## 5 Conclusions  
In this paper, a finite element numerical simulation combined with a practical major bridge structure rock-socketed pile project was employed considering the effect of vertical load on the rock-socketed pile. The influence of different influencing factors on the peak  
Figure 10: Results of the rock-socketed piles with different DCs of karst caves under pile. (**a**) DC at 1, (**b**) DC at 1.5, (**c**) DC at 2, and (**d**) DC at 3.  
strain of the pile were analyzed, and the strain distribution laws of the pile under the condition of complex and multiple karst caves were also explored. The seismic responses of the rock-socketed pile can contribute to the seismic reinforcement measures in actual conditions. The main research conclusions are as follows:  
1. The existence of karst caves in bedrock would significantly affect the strain of the rock-socketed pile. The pile strain at the corresponding position in the karst cave was relatively large. With the increase of the karst cave height and the decrease of the karst cave roof, the restraint effect of the bedrock near the pile gradually decreased and the peak strain of the rock-socketed pile gradually increased.
2. The peak strain of the pile would gradually increase with the increase of the number of beaded karst caves under the condition of beaded karst caves. Moreover, due to the restraint of bedrock between karst caves, the pile presented the distribution characteristics of \"multi-segment and multi-polyline\".
3. In the condition of the underlying karst cave, the existence of the underlying karst cave reduced the restraint effect of the bedrock at the bottom, and made the peak strain of the pile increase to a certain extent. With the increase of the distance coefficients (DC), the influence also decreased. When the DC exceeded 2, the existence of the underlying karst cave had little effect on the pile strain.
4. Due to the same viscous sand layer above 80 m, the peak strain distribution of the pile above 80 m buried depth was almost the same under all conditions. The soil properties would have a great influence on the response of the pile.
5. According to the seismic response of rock-socketed pile, the pile strain at the location of karst cave was considerable; the mentioned dangerous location of the pile should be appropriately reinforced or reinforced with steel hoops. Under the conditions with pile-through and underlying karst cave, the existence of a karst cave reduced the restraint effect of the bedrock at the bottom; it is necessary to strengthen the bedrock between the pile foundation and the karst cave by grouting.  
In the actual engineering, there would be numbers of rock-socketed piles in a large bridge, and the seismic interaction of all the piles would have considerable influences on the seismic responses of the whole soil-pile system. Therefore, the seismic group effect of rock-socketed piles should be further investigated.  
Conceptualization, P.W. and P.Z.; methodology, P.W. and P.Z.; software, D.Q.; validation, W.H.; formal analysis, W.H.; investigation, W.H.; resources, D.Q.; data curation, P.W., writing--original draft preparation, D.Q.; writing--review and editing, P.W. and P.Z.; visualization, P.W. and P.Z.; supervision, P.W. and P.Z.; project administration, P.W.; funding acquisition, P.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by [the Natural Science Foundation of Shandong Province] grant number [ZR2021 ME238].  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to fund requirements.  
The authors express their sincerest gratitude for the support. The authors would like to thank [PERSON], Shandong Jianzhu University, for the constructive suggestions on this paper.  
The authors declare no conflict of interest.