## 1 Introduction
### Motivation  
There were two main motivations for this study. The first was to extend nonlinear static analyses to base-isolated buildings with irregularities, and the second was to predict the nonlinear peak responses of base-isolated buildings according to the concept of energy input. With respect to the first motivation, the authors proposed mode-adaptive bi-directional pushover analysis (MABPA) [39; 40; 41; 42; 43; 44]. This is a variant of nonlinear static analysis and was originally proposed for the seismic analysis of non-isolated asymmetric buildings subjected to horizontal bidirectional excitation. The first version of MABPA was proposed for non-isolated asymmetric buildings with regular elevation [39]; it was then updated following the development of displacement-based mode-adaptive pushover (DB-MAP) analysis [40]. This version has been applied to reinforced concrete asymmetric frame buildings with buckling-restrained braces [41] and to building models with bidirectional setbacks [42]. The applicability of MABPA has been discussed and evaluated based on the effective modal mass ratio of the first two modes [43]. The seismic capacity of an existing irregular building severely damaged in the 2016 Kumamoto earthquake (the former Uto City Hall) has been evaluated using MABPA [44]. Looking back on the development of MABPA, the logical next step should be to extend the method to base-isolated buildings with irregularities. Considering the case in which the seismic isolation period is well separated from the natural period of a superstructure, the behavior of the superstructure may be that of a rigid body, as discussed by several researchers, e.g., [27]. In such cases, the effective modal mass ratio of the first and second modes is close to unity, provided that the torsional resistance at the isolation layer is sufficient. It is expected that the seismic response of such a base-isolated building with plan irregularities will be accurately predicted by MABPA. In the nonlinear static analysis shown in the American Society of Civil Engineers ASCE/SEI 41-17 document [3], the reduced spectrum considering the effective damping is used to predict the target displacement for a nonlinear static analysis. Similarly, as shown in Notification No. 2009 of the Ministry of Construction of Japan [45], the equivalent linearization technique can be used for target displacement evaluations of base-isolated buildings. However, several researchers have examined the responses of long-period building structures subjected to pulse-like ground motions, e.g., [PERSON] examined a base-isolated building [29; 31] and [PERSON] and [PERSON] studied a tall reinforced concrete building [46]. Because the effective damping is calculated based on a steady response having the same displacement amplitude in the positive and negative directions, the use of effective damping for the prediction of the peak responses of base-isolated structures subjected to pulse-like ground motion is questionable. From this viewpoint, an alternative concept for predicting the peak response is required. Accordingly, the second motivation of this study was to investigate the use of the energy concept. The concept of energy input was introduced by [PERSON] in the 1980 s [47] and is implemented in the design recommendation for seismically isolated buildings presented by the Architectural Institute of Japan [2]. In [PERSON]'s theory, the _total input energy_ is a suit able seismic intensity parameter to access the cumulative response of a structure. Instead of the total input energy, [PERSON] et al. proposed the _maximum momentary input energy_[48; 49; 50] as an intensity parameter related to peak displacement; nonlinear peak displacement can be predicted by equating the maximum momentary input energy and the cumulative hysteresis energy during a half cycle of the structural response.