# Dynamic Responses of Asphalt Concrete Waterproofing Layer in Ballastless Track
## 6 Result Analysis and Discussion  
### Analysis of Critical Loading Position  
The train load at different positions along the driving direction causes different stress distributions in the ACWL, so it is necessary to determine the critical loading position when designing the structural layer of ACWL. The critical loading position is the wheelset position where the dynamic responses of ACWL are maximum. Considering the load superposition effect, the rear wheelset of the second bogie is selected as the research target. Three load cases are compared when the train speed is 350 km/h, including the joint of base plates (Load case 1), the joint of track slabs (Load case 2) and the center of base plate (Load case 3), as is illustrated in Figure 14. The vertical stress, vertical displacement, vertical acceleration and the tensile strain at the bottom of ACWL under different load cases are calculated by the model, as summarized in Table 5.  
Figure 14: Three load cases to determine the critical loading position.  
Figure 13: Time history curve for the dynamic responses of ACWL (**a**) Vertical acceleration; (**b**) Vertical displacement.  
Table 5 shows that the order of magnitude of the dynamic responses in the three cases is as follows: Load case 1 > Load case 2 > Load case 3. It is indicated that it is more unfavorable for the dynamic responses of ACWL when the wheelset is located in the structural joint, especially at the expansion joint of base plates. In terms of the tensile strain at the bottom of the ACWL, the longitudinal strain is much greater than the lateral strain, as is shown in Figure 15. It should be noted that there is a significant area of tensile strain concentration below the wheelsets in Figure 15. Overall, the tensile strain level at the bottom of ACWL is relatively low, which guarantees a long service life considering the fatigue performance.  
The lateral distribution of vertical stress and displacement of ACWL under Load Case 1 is shown in Figure 16 to further investigate the unfavorable position for the ACWL in the lateral direction. The position when x = 0 m in Figure 16 denotes the center of double-line railway while x = 6.8 m denotes the outer edge near the shoulder of the roadbed. It can be observed in Figure 16a that the vertical displacement shows a large value over the width range of the base plate and reaches the maximum value near the outer edge of the base plate. In terms of the vertical stress in Figure 16b, it is obvious that the maximum value of vertical stress appears just below the two rails. Therefore, it is necessary to pay more attention to the area of ACWL near the edges of base plate and directly below the rail when designing the structural layer.  
Figure 15: The strain contour at the bottom of ACWL (Load case 1) (**a**) Lateral strain; (**b**) Longitudinal strain.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dynamic Response Indexes** & **Load Case 1** & **Load Case 2** & **Load Case 3** \\ \hline Vertical stress (kPa) & 20.18 & 19.94 & 14.28 \\ Vertical displacement (mm) & 0.37 & 0.33 & 0.26 \\ Vertical acceleration (m/s\({}^{2}\)) & 3.87 & 3.50 & 3.03 \\ \multirow{2}{*}{Tensile strain (\(\mu\varepsilon\))} & Lateral & 9.53 & 6.81 & 5.00 \\  & Longitudinal & 16.18 & 14.39 & 10.63 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Dynamic responses of ACWL under three load cases.