## 4 Nonlinear Analyses
This condition encourages the development and testing of pushover-based procedures also for bridges under near-fault seismic events and it justifies the good performance of IMPA\(\beta\) described in (SS5). In Figures 11-13 are displayed the deformed shapes of the deck when the maximum displacement of the monitoring point (P1) has been recorded. From the analysis, it emerged that, in the near-field case the structural response was characterized by the contemporary achievement of the maximum displacement of the monitoring point and the maximum global base shear, which is different to what usually happens when performing NRHA with FF ground motions. By comparing Figures 11-13 and observing Figure 14 it is clear how the inelastic response, for scale factors greater than 1.0, modifies the response and the deformed shape changes becoming more uniform and less similar to a modal shape. Figure 11: NRHA with scale factor 0.5— deck displacements. Figure 10: Axial load—bending moment @ yielding compared with a variation of axial load of \(\pm\) 40%. From the top: Pier 1, Pier 2 and Pier 3. Figure 15 illustrates that standard pushover methods, that adopt a first mode loading pattern (SPA) or a uniform loading pattern (UPA), differ (for high intensities, SF = 2.0) from the results of the nonlinear dynamic analysis both in terms of values and trend of the deformed shape. By enveloping the modal and the uniform pushover a better response evaluation can be derived; this envelope proposed in IMPA\(\beta\) is well performing if compared with NRHA for an intensity level corresponding to a scale factor 1.0 and therefore to a PGA = 0.32 g, which is very close to the bridge design intensity (PGA = 0.35 g). All the nonlinear static procedures considered underestimate the response at the center (P2) and at the flexible side (P3). This result is consistent to what was obtained in [18], applying IMPA\(\beta\) to the same case study for far-field events (FF). Figure 14: NRHA average response compared with displacements corresponding to plastic hinges yielding. Figure 12: NRHA with scale factor 1.0— deck displacements. Figure 13: NRHA with scale factor 2.0— deck displacements. According to the results presented in other works considering FF events ([PERSON] et al. [35; 36]), the application with NF events confirmed that for bridges having a main translational mode (mode 3) with nonsymmetric modal shapes (mode 1 and 4), it is crucial to first activate the plastic hinges as shown in Figure 16.  
The mode shape changed when the hinge occurred in P2 and, as it can be observed in Figures 15 and 16, the MPA cannot follow this relevant modification of the dynamic response of the bridge. The modal pushover analysis is well performing in the case of an earthquake with a PGA corresponding to the design value (PGA = 0.35 g, the hinge in P2 is active); this latter condition is related to the fact that the bridge response was conditioned by the superstructure. In Figure 17, the MPA-UPAenv curves are compared with results from nonlinear response history analysis, the modal  
Figure 7:0 Fault normal: displacements of the deck derived performing NRHA and IMPA\(\beta\) (scale factor from 0.5 g to 2.0 g). Figure 7:1 Fault normal: displacements of the deck derived performing NRHA and nonlinear static analysis (scale factor from 0.5 g to 2.0 g). pushover and the nonlinear dynamic response are quite similar up to the real average intensity of the seismic event (from PGA 0.16 g to PGA 0.32 g), but the envelope of modal and uniform pushover achieved a better estimation. For higher intensities (scale factor 2.0: PGA = 0.64 g) a good estimation of displacement was only observed at Pier 1, according to the experience from FF events; from Figures 14 and 16, it can be observed that the first plastic hinges occurred, for an intensity level (about PGA = 0.3 g), lower than the design one, first in P2 than in P1. The first hinge in P3 occurred at a very high intensity level (higher than PGA 0.64 g).