# 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 Dimensionless Skeleton Curves
Substituting the stiffness value into Equation (2), the theoretical cracking displacement is calculated as 1.75 mm, which is smaller than that obtained in the test (about 3 mm, as listed in Table 4). 2. Calculation of the peak load For shear walls without embedded steel sections, the ultimate compressive strain of concrete in the constrained edge members is increased because of the stirrup constraints. For shear walls with embedded steel sections, in addition to the constraints from the stirrups and marginal longitudinal reinforcements, the steel sections also contribute to the edge constraint, and the stress-strain relationship of the restrained concrete is calculated by formulas given in the literature [20] as: \[f_{cc}=(1+0.5\lambda_{t})f_{c}\] (7) \[\epsilon_{cc}=(1+2.5\lambda_{t})\epsilon_{c}\] (8) where: \[f_{cc}\] --Compressive strength of the confined concrete. \[\epsilon_{cc}\] --Peak strain of the confined concrete. \[\lambda_{t}\] --Constraint index Figure 9:0 --The volume ratio of stirrups. Figure 9:1 --Design value of concrete axial compressive strength. Figure 9:2 --Elastic limit strain of concrete. Figure 9:3 --Design value of yield strength of stirrup. For the shear wall specimens, when the concrete strain Figure 9:4 in the constrained region reaches the ultimate compressive strain under the constrained state, the steel sections and longitudinal bars in the pull and press edge members have yielded. In this paper, in the prefabricated concrete shear wall with built-in steel sections, symmetrical reinforcements and steel sections are used to establish the equilibrium equation of the force under the ultimate state of the bearing capacity of the section [21], see Figure 10 and Equations (9) and (10). In the equilibrium equations, it is assumed that the reinforcement bars and steel sections in both the tension and compression sides have yielded under the ultimate state. As shown in Figure 10, under the axial load N and the moment M from the lateral load, the steel section (\(A_{a}\)) and the core reinforcements inside the stirrup (\(A_{s}\)) as well as the uniformly distributed vertical rebars in the tension zone provide tensile resistance, and the steel section (\(A_{a}{}^{\prime}\)) and the core reinforcements (\(A_{s}{}^{\prime}\)) as well as the concrete in the compression zone provide compressive resistance. Figure 9:5  
\[ \begin{array}{l}M=r(bx-A_{a})f_{cc}(h_{0}-0.5x)+(f_{y}^{\prime}A_{s}^{\prime }+f_{a}^{\prime}A_{a}^{\prime})(h_{0}-\alpha_{a})\\ -0.5b(h_{0}-1.5x)^{2}\rho_{v}f_{vv}-N(h_{0}-0.5h)\end{array}  \tag{10}\]  
where:  
\(\rho_{v}\)--Vertical distributed reinforcement ratio. \(f_{y}^{\prime}\)--Measured yield strength of the core reinforcements in compression. \(f_{a}^{\prime}\)--Measured yield strength of the steel section in compression. \(f_{y}\)--Measured yield strength of the core reinforcements in tension. \(f_{a}\)--Measured yield strength of the steel section in tension. \(\alpha_{a}\)--Distance of the steel compression force from the wall side. \(f_{yo}\)--Measured strength of the vertical distributed reinforcements. \(\chi\)--Height of the compression zone. \(h_{0}\)--Effective height of the shear wall stem section. \(N\)--Axial compression load. \(M\)--Bending moment caused by the peak load. \(r\)--The ratio of the core area to the constrained area is 0.8. \(b\)--The width of the wall. \(A_{a},A_{a}^{\prime}\)--The area of the embedded steel section. \(f_{cc}\)--Compressive strength of the confined concrete. \(A_{s}^{\prime}\)--Area of the vertical reinforcements in compression. \(A_{a}\)--Area of the vertical reinforcements in tension. \(h\)--Height of the wall section. \(\varepsilon_{cc}\)--Peak strain of the confined concrete. The above equations give the force-balance relationship of the wall section. The compressive strain of concrete in the confined area increases significantly, which is considered in the above equations. For the whole compression area, the average stress should be taken.