# Effective Assessment of Inelastic Torsional Deformation of Plan-Asymmetric Shear Wall Systems
## 4 Comparison between Predictions by the Proposed Method and Time History Analysis Results
### Analysis Results and Interpretation  
In this section, the effectiveness of the inelastic torsional deformation assessment procedure proposed in Section 3 is verified by comparing its results with those of the time history analyses of the model problem. Figure 15 shows the lateral displacement history for \(\alpha=1.3\), \(\beta=1.0\) under El Centro excitation. The lateral displacements of the two \(y\)-directional walls are plotted in Figure 15a. In the figure, the solid line represents the displacement of the flexible side wall \(Y_{1}\), and that of the stiff side wall \(Y_{2}\) (dotted line). The difference between the lateral displacements of the two walls can be interpreted as the rotation of the asymmetric system. Figure 15b shows the lateral displacement history of the two \(x\)-directional walls, in which the lateral displacements of the two walls are of almost the same magnitude, but in the opposite direction. This indicates that the displacement history of the \(x\)-directional walls is mainly determined by the rotational behavior of the asymmetric system. In Figure 15c, the displacement of the flexible side wall \(Y_{1}\) is decomposed into the component induced by lateral shear force and that by torsion, which are differentiated by a dark line and a light one, respectively. The sum of the two displacement components is equivalent to the solid line in Figure 15a. Figure 15: Lateral displacement history of each wall component (El Centro, \(\alpha=1.3\), \(\beta=1.0\)). The results of Figure 15 are replotted in Figure 16, which presents the lateral displacement and rotation history estimated at the mass center of the asymmetric model problem. The maximum lateral displacement of 0.279 m occurs at 4.61 s, while the maximum rotation of \(-0.0119\) rad occurs at 9.30 s. This clearly indicates that the two peak values do not occur simultaneously due to the dynamic effect on the structure by the excitation loading. The maximum rotations of the 63 time history analysis simulations are plotted with respect to the relative eccentricity (\(e_{x}\)/L) and relative stiffness in the transverse direction (\(\beta\)) in Figure 17a,b, respectively. It can be noted from Figure 17a that the maximum rotations of all 63 simulations show an increasing trend with increasing relative eccentricity. As already discussed in Section 4.2, \(\beta\) can be regarded as the torsional constraint ratio, and thus Figure 17b clearly indicates that the maximum rotations of the time history analyses decrease as \(\beta\) is increased. Table 4 provides the results of the time history analysis, the proposed method and the linear elastic torsional response computed for the 9 different combinations of \(\alpha\) and \(\beta\) discussed in Section 4.2. The ground motion applied in the time history analysis is the El Centro excitation. In the time history analysis results of the table, the maximum displacement in the \(y\)-direction at the center of mass (\(\Delta_{\mathit{cnt\_TH}}\)) was given, and then the rotation at the center of mass (\(\varphi_{\mathit{cnt\_TH}}\)) and the displacement on the flexible side wall Y\({}_{1}\) (\(\Delta_{\mathit{flex\_TH}}\)) corresponding to this displacement were listed. The maximum displacement \(\Delta_{\mathit{cnt\_TH}}\) was taken as the target displacement in the results of the proposed method and the linear elastic response, and it was used to compute the rotation at the center of mass (\(\varphi_{\mathit{cnt\_pro}}\) and \(\varphi_{\mathit{cnt\_das}}\)) and the displacement on the flexible side wall Y\({}_{1}\) (\(\Delta_{\mathit{flex\_pro}}\) and \(\Delta_{\mathit{flex\_des}}\)) by each of the two methods, respectively. In addition, the relative error of the displacement on the wall Y\({}_{1}\) was calculated by taking the corresponding time history analysis value \(\Delta_{\mathit{flex\_TH}}\) as the reference value. The average of the absolute relative errors for the other two methods are also provided at the end of the table.