## 2 Theories and Methods  
### Load Model of a Pavement Slab under an Aircraft Load  
To establish a pavement loading model that is consistent with actual conditions, the stress distribution of the aircraft load on the pavement, and the random load under the irregularity excitation, are considered. The theoretical model is established by combining elastic lamination layered with an aircraft vibration model. The establishment procedure is specified as follows:  
(1) Aircraft load model simulation  
The elastic layered Boussinesq calculation theory [32; 33; 34] is used to determine the effect of an aircraft load on the pavement. Based on this, it is evident that the impact load and inertial load of the aircraft on the pavement will be caused by the vibration of the fuselage when the aircraft is taxiing at high speed. By deducing the simplified link between the international roughness index IRI and the pavement power spectral density (PSD), the quarter vehicle model is utilized as a reference [35; 36], and Figure 1 depicts the analytical diagram of the aircraft load operating on the pavement slab. Furthermore, assuming that the geometry and materials of the transverse model are uniform along the runway structure, and that the displacement between the layers of the runway structure is continuous, the aircraft wheel load is simplified as a rectangular load. Figure 1 depicts the analytical diagram of the aircraft load operating on the pavement slab. The analytical expression of displacement and stress at any point obtained from [PERSON]'s stress study in elastic half-space is as follows:  
\[\sigma_{d}=\frac{3P}{2\pi}\bullet\frac{Z^{2}}{R^{5}}=\frac{3P}{2\pi R^{2}}\cos ^{3}\theta \tag{1}\]  
Substituting \(R=\sqrt{x^{2}+y^{2}+z^{2}}\) into the above equation, the following can be obtained. \[\sigma_{d}=\frac{3P}{2\pi\bullet z^{2}\big{[}(x^{2}+y^{2})/z^{2}+1\big{]}^{5/2}} \tag{2}\]  
Figure 1: Schematic diagram of the aircraft load pavement slab and quarter aircraft wheel load vibration model. where \(P\) is the axle load of different types of aircraft; \(x\), \(y\), and \(z\) are the relative coordinates of each wheel load of the aircraft model to any point \(Q\) (\(x\), \(y\), \(z\)) in the base structure. In Equations (1) and (2), the origin of coordinates is located at point \(O\) of Figure 1b, that is, \(\sigma_{d0}\) = (0,0,0). The random load on the runway is regarded as a Gaussian distribution, and the relationship between IRI and DLC (Dynamic Load Coefficient) is established by using the quarter aircraft vibration model [37; 38]:  
\[DLC_{v}=1+\{2.56v^{\frac{w-1}{2}}n_{0}^{\frac{w}{2}} IRI[\int_{0}^{\infty}|(k_{1}+ic_{1}\omega)\bullet H_{1}(\omega)|^{2}f^{- \omega}df]^{\frac{1}{2}}\}/\big{|}a_{0}(m_{1}+m_{2})g] \tag{3}\]  
where \(H_{1}(\omega)\) is the frequency response function of the mass of the non-suspension system, as follows:  
\[ \begin{array}{c}H_{1}(\omega)=[\omega^{4}-i\alpha_{2}(1+h)\omega^{3}-\beta_ {2}(1+h)\omega^{2}]/[\omega^{4}-i(\alpha_{1}+h\alpha_{2}+\alpha_{2})\omega^{3 }\\ -(\alpha_{1}\alpha_{2}+\beta_{1}+\beta_{2}+h\beta_{2})\omega^{2}+i(\alpha_{1} \beta_{2}+\alpha_{2}\beta_{1})\omega+\beta_{1}\beta_{2}]\end{array}  \tag{4}\]  
Considering the natural frequency range of the aircraft, the speed range of the aircraft during take-off and landing is converted into a time-frequency range of 2-16 Hz, with an upper limit of \(f_{u}\) = 35 Hz and a lower limit of \(f_{t}\) = 0.5 Hz, which includes the natural frequency distribution range of the aircraft. To ensure the applicability of the dynamic load factor, the aircraft vibration model's parameters must be harmonized. This paper selects the relevant parameters of B737-800 as the standard parameters of an aircraft model [39; 40; 41], as shown in Table 1.  
The relationship between the dynamic load coefficient \(K\) and IRI when the aircraft is running on the runway is obtained, as follows (5):  
\[K=1+11.5c_{o} IRI\sqrt{v} \tag{5}\]  
where \(c_{0}\) is a constant, and \(c_{0}\) = \(10^{-3}m^{-0.5}s^{0.5}\); \(v\) is the speed of the aircraft.