## 6 Conclusions  
In this study, the main building of the former Uto City Hall was investigated as a case study of the retrofitting of an irregular reinforced concrete building using the base-isolation technique. The nonlinear peak responses of two retrofitted building models subjected to horizontal bidirectional ground motions were predicted by MABPA, and the accuracy of the method was evaluated. The main conclusions and results are as follows:  
* The predicted peak response according to the updated MABPA agreed satisfactorily with the envelope of the time-history analysis results. The peak relative displacement at \(X_{3\Delta}Y_{3}\) at each floor can be satisfactorily predicted. The predicted distribution of the peak displacement at level 0 (just above the isolation layer) approximated the envelope of the nonlinear time-history analysis results, even though in some cases, the predicted distributions differed from the envelope of the nonlinear time-history analysis. A discrepancy between the predicted results and nonlinear time-history analysis may occur because of the lack of a contribution from the higher modal responses. * The relationship between the equivalent velocity of the maximum momentary input energy of the first modal response (\(V_{\Delta E1U}{}^{*}\)) and the peak equivalent displacement of the first modal response (\(D_{1U}{}^{*}\)max) can be properly evaluated from the pushover analysis results. The plots obtained from the nonlinear time-history analysis results fit the evaluated curve from the pushover analysis results well. * The upper bound of the peak equivalent displacements of the first two modal responses can be predicted using the bidirectional \(V_{\Delta E}\) spectrum [52]. Comparisons between the predicted peak equivalent displacements and those calculated from the nonlinear time-history analysis results showed that the predicted peak approximatedthe upper bound of the nonlinear time-history analysis results. The upper bound of \(V_{\Delta E1U}*\) can be approximated by the bidirectional \(V_{\Delta E}\) spectrum. Based on the above findings, the updated MABPA appears to predict the peak responses of irregular base-isolated buildings with accuracy. However, MABPA still has two shortcomings. The first shortcoming is the limitation of the applicability of MABPA. As discussed in a previous study [43], the application of MABPA is limited to buildings classified as torsionally stiff buildings. The current (updated) MABPA has the same restriction. This limitation can be avoided if the torsional resistance of the isolation layer is sufficiently provided, as shown in this study. The second shortcoming involves the contributions of the higher modal responses. In the original MABPA for non-isolated buildings, only the first two modes were considered for the prediction. Therefore, the prediction was less accurate for cases when the response in the stiff-side perimeter was larger than that in the flexible-side perimeter. The contributions of the third and higher modal responses need to be investigated. Another aspect of this study to be emphasized is the application of the bidirectional \(V_{\Delta E}\) spectrum for the prediction of the peak response of a base-isolated building. The results shown in this study imply that the bidirectional \(V_{\Delta E}\) spectrum [52] is a promising candidate for a seismic intensity parameter for the peak response. As discussed in a previous study [53], one of the biggest advantages of the bidirectional momentary input energy is that it can be directly calculated from the Fourier amplitude and phase angle of the ground motion components using a time-varying function of the momentary energy input, without knowing the time-history of the ground motion. This means that researchers can eliminate otherwise unavoidable fluctuations from the nonlinear time-history analysis results.