## 4 Conclusions  
In this paper, a discrete element method was adopted to analyze the impact process and impact force of ablated river ice on bridge abutments by numerical simulation, the damage characteristics and impact force-time curve of ice rows under different ice speed \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Simulation Results** & **Load Modeling Results** & **Inaccuracies** \\ \hline peak impact & & & \\ force/KN & & & \\ Onset time/s & 0.033 & 0.041 & \(-\)5.8\% \\ Load time/s & 0.0054 & 0.005 & 8.7\% \\ Unload time/s & 0.006 & 0.0065 & \(-\)7.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between simulation results and load modeling results. Figure 14: Comparison between simulation results and load modeling results. and porosity conditions were obtained, and the relationship between the river ice porosity and the maximum impact force of the bridge abutment was fitted. The primary takeaways are as follows:  
1. The simulation results indicate that increased ice speed significantly raises river ice impact force. When the speed of the ice reaches a certain critical threshold, the maximum impact force no longer increases with further increases in velocity. Instead, this peak impact force becomes more influenced by the strength of the ice. This finding is crucial for understanding how river ice interacts with bridges under extreme conditions. 2. The effect of river ice porosity on the impact force is also significant. As the porosity increases, the ice load decreases overall, while a lag occurs in the appearance of the peak contact force. In addition, the random loading and unloading characteristics of the ice load are enhanced, indicating that the change in porosity increases the uncertainty of the impact process, which puts forward higher requirements for the ice-resistant design of bridges in cold regions. 3. Regarding the damage pattern of ice rafts, it is found that with the increase in porosity, the damage is gradually concentrated near the area in direct contact with the bridge abutments. Therefore, emphasis should be placed on strengthening the protective capacity of the abutment-susceptible area to effectively resist the threat of ice collision. 4. According to the fitting results of the relationship between the river ice porosity and the maximum impact force, as well as each time parameter, the maximum impact force of the river ice gradually decreases with the increase in river ice porosity, and the fitting results show an obvious nonlinear relationship; the start time of the impact load increases with the increase in the porosity, and the loading time and unloading time decreases with the increase in the porosity and the ice speed. When analyzing and predicting the ice load on the structure, the effect of river ice porosity and the characteristics of the ice load model need to be fully considered. 5. This study focuses on revealing the effect of porosity on ice impact force but is limited by the model simplification: it has not yet covered the actual ice morphology diversity and impact location effect. The subsequent study will combine the discrete element modeling of U-shaped ice and irregular ice rows with the sensitivity analysis of the impact location, coupling multiple parameters, and providing domain-wide theoretical support for the ice-resistant design of bridges in cold regions. Z.X. : writing--original copy and manuscript, writing--review and editing, methodology, software; Y.W. : writing--review and editing; D.X. : writing--review and editing; Y.Z. : writing--review and editing; D.Z. : project administration, funding acquisition, conceptualization. All authors have read and agreed to the published version of the manuscript. This research was funded by Key R&D Program Project of Heilongjiang Provincial, China: Probabilistic Modeling and Dynamic Effects of River Ice Action on Bridges in Cold Regions (GZ20220101). Not applicable.