## 4 Extended Parameter Analysis
### Results of Extended Parameter Analysis
When the concrete strength increased by one grade, the peak load increased by approximately 2-5 kN, and the ductility coefficient decreased by approximately 0.02-0.25. An appropriate increase in the filled concrete strength effectively enhanced the stiffness of the filling layer and the hoop constraint of the core column, further improving the strengthened pier column's overall stiffness and bearing capacity. However, with an increase in the filled concrete strength grade, the growth rate of the peak load decreased from 10.18% to 6.37% to 3.32%, and the growth rate of the ultimate load decreased from 9.34% to 7.31% to 2.40%, as calculated from Table 8. The ductility coefficient decreased during this process, indicating that the excessive strength grade (i.e., >C40) of the filled concrete provided limited improvement in the seismic performance of the pier columns and could lead to rapid ductility degradation. Moreover, the secant stiffness and stiffness degradation trends of MS-2, MS-3, and MS-4 were relatively the same and significantly higher than those of MS-1 (Figure 24). Hence, the high strength of the filled concrete could not significantly improve the secant stiffness. In summary, setting the filled concrete strength of the strengthened pier  
Figure 23: Comparison of the skeleton and stiffness degradation curves for numerical models with different SSAWC reinforcement ratios. (a) Skeleton curves of MR-1 to MR-4; (b) Stiffness degradation curves of MR-1 to MR-4. column to be 1-2 grades higher than the concrete strength of the core column is sufficient. That is, the filled concrete strength grade should not exceed C40 for strengthened piers with a core column of C30 concrete. (4) Influence of SSAWC self-stress on the strengthening effect  
As the self-stress value increased from 0 to 2.8 MPa, the initial stiffness, peak load, ductility coefficient, and secant stiffness of the pier column increased by 34.47%, 37.96%, 30.35%, and 23.33%, respectively (Table 8 and Figure 25). We inferred that an increase in the initial self-stress could significantly enhance the confinement effect of the filling layer on the core pier column. The column was placed in a triaxial compression state to improve its stiffness, bearing capacity, and ductility. However, when the self-stress value of the filled concrete increased from 2.2 to 2.8 MPa, the peak load growth rate decreased from 12.57% to 9.08%, the ductility coefficient growth rate decreased from 5.88% to 1.57%, and the pier column ultimate load began to decrease, indicating that excessive self-stress (i.e., >2.2 MPa) provided limited improvement on the seismic performance of the pier columns. Therefore, under the premise of meeting the strengthened pier columns' bearing capacity design requirements, the initial self-stress value of the filled concrete should not exceed 2.2 MPa. Figure 24: Comparison of the skeleton and stiffness degradation curves for numerical models with different SSAWC strengths. (**a**) Skeleton curves of MS-1 to MS-4; (**b**) Stiffness degradation curves of MS-1 to MS-4. Figure 25: Comparison of the skeleton and stiffness degradation curves for numerical models with different self-stresses. (**a**) Skeleton curves of MS-1 to MS-4; (**b**) Stiffness degradation curves of MS-1 to MS-4. (5) Influence of axial compression ratio on the strengthening effect  
As the axial compression ratio increased from 0.1 to 0.4, the initial stiffness and ductility coefficient of the pier column first increased and then decreased (Table 8); the peak load increased by 18% during this process. The ductility coefficient and initial stiffness reached their respective maximum values of 7.25 and 17.98 kN/mm when the axial compression ratio was 0.2 and 0.3, respectively. We inferred that an increase in the axial compression ratio could delay the tensile steel rebar yield and tensile concrete cracking to a certain extent, improving the pier column's overall stiffness and bearing capacity.