## 4 Nonlinear Analyses
### Analysis of the Irregular Bridge (IB)  
The same analysis performed on the RB were repeated for the IB. All relevant modes (Modes 1, 3 and 4) were taken into account when performing the proposed procedure (Figure 11). From Figures 19-22, it can be observed that the results of the \"standard\" pushover methods (SPA and UPA) differed, even qualitatively, from the results of the NRHA, and the envelope of MPA and UPA emerged as the  
Figure 17: Regular bridge—deck drift (relative displacement between two consecutive joints P\({}_{i}\) and P\({}_{i+1}\)/P\({}_{i}\)-P\({}_{i+1}\) span length) derived by performing nonlinear dynamic analysis and pushover analysis according to the approaches considered (PGA 0.175–0.7 g). Figure 18: Regular bridge—incremental curve (displacement-intensity) and capacity curves derived with IDA (maximum values of ur and Vb,x) or IMPA\(\beta\) (the design PGA was 0.35 g, corresponding to a transversal base shear of V\({}_{b,x}\)\(\sim\) 9200 kN). best approach that, for an intensity level corresponding to the design intensity (0.35 g), gave results similar to NRHA. In general, every pushover procedure considered overestimated the response at the stiffer side (Pier 1), whereas at the center (Pier 2) and at the flexible side (Pier 3), the response was underestimated. According to the results presented herein, also confirmed from the scientific literature (i.e., other applications to this type of bridge--[PERSON] et al. (2019); [PERSON] et al. (2020)), for bridges with asymmetric modes (Modes 1 and 4) and other relevant modes translational (Mode 3, the most relevant one), the occurrence of the first plastic hinges is crucial; in the IB, when the hinge occurred in the central column, the mode shape drastically changed, and it should be recognized that even the multimodal method cannot reflect these sudden and substantial changes of the dynamic properties of the structure. Figure 19: Irregular bridge—deck displacements derived by performing nonlinear dynamic analysis and pushover analysis according to the approaches considered (PGA from 0.175 to 0.7 g). Figure 20: Irregular bridge—deck displacements derived by performing nonlinear dynamic analysis and pushover analysis according to the approaches considered (PGA from 0.175 to 0.7 g). Figure 21: Irregular bridge—pier drift (transversal top displacement/height) derived by performing nonlinear dynamic analysis and pushover analysis according to the approaches considered (PGA from 0.175 to 0.7 g). Figure 22: Irregular bridge—deck drift (relative disp. between two consecutive joints, P\({}_{i}\) and P\({}_{i+1}\)/P\({}_{i}\)-P\({}_{i+1}\) span length) derived by performing nonlinear dynamic analysis and pushover analysis according to the approaches considered (PGA 0.175–0.7 g). On the other hand, the MPA worked relatively well even in the case of the design earthquake, when the hinge in the central column occurs. This, however, was not surprising, since the response was predominantly influenced by one mode dictated by the superstructure. Results are reported in the following figures considering the incremental range. All relevant modes (Modes 1, 3 and 4) were taken into account when performing each MPA. As it can be observed also from Figures 19-23 (in Figure 19, for each intensity step, the curves MPA-UPAenv are compared with the NRHA results), and MPA coincided quite well with the results of NRHA up to the design intensity (from PGA 0.175 g to PGA 0.35 g), but, to achieve a better estimation, the envelope of MPA and UPA should be considered (MPA-UPAenv). For higher intensities (scale factors 1.5 and 2.0: PGA over 0.525 g), a good estimation of displacement was observed. Figure 19 illustrates that the first plastic hinges were in Pier 2, first, and Pier 1 after (at a PGA of 0.175 g that was lower than the design intensity), whereas, in Pier 3, the first hinge emerged only at a very high intensity level (greater than PGA 0.525 g).