## 3 Numerical Validation and Parametric Analysis
### The Timoshenko Model
\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 \\  & Shear modulus & \(G\) & \(7.7\times 10^{10}\) Pa \\  & Mass density & \(\rho\) & \(2373\) kg/m\({}^{3}\) \\  & Cross sectional area & \(A\) & \(0.3\) m\({}^{2}\) \\  & Sectional moment of inertia & \(I\) & \(2.25\times 10^{-3}\) m\({}^{4}\) \\  & Shear coefficients & \(\lambda\) & \(0.4\) \\ \hline \multirow{2}{*}{Roughness} & Wavelength & \(L_{0}\) & \(10\) m \\  & Amplitude & \(A_{0}\) & \(0.01\) m \\ \hline \multirow{4}{*}{Foundation} & Linear stiffness & \(k_{1}\) & \(8\times 10^{6}\) N/m\({}^{2}\) \\  & Nonlinear stiffness & \(k_{3}\) & \(8\times 10^{11}\) N/m\({}^{4}\) \\ \cline{1-1}  & Viscous damping & \(c\) & \(3\times 10^{5}\) N\(\cdot\)s/m\({}^{2}\) \\ \cline{1-1}  & Shear parameter & \(G_{p}\) & \(6.669\times 10^{7}\) N \\ \cline{1-1}  & Rocking stiffness & \(k_{f}\) & \(1\times 10^{8}\) N/m\({}^{2}\) \\ \cline{1-1}  & Rocking damping & \(c_{f}\) & \(1.5\times 10^{6}\) N\(\cdot\)s/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the Timoshenko beam and foundation [11]. Figure 9: The first three terms of \(w_{j}\) and the comparison between analytical solution in Equation (25) and numerical result for Equation (20). (**a**) \(w_{0}\). (**b**) \(w_{1}\). (**c**) \(w_{2}\). (**d**) Comparision of two solutions. However, when using a Timoshenko beam for the simulation, such offset is negligible and almost ignorable. Figures 13 and 14 illustrate the effects of different viscous damping coefficients and nonlinear stiffness coefficients of the foundation on the deformation of the pavement at various vehicle velocities, using all other parameters as specified in Tables 1 and 2. The two figures reveal that, across a range of foundation damping coefficients and vehicle velocities, a critical value for the nonlinear stiffness coefficient persists. From these figures, it is evident that below a certain critical threshold, an increase in \(k_{3}\) has minimal impact on pavement deformation. However, once \(k_{3}\) surpasses this critical value, initially, the pavement deformation experiences a slight reduction, followed by a marked increase as \(k_{3}\) continues to rise. The extent of this critical threshold is affected by the foundation's damping coefficient and the vehicle's velocity, with higher values of velocity and damping leading to a higher critical threshold. Figure 11: The impact of different vehicle velocities and the nonlinear stiffness coefficient 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 10: The effects of varying vehicle velocities and amplitudes of pavement roughness on the pavement’s reactive performance. (**a**) \(v\) = 30 m/s. (**b**) \(v\) = 90 m/s. Figure 12: The impact of different vehicle velocities and the nonlinear stiffness coefficient 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. The subsection's parameter analysis reveals that the effect of the foundation's nonlinear stiffness on pavement deformation during vehicle movement is intricate. For the nonlinear stiffness coefficient \(k_{3}\), a critical threshold, referred to as \(k_{30}\), is identified. Should \(k_{3}\) be less than \(k_{30}\), it minimally affects pavement deformation; nevertheless, if \(k_{3}\) exceeds \(k_{30}\), the deformation will initially drop and then ascend in tandem with the increase in \(k_{3}\). The value of \(k_{30}\) is not static, it fluctuates with variables such as vehicle velocity, the foundation's damping coefficient, and the amplitude of pavement excitation. Specifically, an uptick in velocity and damping within the foundation notably raises \(k_{30}\).