# 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  
(1) Mathematical expression of dimensionless skeleton curves  
The skeleton curves of the specimens (Figure 7) are obtained from the quasi-static tests of the four specimens. It can be seen that although the curves vary in values due to factors including the axial pressure ratio, the shapes of the curves are similar. During the test, the cracking load and cracking displacement were recorded at the occurrence of cracking (close to the base), which was visually observed. The yield load and displacement were obtained by a commonly used secant line method with reference to the strains recorded at the flange of the steel sections. The failure load and ultimate displacement were obtained by taking \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Loading Steps**} & \multirow{2}{*}{\
\begin{tabular}{c} **Load Magnitude** \\ **and Direction** \\ \end{tabular} } & \multirow{2}{*}{**Loading Steps**} & \multirow{2}{*}{
\begin{tabular}{c} **Displacement** \\ **Magnitude and** \\ **Direction** \\ \end{tabular} } \\  & & & & **Direction** \\ \hline
1 & +10 kN & 7 & +3 mm \\
2 & \(-\)10 kN & 8 & \(-\)3 mm \\
3 & +20 kN & 9 & +6 mm \\
4 & \(-\)20 kN & 10 & \(-\)6 mm \\
5 & +30 kN & 11 & +9 mm \\
6 & \(-\)30 kN & 12 & \(-\)9 mm \\  & Crack occurs and & & \\  & switch to & & \\  & displacement control & & wall fails \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Loading scheme. the values corresponding to the point when the horizontal force dropped to 85% of the maximum load. The maximum load and the corresponding displacement were obtained by the measured force of the actuator and the displacement sensor, which are the actual measured values. Because all of the specimens have similar initial stiffness, and before yielding, the deformation and fracture development of specimens are small, no stiffness change is significant. The curves of all specimens coincide before the yielding point, indicating that all the specimens had a relatively large stiffness. After yielding, the stiffness of different specimens varies greatly due to the different axial compression ratios. As a result, the curves shift away from each other, especially after the maximum load point. The dimensionless skeleton curve (Figure 8) is established by normalizing the load values by the maximum load and displacement; that is, the values at the cracking point, yielding point, maximum load point, and failure point are all divided by the value of the maximum load point. In the same dimension, the curves of each specimen converge and have similar changing characteristics. Therefore, on this basis, a changing broken line can be used to approximately express the change rule of the restoring force curve. Figure 8: Dimensionless skeleton curves (loads are normalized by the maximum load). Figure 7: Skeleton curves of specimens. Through the analysis of the dimensionless curves, the curves are transformed into a four-line curve, as shown in Figure 9. Because the curves in the positive and negative directions are similar, only the positive part of the curves are used. In Figure 9, \(P_{c}\), \(P_{y}\), \(P_{m}\), and \(P_{u}\) respectively represent the cracking load, yielding load, maximum load, and failure load; and \(P_{c}\) = 0.295 \(P_{m}\), \(P_{y}\) = 0.822 \(P_{m}\), and \(P_{u}\) = 0.85 \(P_{m}\) according to the average test results from Table 4.  
The slope of the four lines is the stiffness of each part, labelled as \(K_{1}\), \(K_{2}\), \(K_{3}\), \(K_{4}\). Their numerical relations can be obtained from the geometric relations, \(K_{2}\) = 0.821 \(K_{1}\), \(K_{3}\) = 0.145 \(K_{1}\), \(K_{4}\) = 0.129 \(K_{1}\).