted values, is examined through the plot of externally studentized residuals versus predicted values. As depicted in Figure 4, the residuals are symmetrically distributed around zero, with no apparent pattern or systematic deviation. This symmetry indicates that the model's predictions are consistent across the range of data. Autocorrelation can undermine the model's effectiveness by indicating that the residuals are not independent, which is another key assumption for regression models. In Figure 4, the residuals appear to transition randomly between positive and negative values, with no discernible pattern. This randomness suggests that there is no autocorrelation, meaning the stresses are well captured by the independent variables in the model. Finally, the plot of actual versus predicted values demonstrates the model's predictive accuracy. As shown in Figure 3, the data points closely align along the 1:1 line, indicating a strong correlation between the actual values and the values predicted. The strong agreement between actual and predicted values indicates the model's ability to predict accurately.