## 5 Restoring Force Model
### Hysteresis Curve Model  
#### 5.2.1 Hysteresis Rule and Model  
A hysteresis rule is required to describe the hysteresis behavior of a pier column strengthened using the IPCSAM under horizontal cyclic loading. The hysteresis behavior is described through numerous rules for unloading and reloading; hence, establishing a restoring force model of strengthened specimens under cyclic loading requires determining forward and reverse loading and unloading paths, i.e., the hysteresis rule. The typical hysteresis rules, as presented in Figure 29, define the hysteresis behavior as follows:  
Figure 28: Comparison of the theoretical calculated and experimental measured skeleton curve models of the strengthened specimens (**a–e**).  
(a) The stage before loading to yield point A. At this point, the strengthened specimens are primarily in the elastic stage with no residual deformation. The loading and unloading paths follow the elastic section of the skeleton curve. Thus, the loading and unloading stiffnesses at this stage can be considered as the yield stiffness \(K_{y}\), calculated using Equation (15). The loading path is as follows: O\(\rightarrow\)A\(\rightarrow\)O\(\rightarrow\)A'\(\rightarrow\)O.  
(b) The stage where the load exceeds the yield point A but does not reach the peak point F. First, the origin point O is loaded to a certain point B according to the skeleton curve, and the peak stiffness \(K_{p}\) of segment AB can be calculated using Equation (20). Because stiffness degradation occurs during unloading in this stage, point B can be unloaded to point C according to the unloading stiffness \(K_{ei}\) and reversely loaded to point D, which is polarly symmetric to point B in the skeleton curve. The loading stiffness of segment CD is its slope. Subsequently, point D is reversely unloaded to point E, according to \(K_{ei}\), and finally loaded to point B. The loading path is as follows: O\(\rightarrow\)A\(\rightarrow\)B\(\rightarrow\)C\(\rightarrow\)D\(\rightarrow\)E\(\rightarrow\)B. According to [PERSON] et al. (2010), the unloading stiffness \(K_{ei}\) can be calculated using Equation (24), where \(\Delta_{i}\) represents the displacement at the \(i\)-th unloading point. Both \(\alpha\) and \(\omega\) are adjustment coefficients calculated using Equation (25) and Equation (26), respectively. The significance test processes of the regression equations at the inspection level of \(\alpha\) = 0.05 are as follows: for Equation (25), F = 3.68 \(>\) F\({}_{0.05}\)(6, 18) = 2.66, and for Equation (26), F = 3.95 \(>\) F\({}_{0.05}\)(6, 18) = 2.66, indicating that the regression equations were significant and reliable at a 95% confidence level.  
\[K_{ei}/K_{y}=\alpha\left(\Delta_{i}/\Delta_{y}\right)^{\omega} \tag{24}\]  
\[\alpha=5.39+6.62n-0.89\rho_{v}-0.48f_{c}+0.21m-0.71\lambda+5.63u \tag{25}\]  
\[\omega=-2.26-0.81n+0.13\rho_{v}-0.16f_{c}-0.01m+0.29\lambda-0.26u \tag{26}\]  
(c) The loading stage from peak point F to ultimate point L. The hysteresis change and stiffness degradation during loading and unloading are similar to those in the last stage. The ultimate stiffness \(K_{u}\) of the FH segment can be calculated using Equation (23), and the unloading stiffness \(K_{ei}\) can be calculated using Equations (24)-(26). The loading path is as follows: O\(\rightarrow\)A\(\rightarrow\)B\(\rightarrow\)F\(\rightarrow\)H\(\rightarrow\)I\(\rightarrow\)J\(\rightarrow\)K\(\rightarrow\)H\(\rightarrow\)L\(\rightarrow\)M\(\rightarrow\)N\(\rightarrow\)P\(\rightarrow\)L.  
Figure 29: Definition of hysteresis rules for the restoring force model.