## 4 Experimental Analysis
### Envelope Curve
#### 4.1.2 Envelope Curve Generalization  
The envelope curves of the flexural failure of the shear walls are affected by many factors, including the aspect ratio, compression ratio, reinforcement, etc. To obtain the generalized relationship and investigate the post-peak behavior of the PRMSW, the envelope curves of the specimens were simplified and normalized, and a four-line envelope curve, described by dimensionless coordinates, could be obtained through the test data.  
The key points of the generalized envelope curve are the yield point, the peak point, the failure point (the point corresponding to 85% \(Q_{u}\) in the post-peak stage), and the ultimate point (the point corresponding to 65% \(Q_{u}\) in the post-peak stage). The exact yielding point of the shear wall is difficult to acquire from the envelope curve. According to [26; 27; 28], there are several methods for the definition of the yield point, which include the geometric graphic method, the first yield method, and the reduced stiffness equivalent elasto-plastic method. In this paper, the yielding point of the test walls were defined by the equivalent elasto-plastic energy method [29], as shown in Figure 10. The calculated yielding displacement of each specimen is presented in Table 3. The peak point was defined as the point of the maximum lateral load during the test. For its relatively explicit value, the peak point (\(Q_{u},\Delta_{u}\)) was chosen as the datum point, and the envelope curves of the five PRMSWs were normalized based on the datum point.  
Figure 9: Envelope curves of the test walls.  
Through the linear regression analysis of the non-dimensional envelope curves, the coordinate of the yield point A was (0.34, 0.86); the coordinate of the peak point B was (1, 1); the coordinate of the failure point C was (1.80, 0.85), and the coordinate of the ultimate point D was (2.41, 0.65). Thus, the four-line simplified envelope curve can be expressed by Equation (1), and the generalized envelope curve is illustrated in Figure 11.  
\[\left\{\begin{array}{ll}OA:Q/Q_{u}=K_{1}(\Delta/\Delta_{u})&0\leq x\leq 0.34\\ AB:Q/Q_{u}=K_{2}(\Delta/\Delta_{u})+0.79&0.34<x\leq 1\\ BC:Q/Q_{u}=K_{3}(\Delta/\Delta_{u})+1.19&1<x\leq 1.80\\ CD:Q/Q_{u}=K_{4}(\Delta/\Delta_{u})+1.44&1.80<x\leq 2.41\end{array}\right. \tag{1}\]  
where \(K_{1}=2.53,\ K_{2}=0.21,\ K_{3}-0.19,\ K_{4}=-0.33\).  
Figure 11: The generalized four-line envelope curve of the PRMSW.  
Figure 10: Theoretical yielding point of the test walls.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Wall**} & \multicolumn{3}{c}{**Displacement (mm)**} & \multicolumn{3}{c}{**Drift and Displacement Ductility**} \\ \cline{2-9}  & \(\Delta_{y}\) & \(\Delta_{u}\) & \(\Delta_{0.85u}\) & \(\mu_{\Delta u}\) & \(drift_{\Delta u}\) (\%) & \(\mu_{\Delta 0.85u}\) & \(drift_{\Delta 0.85u}\) (\%) \\ \hline PMW-1 & 6.4 & 17.5 & 33.0 & 2.7 & 0.63 & 5.2 & 1.18 \\ PMW-2 & 5.9 & 12.8 & 26.6 & 2.2 & 0.46 & 4.5 & 0.95 \\ PMW-3A & 6.4 & 26.5 & 39.3 & 4.1 & 0.95 & 6.1 & 1.40 \\ PMW-3B & 6.2 & 23.4 & 38.5 & 3.8 & 0.84 & 6.2 & 1.38 \\ PMW-4 & 5.4 & 16.2 & 28.7 & 3.0 & 0.58 & 5.3 & 1.03 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of displacement, drift, and ductility of PRMSW specimens.