## 5 Discussion
### Contribution of the Higher Mode to the Displacement Response at the Edge of Level 0  
As discussed in Section 4.2.2, the accuracy of the predicted horizontal distribution of the peak response at level 0 depends on the ground motion dataset. According to Model-Tf34, the predicted largest peak displacement in the Y direction occurred at X\({}_{1A}\) (the flexible side in the Y direction), while the envelope of the nonlinear time-history analysis results indicates that the largest peak occurred at X\({}_{6A}\) (the stiff side in the Y direction), the opposite side to X\({}_{1A}\) in the case of UTO0414, as shown in Figure 26a. Conversely, in the case of TCU shown in Figure 26b, the envelope of the time-history analysis results indicates that the largest peak occurred at X\({}_{1A}\), which is consistent with the predicted results. In this subsection, the modal response at level 0 is calculated and discussed. From the time-history of the displacement at the center of mass of level 0, the horizontal displacement in the Y direction at point \(j\) on level 0 can be calculated as:  
\[d_{Y0j}(t)=y_{0}(t)-L_{X0j}\theta_{0}(t). \tag{28}\]  
In Equation (28), \(L_{X0j}\) is the location of point \(j\) in the X direction from the center of mass of level 0. Therefore, the modal response of the horizontal displacement at point \(j\) can be calculated as:  
\[d_{Y0j1}(t) =\Gamma_{1U}\big{(}\phi_{Y01}-L_{X0j}\phi_{001}\big{)}D_{1U}{}^{* }(t), \tag{29}\] \[d_{Y0j2}(t) =\Gamma_{2V}\big{(}\phi_{Y02}-L_{X0j}\phi_{002}\big{)}D_{2V}{}^{* }(t),\] (30) \[d_{Y0 jh}(t) =d_{Y0j}(t)-\big{\{}d_{Y0j1}(t)+d_{Y0j2}(t)\big{\}}. \tag{31}\]  
Figure 41 shows comparisons of the modal responses at the edge of level 0. The structural model shown in this figure is Model-Tf34, the input ground motion dataset is UTO0414, and the angle of incidence of the seismic input (\(\psi\)) is \(-75^{\circ}\), the angle at which the largest peak response at X\({}_{1A}\) occurs. Note that \"All modes\" is the response originally  
Figure 40: Peak equivalent displacements of the first two modes for various directions of seismic input (Model-Tf44): (a) UTO0414; (b) UTO0416; (c) TCU; (d) YPT. obtained from the time-history analysis results (\(d_{Y0j}(t)\)), \"First mode\" and \"Second mode\" are the first and second modal responses (\(d_{Y0j1}(t)\) and \(d_{Y0j2}(t)\), respectively), and \"Higher mode\" is the higher (residual) modal response calculated from Equation (31) (\(d_{Y0 jh}(t)\)). In the response of \(X_{\rm IA}\), shown in Figure 41a, the contribution of the higher modal response was non-negligible, even though the contribution of the first modal response was predominant. In addition, the sign of the higher modal response at the time the peak response occurred at \(X_{\rm IA}\) was opposite to that of the \"All mode\" response, with the contribution of the higher mode reducing the peak response at \(X_{\rm IA}\). In the response of \(X_{\rm 6A}\), shown in Figure 41b, the contribution of the first modal response was negligibly small and those of the second and higher modal responses were noticeable. In addition, the sign of the higher modal response at the time the peak response occurred at \(X_{\rm 6A}\) was the same as that of the \"All mode\" response, with the contribution of the higher mode increasing the peak response at \(X_{\rm 6A}\). This indicates that the reason why the largest peak response in the envelope of the time-history analysis results occurred at \(X_{\rm 6A}\) (_not_ at \(X_{\rm IA}\)) in the case of UTO0414 can be explained by the contributions of the higher modal response. In the case of UTO0414, the contribution of the higher modal response was non-negligibly large. Another comparison is made in Figure 42 for the structural model Model-Tf34, the input ground motion dataset TCU, and an angle of incidence of the seismic input (\(\psi\)) of 60\({}^{\circ}\), where the angle of the largest peak response at \(X_{\rm IA}\) occurs.