## 2 Self-Centering Rocking Steel Bridge Pier Structure and Its Finite Element Analysis Model
### Finite Element Analysis Model and Verification
The hysteresis curves are compared in Figure 6. From the computed results in the figure, it can be observed that the shell element size has minimal impact on the calculation results. Therefore, for subsequent studies in this paper, a uniform shell element size of 40 mm will be adopted. The comparison between the hysteresis curves obtained from the FE model and the test results as well as the stress-displacement calculation results of the prestressed tendons are shown in Figure 7. The results indicate that due to the absence of energy dissipation devices in this pier, the area enclosed by the hysteresis curve is small, and there is a slight plastic energy dissipation at the base of the pier. Since the prestress loss caused by prestressing relaxation at the anchorage end [24; 25] is not considered in the computational model used in this study, there is no stress decrement at the displacement zero point. In the figure, \(f_{PT}\) is the stress of the prestressed tendon and \(f_{PT,u}\) is the ultimate tensile stress of the pre-stressed tendon. From the computational results, it can be observed that the load-displacement curve and the stress-displacement curve are very close to the experimental results. Therefore, adopting the FE model proposed in this study can achieve accurate simulation results of the seismic performance of rocking steel bridge pier. The same FE \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specimen No.** & \(R\times t\times h\) **(mm)** & \(R_{t}\) & \(\frac{-}{\lambda}\) & \(\alpha\) & \(\lambda\) & \(N/N_{y}\) & \(i_{t}\) **(\%)** & \(i_{R}\) **(\%)** \\ \hline RP3-DT43-PT44-BP20 & 203 \(\times\) 9.53 \(\times\) 1677 & 0.071 & 0.335 & 0.162 & 0.0 & 0.0 & 166.5 & 25.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design parameters of experimental rocking steel bridge pier [24]. Figure 5: Quasi-static loading system. modeling approach will be employed in the subsequent study on the seismic performance of the piers. Comparison of Seismic Performance between Concrete-Filled Rocking Steel Bridge Piers and Hollow Rocking Steel Bridge Piers  
The concrete-filled rocking steel bridge pier have received significant attention from experts and scholars [13; 16]. This paper takes the Pier 2 and Pier 12 in Table 1 as examples to carry out a comparative analysis of the seismic performance of the concrete-filled and the hollow rocking steel bridge pier. The piers No. 2 and No. 12 only differ in terms of their slenderness ratios, representing relatively low and high piers, respectively. C30 concrete, commonly used in China, is selected to fill the steel bridge pier, forming concrete-filled rocking steel bridge pier. The axial compressive strength of C30 concrete is 20.1 MPa and the elastic modulus is 22,500 MPa. The elastoplastic constitutive relationship of the concrete adopts the concrete plastic damaged model. The interface between the inner filled concrete and the outer steel tube is modeled using hard contact and tangential friction, with a  
Figure 6: Influence of shell element size on the hysteresis curve of the rocking steel bridge pier. Figure 7: Comparison of FE analysis results and experimental results: (**a**) Comparison of load-displacement hysteresis curves (**b**) Comparison of prestressed tendon stress-displacement results. friction coefficient of 0.2 [28]. The computational models take into account the self weight of the piers. Figure 8 presents a comparison of the hysteresis curves between concrete-filled rocking steel bridge piers and hollow rocking steel bridge piers. The results indicate that under horizontal cyclic loading, both rocking steel bridge piers without filled concrete and concrete-filled rocking steel bridge pier exhibit typical flag-shaped hysteresis curves with small residual displacements. This signifies that the piers possess good energy dissipation capacity and self-centering performance. Furthermore, whether it is a low-height or high-height rocking pier, the hysteresis curve results with and without concrete filling are very similar.