## 1 Introduction
It is also worth remembering that, for seismic assessment of structures by means of NRHA, a set of GMs compatible with the seismic hazard spectrum of the site must be generated or selected; this, as described by [PERSON] and [PERSON] [24], is a complex task, since there are no clear procedures to generate artificial spectrum-compatible records or select appropriate suites of real GMs. It is also important to notice that, if IDA is used, it may result in being very difficult to find representative records for high intensities in the catalogues (given the few natural records available in databases for high intensities). Conversely, in pushover analysis, based on one or more modes, the seismic input can be easily defined by selecting appropriate spectral shapes for any specific seismic intensity. In fact, in pushover-based procedures, modeling the nonlinear behavior essentially follows NRHA without any requirement for hysteretic properties; the pushover analysis requires a simple bridge model that includes beam-column elements with plastic hinges [25]. Moreover, NSAs can provide all the relevant information required for investigating structural response: With a pushover, the identification of critical regions and prediction of the sequence of yielding and/or failure of structural members is very simple. Furthermore, many fundamental data for structural analysis are explicit and summarized in the capacity curve. According to this scientific background and developing previous studies of the same authors dealing with a nonlinear static analysis of buildings [26; 27; 28; 29], this work aims at proposing an innovative pushover-based procedure, named IMPA\(\beta\), for the assessment of bridges under seismic loads. With IMPA\(\beta\), detailed in Section 2.2, the IDA's NRHAs are replaced by a set of pushover analyses (IMPA\(\beta\) is incremental; therefore, a wide range of intensities is considered). The method, keeping the conceptual simplicity and computational attractiveness of standard nonlinear static analyses, is also suitable for irregular bridges and for high seismic intensity (advanced plastic structural stage), since different loading patterns, including the effects of higher modes, are adopted and integrated. In the development of IMPA\(\beta\), three pushover loading patterns were considered: (a) proportional to the first mode (SPA); (b) uniform (UPA); and (c) modal pushover (MPA). Patterns (a) and (b) are commonly adopted in pushover, according to FEMA-356 [19] or Eurocode 8 [30; 31]. The IMPA\(\beta\), finally proposed and discussed in Section 5, is an incremental procedure defined by a double pushover approach (both MPA and UPA are integrated in the procedure). In the paper, IMPA\(\beta\) is validated through the application to two case studies and a comparison with IDA (Section 4). The analyzed case studies are girder bridges with continuous deck, one regular (RB) and the other irregular (IB), according to the regularity definition made by Eurocode. These cases were already investigated by many authors in previous comparison works since the 1990s [32]. The bridge responses to the mean spectrum at different intensities obtained with pushover versus mean responses with NRHA are compared. The incremental curves are obtained by scaling the same seismic input in the two methods (pushover and NRHA) and using the mean spectrum for the pushover and the mean response for the NRHA.