## 5 Restoring Force Model
### Skeleton Curve Model
#### 5.1.2 Skeleton Curve Model Verification  
The accuracy and reliability of the skeleton curve model were verified by comparing the experimentally measured and theoretically calculated values for each characteristic point of the five strengthened specimens: RP-0, RP-1, RP-2, RP-4, and RP-5. Since the initial damage to the pier column was not considered during the establishment of the skeleton curve model, strengthened specimen RP-3 was not selected for model verification.  
As shown in Tables 9-11, the relative errors between the measured and calculated values of the characteristic points for each strengthened specimen were within 10%, and 5/30 groups from the contrast data had an error \(>\) 5%. Moreover, the theoretically calculated skeleton curve models of each strengthened specimen agreed well with the skeleton curves measured in the quasi-static test (Figure 28). The primary causes of the apparent deviation after the yield point between the measured and calculated models under reverse loading are the asymmetry of the skeleton curve caused by the asymmetry of the specimen loading and damage, as well as the discreteness of the material (Figure 28b,d,e). Therefore, the skeleton curve model obtained using the proposed calculation method for the characteristic points can accurately reflect changes in the bearing capacity, deformation, and stiffness of the pier column strengthened using the IPCSAM under horizontal cyclic loading.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & \multicolumn{3}{c}{**Utimate Load**} & \multicolumn{3}{c}{**Ultimate Displacement**} \\ \cline{2-7}  & \(F_{\text{u,m}}\) **(kN)** & \(F_{\text{u,c}}\) **(kN)** & **RE (\%)** & \(\Delta_{\text{u,m}}\) **(mm)** & \(\Delta_{\text{u,c}}\) **(mm)** & **RE (\%)** \\ \hline RP-0 & 40.93 & 39.37 & 3.81 & 56.02 & 57.98 & 3.50 \\ RP-1 & 48.01 & 45.83 & 4.54 & 76.58 & 73.70 & 3.76 \\ RP-2 & 43.74 & 44.73 & 2.26 & 64.37 & 67.19 & 4.38 \\ RP-4 & 31.97 & 33.12 & 3.60 & 95.70 & 89.57 & 6.41 \\ RP-5 & 22.18 & 21.57 & 2.75 & 123.61 & 119.92 & 2.99 \\ \hline \hline \end{tabular} Note: (a) \(F_{\text{u,m}}\) and \(F_{\text{u,m}}\) represent the measured and calculated ultimate load, respectively. (b) \(\Delta_{\text{u,m}}\) and \(\Delta_{\text{u,c}}\) represent the measured and calculated peak displacement, respectively. (c) RE represents the relative error, calculated as RE = \((F_{\text{u,c}}-F_{\text{u,m}})/F_{\text{u,m}}\)\(\times\) 100% or RE = \((\Delta_{\text{u,c}}-\Delta_{\text{u,m}})/\Delta_{\text{u,m}}\)\(\times\) 100%.  
\end{table}
Table 11: Calculated and measured strengthened specimens’ ultimate points.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & \multicolumn{3}{c}{**Yield Load**} & \multicolumn{3}{c}{**Yield Displacement**} \\ \cline{2-7}  & \(F_{\text{u,m}}\) **(kN)** & \(F_{\text{u,c}}\) **(kN)** & **RE (\%)** & \(\Delta_{\text{u,m}}\) **(mm)** & \(\Delta_{\text{u,c}}\) **(mm)** & **RE (\%)** \\ \hline RP-0 & 38.46 & 37.05 & 3.67 & 9.56 & 9.28 & 2.93 \\ RP-1 & 45.25 & 42.18 & 6.78 & 9.11 & 9.47 & 3.95 \\ RP-2 & 41.28 & 40.32 & 2.33 & 9.36 & 9.65 & 3.10 \\ RP-4 & 30.38 & 29.20 & 3.88 & 13.06 & 12.59 & 3.60 \\ RP-5 & 20.48 & 20.91 & 2.10 & 20.67 & 19.75 & 4.45 \\ \hline \hline \end{tabular} Note: (a) \(F_{\text{u,m}}\) and \(F_{\text{u,m}}\) represent the measured and calculated yield load, respectively. (b) \(\Delta_{\text{u,m}}\) and \(\Delta_{\text{u,c}}\) represent the measured and calculated yield displacement, respectively. (c) RE represents the relative error, calculated as RE = \((F_{\text{u,c}}-F_{\text{u,m}})/F_{\text{u,m}}\)\(\times\) 100% or RE = \((\Delta_{\text{u,c}}-\Delta_{\text{u,m}})/\Delta_{\text{u,m}}\)\(\times\) 100%.  
\end{table}
Table 9: Calculated and measured strengthened specimens’ yield points.