## 3 Numerical Validation and Parametric Analysis
### The E-B Model  
The beam, foundation, and vehicle load parameters for the numerical simulation of the E-B beam model are shown in Table 1.  
Figure 2 presents the variation in tire contact force over time. It can be observed from the figure that when the pavement roughness is a sine function, the tire contact force is a harmonic load that varies with time. Using the parameters from Table 1, we calculate the vertical displacement of the pavement through Equation (19). According to Equation (18), it can be derived that  
\[ \begin{split}\|w_{0}\|=1.83\times 10^{-3},\quad\|w_{1}\|=3.08 \times 10^{-4},\quad\|w_{2}\|=1.56\times 10^{-4},\\ \phi_{0}=\frac{\|w_{1}\|}{\|w_{0}\|}=\frac{3.08\times 10^{-4}}{1.83 \times 10^{-3}}\approx 0.168<1,\\ \phi_{1}=\frac{\|w_{2}\|}{\|w_{1}\|}=\frac{1.56\times 10^{-4}}{3.08 \times 10^{-4}}\approx 0.506<1.\end{split}  \tag{26}\]  
Thus, the analytical solution derived from Equation (19) quickly converges to the precise solution of Equation (10). Figure 3 illustrates a comparison of the analytical solution for Equation (10) with numerical integration outcomes. It demonstrates that for this problem, summing the initial three terms of \(w_{j}\) provides a close approximation to the actual value. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Item** & **Notation** & **Value** \\ \hline \multirow{4}{*}{Beam} & [PERSON]’s modulus & \(E\) & \(6.998\times 10^{9}\) Pa \\  & Mass per unit Length & \(m\) & \(711.9\) kg/m \\  & Sectional moment of inertia & \(I\) & \(2.25\times 10^{-3}\) m\({}^{4}\) \\ \hline \multirow{2}{*}{Roughness} & Wavelength & \(L_{0}\) & \(10\) m \\  & Amplitude & \(A_{0}\) & \(0.01\) m \\ \hline \multirow{4}{*}{Foundation} & Linear stiffness & \(k\) & \(8\times 10^{6}\) N/m\({}^{2}\) \\  & Nonlinear stiffness & \(k_{N}\) & \(8\times 10^{11}\) N/m\({}^{4}\) \\  & Viscous damping & \(c\) & \(3\times 10^{5}\) N/s/m\({}^{2}\) \\ \hline \multirow{4}{*}{Vehicle} & Wheel mass & \(m_{1}\) & \(550\) kg \\  & Suspension mass & \(m_{2}\) & \(4450\) kg \\  & Tire stiffness & \(K_{1}\) & \(1.75\times 10^{6}\) N/m \\  & Suspension stiffness & \(K_{2}\) & \(1\times 10^{6}\) N/m \\  & Tire damping & \(C_{1}\) & \(2\times 10^{3}\) N/s/m\({}^{2}\) \\  & Suspension damping & \(C_{2}\) & \(1.5\times 10^{4}\) N/s/m\({}^{2}\) \\  & velocity & \(v\) & \(30\) m/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the E-B beam, foundation [11] and vehicle [45]. Figure 2: The tire contact force exerted by the vehicle on the pavement. Next, we focus on analyzing the impact of parameters on the vertical displacement of the beam using Equation (19). Figure 4 shows the impact of pavement roughness amplitude variation on pavement deformation at different vehicle velocities, with all other parameters taking the values from Table 1. It can be seen from the figure that at lower vehicle velocity, the pavement deformation increases as the amplitude \(A_{0}\) increases. However, when \(A_{0}\) continues to increase, the pavement deformation remains almost unchanged. At higher vehicle velocity, the pavement deformation decreases as \(A_{0}\) increases. But when \(A_{0}\) continues to grow, the pavement deformation is hardly affected. Figure 5 demonstrates how varying vehicle velocities and the nonlinear stiffness of the foundation affect the pavement response, given that \(A_{0}\) = 0.01 m and all other parameters are fixed as per the values in Table 1. From Figure 5, it can be observed that when the coefficient \(k_{N}\) is relatively small, it has almost no effect on the vertical deformation of the pavement; as \(k_{N}\) increases, the maximum displacement of the pavement decreases slightly; when \(k_{N}\) is further increased, there is a significant and substantial increase in the maximum displacement of the pavement. Comparing the pavement displacement at different velocities in Figure 5, the value of the nonlinear stiffness that affects displacement increases with increasing velocity.