## 4 Numerically Simulated Dynamic Response of JPCP  
### Structure Model  
The geometric and the material information of each structural layer and the dowel bar parameters of JPCP are summarized in Table 4, which refer to the Chinese concrete pavement design guide [3]. Slab thickness of 220-280 mm is recommended for heavy traffic roads, and correspondingly the elastic modulus and the flexural strength of concrete are recommended to be 31.0 GPa and 5.0 MPa, respectively. The thicknesses of the base layer and the subbase layer are recommended to be between 150-200 mm if their material category is inorganic binder-stabilized aggregate. The elastic modulus of the cement-stabilized aggregate after shrinkage cracking ranges from 1.0 GPa to 2.5 GPa. The elastic modulus of the foundation is recommended to be greater than 60 MPa. The schematic diagram of the numerical model of JPCP is shown in Figure 5.  
Three successive slabs with size of 4 m \(\times\) 5 m are arranged along the vehicle traveling direction to simulate the dynamic response of JPCP under the moving vehicle load. Slab1 and Slab3 are used for initial loading and final unloading, and Slab2 is selected for monitoring the dynamic response at the specific locations of P1 (bottom of mid-slab edge), P2 (bottom of slab center), P3 (top of slab edge 0.25 m from the joint), and P4 (top of joint 1 m from the longitudinal joint). The coordinate system of the numerical model is shown in Figure 5, the x direction is parallel to the direction of vehicle travel, the y direction is parallel to the direction of the transverse joints, and the z direction is perpendicular to the pavement. The damping ratio of the model is taken as 0.05. The effects of both gravity and the dynamic axle load are considered, and the time increment is fixed as 0.002 s. Thus, the signal sampling rate is fixed as 500 Hz. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline \multirow{3}{*}{\(m_{1}\)} & 4450 kg (axle weight = 10 tons, and first natural frequency = 11.86 Hz) \\  & 9450 kg (axle weight = 20 tons, and first natural frequency = 8.17 Hz) \\  & 14,450 kg (axle weight = 30 tons, and first natural frequency = 6.62 Hz) \\ \multirow{3}{*}{\(m_{2}\)} & 550 kg \\  & 1 \(\times\) 10\({}^{6}\) N/m \\ \multirow{3}{*}{\(k_{2}\)} & 1.75 \(\times\) 10\({}^{6}\) N/m \\  & 15 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m \\ \multirow{3}{*}{\(c_{2}\)} & 2 \(\times\) 10\({}^{3}\) N\(\cdot\)s/m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of quarter-car model of medium trucks. A tie connection is used between the soil base and the subbase as well as between the subbase and the base. The friction coefficient between the surface slab and the base is set as 1.5; it was proved that the magnitude of friction coefficient has no significant effect on the calculated stress under dynamic loads [26]. The width of the transverse joint is 4 mm, and a total of 13 dowel bars are arranged at the transverse joint, with spacing of 300 mm. One end of the dowel bar is tied to one concrete slab and the other end is in frictional contact with the other concrete slab, with a friction coefficient of 0.05. According to previous study [21], the width of the transverse joint is of little significance to dynamic response of the pavement under conditions of no faulting. Therefore, the effect of transverse joints is not considered in this study. The finite element method is used to calculate the stress generated in JPCP, and dynamic analysis is conducted to calculate the dynamic response of JPCP, following the motion equation as shown in Equation (16). It has been shown that second-order elements in the finite element analysis can significantly reduce the number of elements along the thickness of the slab [27].