## 4 Nonlinear Analyses  
The bridge was analyzed at several seismic intensity levels: considering a scale factor from 0.5 to 2.0 being the average PGA of the GM selected 0.32g and being PGA = 0.35 g the design value of the bridge analyzed. The nonlinear dynamic analyses were preceded by the application of the gravity loads performing a nonlinear static analysis. The nonlinear response-history analysis (NRHA), performed using time increments of 0.1 s, was performed considering both the vertical and the horizontal component (transversal direction) of the ground motion. Among the common methods provided by SAP2000 software for conducting direct-integration time history analysis, the Hibler-Hughes-Taylor alpha (HHT) technique [36; 37] was used in this study; the large displacement effects and material nonlinear properties were also considered. Figure 6: Bridge modal shape: most relevant modes in the transversal direction. From Figures 7-9, the structural response to each GM is plotted in terms of deck displacement (transversal displacement of the deck at Pier 1, Pier 2 and Pier 3) and the variation of the axial load on each pier. The axial load variation is useful to evaluate the relevance of the vertical component of the GM in terms of axial load-bending moment interaction (the hinge of the piers are PM being P = axial load and M = bending moment) and therefore for the structural response; in a pushover analysis this loading variation cannot be considered adopting procedures such as SPA, UPA or MPA. In all the cited pushover methods the analysis is performed starting from a preliminary nonlinear step in which the vertical static loads are imposed. As already discussed in the introduction, in a preliminary phase of this work, the use of EMPA (a modal pushover that includes also the vertical component of the seismic action) as an alternative to MPA was evaluated. This modification of the IMPA\(\beta\) was discarded because the results, using MPA or EMPA were very similar as discussed below. Figure 7: NRHA with scale factor 1.0 (considering both vertical and transversal components: top). Variation of the axial load (from the static load on piers during the seismic event; bottom). Deck displacements. As one can observe from Figures 7-9 (only the scale factor SF = 1.0 is plotted for the sake of brevity), the response to AMT record is characterized by the highest variation of the piers axial load (about \(\pm 40\%\) of variation from the static load determined in seismic condition and used for the pushover analyses). This condition may suggest that the vertical action is relevant, and therefore, it can compromise results of a static pushover analysis. However this remark, that is true only for some typologies of structures  
Figure 8: NRHA with scale factor 1.0 (considering both vertical and transversal components: top) Variation of the axial load (from the static load on piers during the seismic event; bottom). Deck displacements. Figure 9: NRHA with scale factor 1.0 (considering both vertical and transversal components: top) Variation of the axial load (from the static load on piers during the seismic event; bottom). Deck displacements. and in particular only for some specific typologies of bridges (e.g., cable-stayed bridges [19]), cannot be applied to bridges, like the case study of this work, since the gravity loads influenced the design of the deck but not that of the pier. In fact, piers are usually designed with the bending action and ductility requirements (Eurocode 8 in this case [38]). Moreover, the axial load capacity is determined considering loading conditions (e.g., traffic load) that are much stronger than the seismic vertical action. The case study analyzed is an example of this condition; Figure 10 demonstrates that even a variation of axial load of about \(\pm 40\%\) (that is the highest of the sets of time histories selected) does not influence the cross-section plastic behavior and therefore, the capacity of the piers.