## 5 Discussion
### Accuracy of the Predicted Peak Equivalent Displacements of the First and Second Modal Responses  
Figure 37 shows the accuracy of the predicted peak equivalent displacement of the first and second modal responses, \(D_{1U}\)\({}^{*}\)\({}_{\rm max}\) and \(D_{2V}\)\({}^{*}\)\({}_{\rm max}\), respectively, for Model-Tf34. The predicted peak in the case of complex damping \(\beta\) = 0.10 approximated the upper bound of  
Figure 36: Prediction of \(V_{\Delta E1U}\)\({}^{*}\) from the bidirectional \(V_{\Delta E}\) spectrum and its accuracy (Model-Tf44): (**a**) Art-1 series; (**b**) Art-2 series; (**c**) UTO0414; (**d**) UTO0416; (**e**) TCU; (**f**) YPT.  
the time-history analysis results, except when the ground motion dataset was the Art-2 series (Figure 37b) or YPT (Figure 37f).  
Figure 38 shows the accuracy of the predicted \(D_{1U}{}^{*}{}_{\text{max}}\) and \(D_{2V}{}^{*}{}_{\text{max}}\) for Model-Tf44. The predicted peak in the case of \(\beta=0.10\) approximated the upper bound of the time-history analysis results in all cases.  
It is also observed from Figures 37c-f and 38c-f that the largest \(D_{1U}{}^{*}{}_{\text{max}}\) and \(D_{2V}{}^{*}{}_{\text{max}}\) did not occur simultaneously. To understand this phenomena, Figures 39 and 40 show \(D_{1U}{}^{*}{}_{\text{max}}\) and \(D_{2V}{}^{*}{}_{\text{max}}\) for varying angles of incidence of the seismic input (\(\psi\)) for both models. As shown here, the angle where the largest \(D_{2V}{}^{*}{}_{\text{max}}\) occurred was different from the angle where the largest \(D_{1U}{}^{*}{}_{\text{max}}\) occurs, with the difference between the two angles being approximately \(90^{\circ}\). This was clearly observed in the cases of TCU and YPT for both models.  
Based on the above discussion, the upper bound of the peak equivalent displacements of the first and second modal responses, \(D_{1U}{}^{*}{}_{\text{max}}\) and \(D_{2V}{}^{*}{}_{\text{max}}\), can be predicted by the updated MABPA.  
The variation in \(D_{1U}{}^{*}{}_{\text{max}}\) and \(D_{2V}{}^{*}{}_{\text{max}}\) due to the angles of incidence of the seismic input \(\psi\) may explain the variation in the peak displacement of the isolators shown in Figures 29 and 30. As the example, the response of Model-Tf44 subjected to YPT ground motion was focused on. From Figure 40c, the angle (\(\psi\)) at which the largest \(D_{1U}{}^{*}{}_{\text{max}}\) occurred was \(-30^{\circ}\). While the peak displacement at isolator \(X_{1A}Y_{6}\) shown in Figure 30c, the peak displacement at the angle (\(\psi\)) of \(-30^{\circ}\) was close to the largest peak value. On the contrary, the peak displacement at isolator \(X_{6A}Y_{1}\) at the angle (\(\psi\)) of \(-30^{\circ}\) was close to the smallest peak value.  
Figure 37: Accuracy of the predicted peak equivalent displacements of the first two modes (Model-Tf34): (**a**) Art-1 series; (**b**) Art-2 series; (**c**) UTO414; (**d**) UTO0416; (**e**) TCU; (**f**) YPT.  
Figure 39: Peak equivalent displacements of the first two modes for various directions of seismic input (Model-Tf34): (**a**) UTO0414; (**b**) UTO0416; (**c**) TCU; (**d**) YPT.  
Figure 38: Accuracy of the predicted peak equivalent displacements of the first two modes (Model-Tf44): (**a**) Art-1 series; (**b**) Art-2 series; (**c**) UTO0414; (**d**) UTO0416; (**e**) TCU; (**f**) YPT.