## 2 Numerical and Analysis Methods
### Inhomogeneous Distribution of Horizontal Subgrade  
To simulate the spatial effects of the horizontal distribution of the inhomogeneous subgrade, the factors of the distribution range and reaction modulus of the inhomogeneous subgrade are considered in this paper. The pavement structure model is divided into 2 \(\times\) 2 areas, as shown in Figure 2, A, B, C and D, which represent the different subgrades with various reaction moduli. In this model, Y1 and Y2 represent the sizes of different subgrades along the vertical direction, and X1 and X2 represent the sizes of different subgrades along the horizontal direction. Table 5 presents the reaction modulus of the inhomogeneous subgrade based on various conditions.  
The change rate of inhomogeneous subgrade support \(\beta\) is introduced to represent the level of the horizontal inhomogeneous distribution in the pavement subgrade:  
\[\beta=\frac{\mathrm{E}_{max}-\mathrm{E}_{min}}{\mathrm{E}_{max}} \tag{2}\]  
where \(\beta\) represents the change rate of inhomogeneous subgrade support, \(\beta\) = 0 represents the uniform subgrade, \(\mathrm{E}_{max}\) represents the maximum value of the reaction modulus in the inhomogeneous subgrade, and \(\mathrm{E}_{min}\) represents the minimum value of the reaction modulus  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Aircraft Model** & **Tire Pressure (MPa)** & **Wheel Print Area (m\({}^{2}\))** & **Wheel Print Length (m)** \\ \hline A320 & 1.14 & 0.16 & 0.4 \\ B737-800 & 1.47 & 0.13 & 0.36 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Load size and value of aircraft wheels.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pavement Layer** & \begin{tabular}{c} **Elasticity** \\ **Modulus** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **(kg\(\cdot\)m\({}^{-3}\))** \\ \end{tabular} & **Thickness (m)** & \
\begin{tabular}{c} **Resilience** \\ **Modulus** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Reaction Modulus** \\ **(10\({}^{7}\) MPa)** \\ \end{tabular} \\ \hline cement concrete & 36,000 & 0.15 & 2400 & 0.36, 0.38, 0.40 & / & / \\ pavement layer & & & & & & \\ cement stabilized & & & & & & \\ macadam base & / & 0.25 & / & 0.2 & 1500 & / \\ natural sand cushion & / & 0.3 & / & 0.3 & 200 & \\ subgrade & / & / & / & / & / & 10, 20, 40, 60, 70 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties of pavement structures.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Aircraft Model** & **B737-800** & **A320** \\ \hline Weight (kN) & 792.60 & 774.00 \\ Partition coefficient of landing gear & 0.95 & 0.95 \\ Distance of landing gear (m) & 5.72 & 7.60 \\ Number of landing gear wheel & 2 & 4 \\ Number of landing gear & 2 & 2 \\ S\({}_{\mathrm{t}}\) & 0.86 & 0.78 \\ Spacing of landing gear wheel (m) & S\({}_{\mathrm{L}1}\) & / & 1.01 \\ S\({}_{\mathrm{L}2}\) & / & / \\ Tire pressure (MPa) & 1.47 & & 1.14 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Wheel load parameters of different aircraft types.  
in the inhomogeneous subgrade. The thicknesses of the pavement slab are considered as 0.36 m, 0.38 m and 0.40 m, respectively.