## 5 Seismic Strengthening Mechanism Analysis of the Plastic Region  
To analyze the seismic strengthening mechanism of the plastic region at the bottom of the piers, a comparative analysis was conducted between the unreinforced Pier 2 and the reinforced Piers 6 and 7 with thickened wall plates, as well as Piers 9 and 10 with additional longitudinal stiffeners. The hysteresis curves are shown in Figure 16. It can be observed that both the reinforcement methods, i.e., thickened wall plates and additional stiffeners, resulted in a decrease in the thickness-to-radius ratio \(R_{td}\) in the bottom region of the piers. This significantly enhanced the ductility performance of the piers, reduced the residual displacement, and mitigated the stiffness degradation effects. Figure 14: Impact of slenderness ratio. Figure 15: Impact of diameter-to-thickness ratio. To analyze the effect of reinforcement zone length on the hysteresis performance, a comparative analysis was conducted between the unreinforced Pier 2 and the reinforced Piers 5 and 6 with different heights of thickened wall plates, as well as reinforced Piers 8 and 9 with different heights of additional stiffeners. The hysteresis curves are presented in Figure 17. It can be seen that the reinforcement within a range of 0.3 m exhibited further improved performance compared to the reinforcement within a length of only 0.15 m.  
To further compare the effectiveness of the additional stiffeners and thickened wall plates at the bottom of the piers in enhancing the seismic performance of bridge piers, a comparison of the hysteresis curves was carried out between the unreinforced Pier 2 and the reinforced Piers 5 and 8 with thickened wall plates and additional stiffeners within a range of 0.15 m, as well as Piers 6 and 9 with thickened wall plates and additional stiffeners within a range of 0.3 m. This is illustrated in Figure 18. From the figure, it can be observed that under the same reinforcement length and radius-to-thickness ratio, the load-carrying capacity of the piers with thickened wall plates is significantly higher than that of the piers with additional longitudinal stiffeners. This indicates that the thickened wall plate approach is a more effective method for seismic strengthening. Figure 19 further presents the comparative results of the load-carrying capacity, energy dissipation capacity, initial lateral stiffness, and residual displacement under the condition of bottom reinforcement within a range of 0.15 m. Based on the comparison between Piers 2 and 5, it can be observed that the performance of the proposed  
that when thickened wall plates are applied, the maximum load-carrying capacity of the rock piers increases by 5.5%, energy dissipation value increases by 0.18%, initial stiffness increases by 8.9%, and residual displacement decreases by 15.6%. On the other hand, based on the comparison between Piers 2 and 8, when additional stiffeners are utilized, the maximum load-carrying capacity of the rock piers increases by 0.18%, energy dissipation value decreases by 1.4%, initial stiffness increases by 5.1%, and residual displacement decreases by 9.8%. Figure 19: Comparison of calculation results using different reinforcement methods within a range of 0.15 m. (**a**) Load-carrying capacity; (**b**) Energy dissipation capacity; (**c**) Initial lateral stiffness; (**d**) Residual displacement. Figure 18: Comparison of hysteresis curves using different reinforcement methods.