## 5 Restoring Force Model
### Skeleton Curve Model
#### 5.1.2 Calculation of Yield Point  
(1) Yield Load  
According to the quasi-static test and extended parameter analysis results, the ratio of peak load to yield load decreases with increased segment thickness, length-diameter ratio, axial compression ratio, and filled concrete strength, and increases with an increased  
Figure 27: Trilinear skeleton curve model. reinforcement ratio of the filled concrete and initial self-stress. Accordingly, and considering previous studies [71; 72], Equation (5) was obtained:  
\[\frac{F_{p}}{F_{y}}=A-k_{1}k_{2}Bn+C\rho_{v}-k_{3}Df_{c}+Em-F\lambda-Gu \tag{5}\]  
where \(F_{y}\) and \(F_{p}\) are the yield and ultimate loads, respectively; \(n\) represents the ratio of the segment thickness to the filling-layer thickness; \(\rho_{v}\) represents the reinforcement ratio of the filling layer; \(f_{c}\) represents the ratio of the filled concrete strength to the C40 concrete strength; \(m\) is the initial self-stress of filled concrete; \(\lambda\) and \(u\) represent the length-diameter ratio and axial compression ratio of the pier column, respectively. Parameters A-E are regression coefficients; \(k_{1}\), \(k_{2}\), and \(k_{3}\) represent the adjustment coefficients of the concrete strength of the segment, reinforcement ratio of the segment, and increased area of the filling layer, respectively, which can be calculated using Equations (6)-(8):  
\[k_{1}=1+\left(\frac{f_{s}}{40}-1\right)\times 0.16 \tag{6}\]  
\[k_{2}=1+\left(\frac{\rho_{s}}{0.5}-1\right)\times 0.08 \tag{7}\]  
\[k_{3}=1+\left(\frac{h_{s}}{d_{s}}-0.1\right)\times 0.8 \tag{8}\]  
where \(f_{s}\) and \(\rho_{s}\) represent the concrete strength and the reinforcement ratio of the segment, respectively; \(h_{s}\) is the thickness of the filling layer; \(d_{s}\) is the diameter of the strengthened pier column. The relationship between the peak and yield loads, expressed as a multiple linear equation, was obtained by linear regression fitting, as follows:  
\[\frac{F_{p}}{F_{y}}=1.53-0.81n+0.08\rho_{v}-0.16f_{c}+0.02m-0.01\lambda-0.12u \tag{9}\]  
The significance test process of the regression represented by Equation (9) at the inspection level of \(\alpha\) = 0.05 is as follows: F = 4.15 \(>\) F\({}_{0.05}\)(6, 18) = 2.66, indicating that the regression equation was significant and reliable at a 95% confidence level. (2) Yield Displacement  
The yield displacement \(\Delta_{y}\) of the strengthened pier column consists primarily of the bending deformation \(\Delta_{by}\) and shear deformation \(\Delta_{sy}\), expressed as  
\[\Delta_{y}=\Delta_{by}+\Delta_{sy} \tag{10}\]  
\(\Delta_{by}\) and \(\Delta_{sy}\) are expressed as  
\[\Delta_{by}=\frac{1}{3}\phi_{y}L^{2} \tag{11}\]  
\[\Delta_{sy}=1.2\frac{F_{y}L}{Gd_{w}2} \tag{12}\]  
where \(L\) is the effective calculated height of the strengthened pier column; \(G\) represents the shear modulus of the concrete of the column body, which can be set to 0.4 times the elastic modulus; \(d_{w}\) represents the diameter of the plastic hinge region, considered as the diameter of the unreinforced core pier column; \(\phi_{y}\) represents the yielding curvature, calculated using Equation (13). According to a previous study [75], parameter \(\beta\) was obtained through linear regression fitting based on the results of the quasi-static test and extended parameter analysis, as presented in Equation (14):  
\[\phi_{y}=\beta\frac{f_{y}}{E_{s}d_{w}} \tag{13}\]  
\[\frac{F_{p}}{F_{y}}=A-k_{1}k_{2}Bn+C\rho_{v}-k_{3}Df_{c}+Em-F\lambda-Gu \tag{5}\]0  
where \(f_{y}\) represents the design value of the rebar compressive strength; \(E_{s}\) represents the elastic modulus of the rebar. The significance test process of the regression Equation (14) at the inspection level of \(\alpha=0.05\) is as follows: F = 21.11 > F\({}_{0.05}\)(6, 18) = 2.66, indicating that the regression equation was significant and reliable at a 95% confidence level.