## 3 Numerical Validation and Parametric Analysis
### The E-B Model
In other words, at higher velocities, a greater nonlinear stiffness is required to affect the maximum displacement of the pavement. Figure 6 depicts the influence of varying vehicle velocities and the nonlinear stiffness on the pavement deformation, under the condition that \(A_{0}\) = 0.05 m and with all remaining parameters aligning with the specifications detailed in Table 1. Comparing Figures 5 and 6, the influence of vehicle velocity and nonlinear stiffness on pavement deformation is consistent under different pavement excitation amplitudes. Additionally, the position where  
Figure 4: The impact of different vehicle velocities and pavement roughness amplitudes on the pavement response. (**a**) \(v\) = 30 m/s. (**b**) \(v\) = 90 m/s. Figure 3: Comparison between the approximate analytical solutions of Equation (10) and numerical results when the vehicle velocity is \(v\) = 30 m/s. (**a**) The first three terms of \(w_{j}\). (**b**) Comparison of two solutions. the tire is located is not where the pavement experiences the greatest deformation, and the larger the \(k_{N}\) value, the more pronounced this effect becomes. Figure 5: The impact of different vehicle velocities and the nonlinear stiffness of the foundation on the pavement response when the pavement roughness amplitude is \(A_{0}\) = 0.01 m. (a) \(v\) = 30 m/s. (b) \(v\) = 60 m/s. (c) \(v\) = 90 m/s. (d) \(v\) = 100 m/s. Figure 6: The impact of different vehicle velocities and the nonlinear stiffness of the foundation on the pavement response when the pavement roughness amplitude is \(A_{0}\) = 0.05 m. (a) \(v\) = 30 m/s. (b) \(v\) = 60 m/s. (c) \(v\) = 90 m/s. (d) \(v\) = 100 m/s. Figures 7 and 8 display the impact of varying viscous damping coefficients and the nonlinear stiffness of the foundation on pavement deformation at different vehicle velocities, with all other parameters set to the values listed in Table 1. It can be observed from the two figures that under varying foundation damping coefficients and vehicle velocities, there still exists a critical value for the nonlinear stiffness coefficient. When \(k_{N}\) is less than the critical value, the pavement deformation remains largely unaffected by increases in \(k_{N}\). When \(k_{N}\) exceeds the critical value, the pavement deformation initially decreases and subsequently experiences a significant increase as \(k_{N}\) grows. The critical value is influenced by both the foundation damping coefficient and the vehicle velocity; the greater the velocity and damping coefficient, the larger the critical value. From the analysis of parameter influences in this subsection, it can be concluded that the impact of the foundation's nonlinear stiffness on pavement deformation during vehicle travel is quite complex. For the nonlinear stiffness coefficient \(k_{N}\), there exists a critical value, denoted as \(k_{N0}\). If \(k_{N}<k_{N0}\), changes in \(k_{N}\) have almost no effect on pavement deformation; if \(k_{N}>k_{N0}\), the deformation of the pavement initially diminishes and later escalates with the escalation of \(k_{N}\). The critical value \(k_{N0}\) is not fixed and is mainly related to vehicle velocity, foundation damping coefficient. An increase in vehicle velocity and foundation damping coefficient will significantly increase the value of \(k_{N0}\). Changes in the pavement excitation amplitude do not significantly affect the critical value \(k_{N0}\).