## 2 Short Description of IMPA\(\beta\) Procedure  
It is assumed that NRHA represents the \"correct\" response evaluation, even for NF input. The incremental modal pushover analysis for bridges (IMPA\(\beta\)) [18] is carried out and results are compared to IDAs. IMPA\(\beta\) requires the execution of several pushover analyses according to two different load patterns: a uniform loading profile (UPA) and a modal pushover analysis (MPA) according to the procedure presented in [16; 17]. As previously mentioned, the use of EMPA (which includes vertical action) instead of MPA was tested and verified as irrelevant for the scope of the analysis. In EMPA the seismic excitation vector is the sum of three terms, each one corresponding to the components of the accelerogram record. The pushover capacity curve is obtained taking into account the contributions of the cited excitation vector in the considered degrees of freedom (transversal and vertical displacement): two capacity curves were recorded in a single pushover analysis of each mode (the dominant modes associated with the selected directions) and the component along the investigated direction (the transversal for the case study discussed herein) were successively combined according to the SRSS rule in order to obtain the final capacity curve. Nonetheless, the prediction of the structural response of the bridge with EMPA, also for NF events, is close to MPA (the same conclusion is discussed in [19] for the case study performed with FF events) and therefore the original version of IMPA\(\beta\), developed and tested considering far-field ground motions (FF), was tested and discussed herein and validated in case of NF. The irrelevance of the vertical input in the pushover analysis, according to the scope and the results of this paper, is discussed in Section 4. In addition, in this work a standard pushover analysis (SPA), with a load profile proportional to the dominant modal shape has been performed to compare results from different pushover procedures with NRHA. The IMPA\(\beta\) procedure can be summarized in the following main steps:  
* Definition of seismic demand in terms of a response spectrum (RS);
* Discretization of the intensity range and definition of a set of intensity levels I;
* Traditional pushover analysis adopting a uniform loading profile (UPA);
* Modal analysis of the bridge in order to select the relevant modes;
* Modal pushover analysis for each intensity level and, selection of the performance point (P.P.). Combining what obtained according to a combination rule (e.g., CQC or SRSS) a performance point that can be considered \"multimodal\" (P.P.m,i) can be defined. Therefore, IMPA\(\beta\) requires to perform an incremental modal pushover analysis (IMPA) and an incremental UPA (IUPA). Afterwards, in order to obtain a univocal capacity curve, for each intensity step \(i\), the IMPA\(\beta\) procedure requires to evaluate the \(P_{i,IMPA\beta}\) (Figure 1) according to (1). The capacity curve obtained is therefore an envelope of the results obtained from UPA and MPA. \[P_{i,IMPA\beta}=(\max\left(\mathtt{u}_{\mathrm{r}i,\mathrm{J}\mathrm{P}A}; \mathtt{u}_{\mathrm{r}i,\mathrm{U}\mathrm{P}A}\right);\max\left(\mathtt{V}_{ \mathrm{b}i,\mathrm{M}\mathrm{P}A};\mathtt{V}_{\mathrm{b}i,\mathrm{U}\mathrm{P }A}\right)) \tag{1}\]  
being:  
\(\mathtt{u}_{\mathrm{r}i,\mathrm{M}\mathrm{P}A}\) the displacement at the P.P.m,i obtained performing MPA;  
\(\mathtt{V}_{\mathrm{b}i,\mathrm{M}\mathrm{P}A}\) the base shear at the P.P.m,i obtained performing MPA;  
\(\mathtt{u}_{\mathrm{r}i,\mathrm{U}\mathrm{P}A}\) the displacement at the P.P.i obtained performing UPA;  
\(\mathtt{V}_{\mathrm{b}i,\mathrm{U}\mathrm{P}A}\) the base shear at the P.P.i obtained performing UPA. The seismic demand is expressed in terms of Response Spectrum (RS): a RS can be selected and scaled according to different criteria. In the application discussed herein the RS has been derived from the set of near-fault ground motion records selected (SS 3.2).