## 2 FEM-SPH Conversion Coupling Algorithm
### Analysis of Pier Deformation Characteristics
#### 4.2.2 Displacement and Deformation Characteristics of the Pier  
Under the circulation of the reservoir water level, the deformation and instability of the bank slope will produce lateral thrust on the pile pier, resulting in the inclination and deviation of the pile pier towards the river center. The displacement and deformation characteristics of the pier are shown in Figure 10.  
As can be seen from Figure 10, because of the rising and falling cycles of the reservoir water level, the pile pier line shape increasingly deviated from the original design position, and the overall deformation of the pile pier was characterized as follows: the height of the embedded foundation part of the pile displacement (0-3.5 m) was almost 0, the pile body from the soil-rock interface to the pile-soil interface (body height 3.5-26.36 m) under the action of horizontal deflection soil thrust, and the pier body from the pile-soil interface to the pier. The displacement of the pier body from the pile-soil interface to the top section (height 22.86-81.5 m) increases linearly. Near the soil-rock interface, the displacement of pile foundation increases sharply, and it was assumed that the most unfavorable position of pile foundation was near the soil-rock interface. When N = 20, there was an obvious inflection point in the pile foundation line shape from the soil-rock interface to the pile-soil interface. It can be seen that the pile foundation will be seriously deformed and damaged in the 20 th reservoir water level lifting cycle. In order to more clearly explore the indirect impact of the reservoir water level rise and fall cycle on the pier deflection, the variation law of pier top deflection under different times of the reservoir water level rise and fall cycle will be analyzed in combination with the deformation and instability of the bank slope. The variation of pier top deflection is shown in Figure 11.  
Figure 10: Deformation characteristics of the bridge pier. According to Figures 8 and 11, the results show that the deviation of the pier top increases with the increase in the number of reservoir water level rise and fall cycles. In the first 15 cycles of reservoir level rise and fall, the bank slope was in a stable state, and the displacement of the top of the pier increased slowly during this period. After the 16 th reservoir level rise and fall cycle, that is, after the first destabilization of the shore slope, the increment of the pier top offset suddenly changes, and its growth trend changes from a slow increase to a sharp increase. In the 17 th to 19 th water level rise and fall cycles, it was known from the previous analysis that although the slope was in a new stable state, the pile foundation played a certain anti-slip role in the deformation of the bank slope, the unbalanced thrust of the soil behind the pile gradually increased, and the deflection trend of the pier top was not moderated. In the 20 th water level lift cycle, the second destabilization damage occurred on the bank slope behind the pile. According to the previous analysis, at this time the pile foundation had already undergone serious deformation damage. Therefore, the deflection tendency of the top of the bridge pier in this process was stronger than before. Failure Mode of Bank Slope-Bridge Pile Foundation Interactions under Reservoir Water Level Cycling Conditions  
The damage distribution and cracking of bridge pile foundations in the model were monitored to explore their damage patterns. The tensile damage on the back side of the pile foundation (back from the river center side) is shown in Figure 12. When N = 14, the pile foundation already showed obvious tensile damage and transverse cracks with exposed reinforcement near the soil-stone interface. The damage and cracks were initially concentrated near the soil-stone interface, then gradually developed towards the upper part of the pile and continued to produce transverse cracks.