# A Restoring Force Model for Prefabricated Concrete Shear Walls with Built-In Steel Sections
## 5 Mathematical Fitting of the Restorative Force Curve Models
### Determination of Hysteresis Rules  
Besides determining the skeleton curves of the restoring force model, the hysteretic rule should also be determined, including the selection of the stiffness at each stage and the path of positive and negative loading and unloading. (1) Post-yielding rigidity  
The data of this test show that the post-yielding rigidity of the test structures is different from that before yielding. When the load exceeds the yield load, the post-yielding stiffness is no longer consistent with the loading stiffness, and each increase of the load level leads to a gradual decrease in the post-yielding stiffness of the specimen. By measuring the wall-top deformation and the corresponding stiffness of each hysteresis loop after yielding, the ratio of the stiffness to the pre-cracking stiffness \(K_{1}\) and the ratio of the wall-top deformation to the yield displacement is determined. According to the test results of the four wall specimens, the displacement and corresponding stiffness of each hysteresis loop apex after yielding were obtained for all hysteresis curves. In Figure 12, the ratios of the stiffness after yielding to that before cracking (\(K_{1}\)) was calculated as the \(y\)-axis values, and the ratio of the displacement at the apex of the hysteresis curve to the yielding displacement \(D_{y}\) was calculated as the \(x\)-axis coordinate values. By a nonlinear curve fitting (Figure 12), an equation of the post-yielding stiffness of the specimen can be obtained (Equation (11)). Figure 11: Comparison between theoretical and experimental skeleton curves. Figure 12: The relationship between the deformation ratio and stiffness ratio. The post-yielding stiffness calculation formula obtained from the fitting curve:  
\[K_{i}=1.07(\frac{D_{i}}{D_{y}})K_{1} \tag{11}\]  
where:  
\(K_{i}\)--Unloading stiffness. \(K_{1}\)--Cracking stiffness, taking the value of Kc in Figure 12.  
\(D_{i}\)--Displacement corresponding to a loading point. \(D_{y}\)--The displacement at the yield point. (2) The stiffness and change law of each stage. Before cracking, the prefabricated shear wall specimens are in the elastic stage. In this stage, there is no need to consider the stiffness degradation or residual deformation. The loading and reverse loading stiffness (forward and reverse loading applied on the top of specimens by the actuator) and unloading stiffness are the same, which is smaller than the cracking stiffness (\(K_{1}\)). From cracking occurrence to yielding, the loading stiffness is designated as pre-yield stiffness \(K_{2}\); the corresponding unloading stiffness is close to \(K_{1}\); and the unloading curve is parallel to the curve before cracking. In reverse loading, the pre-yield stiffness is \(K_{2}\), while the unloading stiffness is taken as the pre-cracking stiffness \(K_{1}\). From yielding occurrence to peak load, when the load exceeds the yielding load, the loading stiffness of the curve is taken as the post-yield stiffness \(K_{3}\), the unloading stiffness is \(K_{5}\); and in the case of reverse loading, the post-yield stiffness is \(K_{3}\), and the unloading stiffness is \(K_{5}\). Post peak load, when the load exceeds the peak load, the loading stiffness is taken as \(K_{4}\), and the unloading stiffness is taken as \(K_{5}\); and in the case of reverse loading, the loading stiffness is \(K_{4}\), and the unloading stiffness is \(K_{5}\). (3) Positive and negative loading and unloading paths  
The paths are shown in Figure 13. Before cracking, the specimen was in the elastic stage, loaded and unloaded along a straight line, and the stiffness was all \(K_{1}\). Before yielding and after cracking, the pre-yield stiffness was taken as \(K_{2}\) to point 1 for the loading stiffness, the unloading stiffness at the cracking stage is \(K_{1}\) to point 2, reverse loading to point 3 as \(K_{2}\), and the unloading stiffness before cracking is \(K_{1}\) to point 4. Figure 13: Schematic diagram of loading and unloading paths.