# Fracture Threshold Analysis and Parameter Matching of Cut-Out Induced Bolts for Subway Couplers
## 3 Effect of Cut-Out Zone Structural Parameters
### Analysis of Bol Failure Modes  
By extracting the force-time curves from the simulation results and combining them with the element failure times, the force-displacement curves were filtered to obtain the fracture time and fracture force for each set of bolts. The results are shown in Appendix A Table 12. The simulation results can be categorized into three main types: bolts breaking before the expansion tube starts working; bolts breaking during the operation of the expansion tube; and bolts breaking after the expansion tube has completed its work. Taking the case of bolts breaking during the operation of the expansion tube (the 20 th set of bolt conditions in the DOE) as an example, as shown in Figure 17, the simulation calculation provides a time series graph of the total bolts area, the coupler contact force, and the total cross-sectional force of the bolts. From the graph, it can be observed that during the collision process, the change in the cross-sectional area of the bolts undergoes a total of five stages. The first stage is from the beginning of the collision to the moment when the cross-sectional force of the bolts reaches the initial peak force. During this stage, the maximum stress generated in the bolts does not reach the yield stress of the material, and the cross-sectional area of the bolts remains essentially unchanged. Comparing the magenta line and the light blue line in the graph, it can be seen that due to the presence of mass points and shock absorbers, there is a noticeable lag in the transmission of the initial peak force from the coupler to the bolts cross-section. Therefore,  
Figure 16: Distribution of DOE sampling point. the change in the cross-sectional area of the bolts does not follow the change in the coupler contact force. The second stage is the initial peak force phase. During this stage, the cross-sectional force of the bolts begins to rise sharply, and the material undergoes permanent plastic deformation as it cannot withstand the corresponding stress, causing the area of each bolt to suddenly decrease. However, the bolts do not break at this point. The third stage is the transition from the initial peak force to the average crush force. After passing the initial peak force, the force on the bolts' cross-section decreases, the deformation of the bolts stops, and the cross-sectional area remains unchanged. The fourth stage is the average crush force phase. In this stage, due to the improper design of the cut-out zone, the structure of the bolts after being impacted by the initial peak force can no longer withstand the load of the average crush force. During this process, the cross-sectional area of the bolts continues to decrease until it eventually cannot withstand the corresponding load and breaks at a certain moment. The fifth stage is the post-fracture phase. In this stage, since the bolts have already broken, they can no longer withstand the corresponding load, so the cross-sectional area remains at the size it was at the moment of fracture. Additionally, the unloading of the cross-sectional force of the bolts, after passing through the shock absorber, also experiences a corresponding lag in being transmitted to the coupler contact surface. Taking the case of bolts breaking before the expansion tube starts working (the 29 th set of bolt conditions in the DOE) as an example, as shown in Figure 18, the first and second stages are the same as before. However, in the third stage, after the cross-sectional force of the bolts reaches the initial peak force and starts to decline before returning to the platform  
Figure 17: The area–time and force–time curve of bolts breaking during the expansion tube operation (the 20 th set of bolt conditions in the DOE). force, there is still a reduction in the cross-sectional area of a certain bolt within the bolt group.