## 4 Results and Discussions  
### Pavement Parameters and Random Variable Parameters  
Taking a viscoelastic three-layered system under a rectangular uniform moving load as an example, the asphalt pavement material and structural parameters are listed in Table 1. The viscoelastic properties of the asphalt layer were described by Prony series and WLF equations which are displayed in Tables 2 and 3, respectively. The instantaneous elastic modulus of the asphalt layer is 13,551.76 MPa. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Modulus (MPa)** & **Poisson’s Ratio** & **Density (kg/m3)** & **Depth (m)** \\ \hline Asphalt layer & - & 0.35 & 2400 & 0.2 \\ Base layer & 1000 & 0.3 & 2000 & 0.4 \\ Subgrade & 200 & 0.4 & 1500 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Asphalt pavement material and structural parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(X_{1}\) (MPa) & \(X_{2}\) (m) & \(X_{3}\) (m/s) & \(X_{4}\) (\({}^{\circ}\)C) \\ \hline Mean value & 0.7 & 0 & 25 & 20 \\ Standard deviation & 0.1 & 0.2 & 3 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean values and standard deviations of random variables. \begin{table}
\begin{tabular}{c c} \hline \hline \(T_{r\prime}\) (\({}^{\circ}\)C) & \(C_{1}\) & \(C_{2}\) \\ \hline
20 & 21.6 & 170.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: WLF parameters of the asphalt layer. Figure 3: Double rectangular uniform moving load. \begin{table}
\begin{tabular}{c c} \hline \hline \(\tau_{j}\) & \(g_{j}\) \\ \hline
1000 & 0.017837 \\
100 & 0.022262 \\
10 & 0.044653 \\
1 & 0.066893 \\
0.1 & 0.104727 \\
0.01 & 0.105207 \\
0.001 & 0.164525 \\
0.0001 & 0.002653 \\
0.00001 & 0.451526 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Prony series parameters of the asphalt layer. The results of the cumulative probability distributions of longitudinal tensile strain at \(A_{1}\) and vertical compressive strain at \(B_{1}\) are displayed in Figure 5. This figure presents the trends of the cumulative probability distributions of longitudinal tensile and vertical compressive strains calculated through the combined RSM and FORM, which are consistent with those using MCS. Thus, the combined RSM and FORM is proven to be validated for the probability calculation of mechanical responses. Compared with MCS, the probability calculation procedure based on the combined RSM and FORM is recommended for reasons of computational efficiency. Specifically, the calculation process for the probability of being less than the target mechanical response \(R_{\text{f}}\) based on the combined RSM and FORM only takes approximately one minute, while the calculation process for the cumulative probability distribution of mechanical responses by utilizing the MCS of 1000 random samples requires more than one hour. Deterministic results may be acquired for the cumulative probability distributions of mechanical responses when standard deviations of the random variables equal zero (i.e., deterministic problem), and are also shown in Figure 5. These deterministic results could not fully reflect the random characteristics of pavement mechanical responses under realistic random moving loads. The results of the probability of being less than the target mechanical responses for longitudinal tensile and vertical compressive strains at different calculation points are depicted in Figure 6. Figure 7 displays the deterministic results for longitudinal tensile and vertical compressive strains at different calculation points. As can be observed from Figures 6 and 7, deterministic results are a special case of probabilistic results. Moreover, the probability that the point sustains a significant mechanical response is high when the probability of being less than a target mechanical response at a certain point is low.