## 4 Nonlinear Analyses  
Standard pushover (SPA, according to the first modal shape), uniform pushover analysis (UPA) and modal pushover analysis (MPA) were performed to assess the inelastic response of the bridges. The results of these analyses are presented and evaluated, comparing results with NRHA.  
According to the capacity spectrum method (ATC 40 [20]), the base shear forces and the corresponding displacements (being \(V_{bn}-u_{m}\) the pushover curve of the multi-degree of freedom) were converted to ADRS format, using the following relationships (being \(Sa\) and \(Sd\) the spectral accelerations and spectral displacements of an equivalent single degree of freedom (SDOF) system):  
\[Sa=\frac{V_{bn}}{M_{n}^{\star}} \tag{5}\]  
\[Sd=\frac{u_{m}}{\Gamma_{n}\varphi_{m}} \tag{6}\]  
where f\({}_{\rm m}\) is the value of f\({}_{\rm n}\) at the monitoring point; \(M^{\star}{}_{\rm n}=L_{\rm n}G_{n}\) is the effective modal mass; \(L_{\rm n}=f^{\rm T}{}_{\rm n}\ m\)\(l\), \(G_{\rm n}=L_{\rm n}/M_{\rm n}\) and \(M_{\rm n}=f^{\rm T}{}_{\rm n}\ m\!f_{\rm n}\) were the generalized mass for the natural mode, \(n\).  
As discussed by [PERSON] et al., in 2010 [11], if the structure remains elastic for the given earthquake intensity, the spectral displacement, \(Sd\), and the product G\({}_{n}\)\(\cdot\)f\({}_{\rm n}\) will be independent of the selection of the monitoring point; this means that deck displacements are independent from the location of the monitoring point.  
Conversely, in some applications of the same authors, it was found that deck displacements derived with respect to different monitoring points, for inelastic behavior of the structure, were not identical, but rather the estimated deformed shape of the bridge depended on the monitoring point selected for drawing the pushover curve for each mode.  
In the case of inelastic behavior, the conversion of the displacement demand of the \(n\)-mode SDOF system to the peak displacement of the monitoring point, \(u_{mi}\), of the bridge gives a different value, not only because of the deviation of the elastic mode shape, f\({}_{\rm n}\), from the actual deformed shape of the structure, but also due to the fact that the spectral displacement, \(Sd\), is dependent on the selection of monitoring point if the structure exhibits inelastic behavior (due to the bilinearization of the capacity curve). For the applications conducted in this work (RB and IB), it was noted that the approximations involved in the capacity-demand spectra procedure, deriving deck displacements with respect to different monitoring points, were neglectable, and results were deemed acceptable, analogously to what was concluded by other authors [36; 37], for all practical purposes. Figures 12 and 13 illustrate the deck displacements of the RB and the IB, derived by using pushover analysis for each mode independently, as well as the MPA, considering the monitoring point at the different positions: top of Pier 1 (P1), Pier 2 (P2) and Pier 3 (P3).  
Figure 11: Irregular bridge—mode shapes (transverse direction).  
According to what we discussed in Section 2, the same results can be finally obtained by selecting one or the other MP (it should be considered that a wrong selection can return a complex analysis calibration). According to some modal shapes, a null or quasi-null modal coefficient (or with an opposite sign if compared with the main bridge incremental deformation) can compromise the analysis (e.g., Pier 2 @ Mode 1 of the RB--Figure 11 and IB--Figure 12 and Pier 2 @ Mode 4 for both the IB). Therefore, the selection of the top of Pier 1 as the most suitable monitoring point was adopted for both cases.