# 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
\(D_{c}\), \(D_{y}\), \(D_{m}\), \(D_{u}\) are the cracking displacement, yielding displacement, maximum-load displacement, and failure displacement, respectively, which are obtained from the average test results from Table 4.  
The obtained four-fold-line skeleton curve is expressed as Equation (1), and the relationships between load and stiffness are defined in Equations (2)-(5):  
\[P(x)=\left\{ \begin{array}{cc}K_{1}x&0\leq x\leq D_{c}\\ P_{c}+K_{2}(x-D_{c})&D_{c}<x\leq D_{y}\\ P_{y}+K_{3}(x-D_{y})&D_{y}<x\leq D_{m}\\ P_{m}+K_{4}(x-D_{m})&D_{m}<x\leq D_{u}\end{array} \right. \tag{1}\]  
\[D_{c}=\frac{P_{c}}{K_{1}}=\frac{0.295P_{m}}{K_{1}} \tag{2}\]  
\[D_{y}=\frac{P_{c}}{K_{1}}+\frac{P_{y}-P_{c}}{K_{2}}=\frac{0.295P_{m}}{K_{1}}+ \frac{0.527P_{m}}{0.821K_{1}}=\frac{0.936P_{m}}{K_{1}} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Specimens**} & **Cracking Load** & **Cracking Deformation** & **Yield Load** & **Yield Deformation** & **Yield Load** & **Yield Deformation** & **Yield Deformation** & **Yield Load** & **Yield Load** & **Yield Load** & **Yield Load** \\  & **Load** & **Deformation** & **Yield Load** & **Displace-** & **Displace-** & **Displace-** & **Load** & **Maximum Load** & **Load** & **Displace-** & **Coatelli** \\  & **(KN)** & **(mm)** & **(mm)** & **(mm)** & **ratio** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline XZW1 & 58 & 3.2 & 203 & 10.2 & 1/181 & 245.6 & 24.2 & 208.8 & 39.4 & 3.86 \\ XZW2 & 109 & 3.0 & 258 & 11 & 1/168 & 307.1 & 22.3 & 261.0 & 36.1 & 3.28 \\ XZW3 & 60 & 3.1 & 195.9 & 9.2 & 1/201 & 252.3 & 27.0 & 215.1 & 37.5 & 4.08 \\ XZW 4 & 110 & 3.3 & 265 & 10.5 & 1/176 & 313.0 & 25.9 & 268.0 & 34.6 & 3.29 \\ \hline \hline \end{tabular}  
* * Duclitity factor: The ratio of the ultimate displacement to the yield displacement of the specimen.  
\end{table} 
Table 4: Test results. Figure 9: Four-fold skeleton curve model. \[D_{m}=\frac{P_{c}}{K_{1}}+\frac{P_{y}-P_{c}}{K_{2}}+\frac{P_{m}-P_{y}}{K_{3}}= \frac{0.295P_{m}}{K_{1}}+\frac{0.527P_{m}}{0.821K_{1}}+\frac{0.178P_{m}}{0.145K_ {1}}=\frac{2.164P_{m}}{K_{1}} \tag{4}\]  
\[D_{u}=\frac{P_{c}}{K_{1}}+\frac{P_{y}-P_{c}}{K_{2}}+\frac{P_{m}-P_{y}}{K_{3}}- \frac{P_{u}-P_{m}}{K_{4}}=\frac{0.295P_{m}}{K_{1}}+\frac{0.527P_{m}}{0.821K_{1 }}+\frac{0.178P_{m}}{0.145K_{1}}-\frac{-0.15P_{m}}{0.129K_{1}}=\frac{3.327P_{m} }{K_{1}} \tag{5}\]  
1. Relationship between the theoretical initial stiffness and cracking stiffness The specimens studied in this paper are all straight shear walls, which are like vertically placed cantilever beams. By observing the skeleton curves of the specimens, it is close to an elastic state before cracking. The theoretical initial stiffness is derived by applying the basic theory of mechanics of materials. According to the design code [19], the rigidity of a steel-concrete composite wall is similar to that of a reinforced concrete wall of the same size. The steel sections can improve the flexural bearing capacity of the structure but make little contribution to its rigidity. Thus, the stiffness of a reinforced concrete shear wall with steel sections can be calculated approximately according to the stiffness of a reinforced concrete shear wall with the same section, and the corresponding shear deformation stiffness can be calculated with: \[\text{K}_{0}=\left(\frac{H^{3}}{3E_{c}I_{c}}+\frac{\mu H}{G_{c}A}\right)^{-1}\] (6) where \(H\) is the height of the wall, which is 1850 mm for the walls in this study (the height from the center of the actuator loading pad to the bottom of the wall); \(E_{c}\), \(I_{c}\)--The elastic modulus of concrete and the moment of inertia of the rectangular cross section of the wall, respectively. \(E_{c}\) is obtained from the concrete test results; \(G_{c}\)--The shear modulus of concrete, generally 0.4 of the elastic modulus of concrete; \(\mu\)--The non-uniformity coefficient of shear stress, for a rectangular section, is 1.2; \(A\)--Section area of the concrete wall (mm\({}^{2}\)).