**Figure 1.**
IMPA procedure. (**a**) Evaluation of the performance points (P.P.) for each capacity curve that belongs to the pushover analysis: proportional to Mode 1. Mode n. (**b**) Evaluation of the P.P. for each capacity curve: e.g., via C.S.M. (**c**) Capacity curve plotted in the u_{r}-intensity plane (u_{r} is the displacement of the monitoring point along the investigated direction, e.g., the transverse direction).

**Figure 1.**
IMPA procedure. (**a**) Evaluation of the performance points (P.P.) for each capacity curve that belongs to the pushover analysis: proportional to Mode 1. Mode n. (**b**) Evaluation of the P.P. for each capacity curve: e.g., via C.S.M. (**c**) Capacity curve plotted in the u_{r}-intensity plane (u_{r} is the displacement of the monitoring point along the investigated direction, e.g., the transverse direction).

**Figure 2.**
Degree of freedom of the bridge: by performing the NSA, the displacement u_{r} of the monitoring point is controlled. In this work, u_{r} is the transversal displacement.

**Figure 2.**
Degree of freedom of the bridge: by performing the NSA, the displacement u_{r} of the monitoring point is controlled. In this work, u_{r} is the transversal displacement.

**Figure 3.**
Application of the Square Root of the Sum of Squares rule to a bridge.

**Figure 3.**
Application of the Square Root of the Sum of Squares rule to a bridge.

**Figure 4.**
Evaluation of the IMPAβ capacity curve: envelope of IUPA and IMPA.

**Figure 4.**
Evaluation of the IMPAβ capacity curve: envelope of IUPA and IMPA.

**Figure 5.**
Flowchart of IMPAβ procedure.

**Figure 5.**
Flowchart of IMPAβ procedure.

**Figure 6.**
Layout of the bridges: regular bridge (RB) and irregular bridge (IB).

**Figure 6.**
Layout of the bridges: regular bridge (RB) and irregular bridge (IB).

**Figure 7.**
(**a**) Regular bridge: pier sections. (**b**) Irregular bridge: pier sections.

**Figure 7.**
(**a**) Regular bridge: pier sections. (**b**) Irregular bridge: pier sections.

**Figure 8.**
Individual response spectra ζ = 5%, “component” of the ground motion records (transverse direction of bridge models) for the seven unscaled ground motions (RS1, …, RS7) and their median response spectrum (RSm), used for the pushover analyses, with the RS being the code elastic spectrum used for the bridges (RB and IB) design.

**Figure 8.**
Individual response spectra ζ = 5%, “component” of the ground motion records (transverse direction of bridge models) for the seven unscaled ground motions (RS1, …, RS7) and their median response spectrum (RSm), used for the pushover analyses, with the RS being the code elastic spectrum used for the bridges (RB and IB) design.

**Figure 9.**
(**a**) Bridge structural model, (**b**) SAP 2000 model of the regular bridge and (**c**) SAP 2000 model of the irregular bridge.

**Figure 9.**
(**a**) Bridge structural model, (**b**) SAP 2000 model of the regular bridge and (**c**) SAP 2000 model of the irregular bridge.

**Figure 10.**
Regular bridge—mode shapes (transverse direction).

**Figure 10.**
Regular bridge—mode shapes (transverse direction).

**Figure 11.**
Irregular bridge—mode shapes (transverse direction).

**Figure 11.**
Irregular bridge—mode shapes (transverse direction).

**Figure 12.**
Regular bridge—deck displacements derived by performing pushover analysis (ag = 0.7 g), with a load pattern proportional to the main mode shapes, with respect to different monitoring points (M.P.). Pushover with Mode 2 (**a**) or Mode 4 (**b**).

**Figure 12.**
Regular bridge—deck displacements derived by performing pushover analysis (ag = 0.7 g), with a load pattern proportional to the main mode shapes, with respect to different monitoring points (M.P.). Pushover with Mode 2 (**a**) or Mode 4 (**b**).

**Figure 13.**
Irregular bridge—deck displacements derived by performing pushover analysis (ag = 0.7 g), with a load pattern proportional to the main mode shapes, with respect to different monitoring points (M.P.). Pushover with Mode 1 (**a**), Mode 3 (**b**) or Mode 4 (**c**).

**Figure 13.**
Irregular bridge—deck displacements derived by performing pushover analysis (ag = 0.7 g), with a load pattern proportional to the main mode shapes, with respect to different monitoring points (M.P.). Pushover with Mode 1 (**a**), Mode 3 (**b**) or Mode 4 (**c**).

**Figure 14.**
Regular 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 14.**
Regular 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 15.**
Regular 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 15.**
Regular 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 16.**
Regular 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 16.**
Regular 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 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 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 IMPAβ (the design PGA was 0.35 g, corresponding to a transversal base shear of V_{b,x} ~ 9200 kN).

**Figure 18.**
Regular bridge—incremental curve (displacement-intensity) and capacity curves derived with IDA (maximum values of ur and Vb,x) or IMPAβ (the design PGA was 0.35 g, corresponding to a transversal base shear of V_{b,x} ~ 9200 kN).

**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 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 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 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).

**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).

**Figure 23.**
Case study—incremental curve (displacement-intensity) and capacity curves derived with IDA (maximum values of ur and Vb,x) or IMPAβ (the design PGA was 0.35 g, corresponding to a transversal base shear of V_{b,x} ~ 13,000 kN).

**Figure 23.**
Case study—incremental curve (displacement-intensity) and capacity curves derived with IDA (maximum values of ur and Vb,x) or IMPAβ (the design PGA was 0.35 g, corresponding to a transversal base shear of V_{b,x} ~ 13,000 kN).

**Table 1.**
Loads and actions.

**Table 1.**
Loads and actions.

Load | kN/m | kN |
---|

Dead | Self-weight | 200 | - |

Live | Vehicle loads (Qik) | - | 1200 |

Live | Distributed load (qik) | 54.5 | - |

**Table 2.**
List of the selected ground motions.

**Table 2.**
List of the selected ground motions.

Etq ID | Earthquake Name | Waveform | Date | PGA (g) |
---|

1635 | South Iceland | 4674-xa | 17/06/2000 | 0.31 |

1635 | South Iceland | 4674-ya | 17/06/2000 | 0.31 |

2309 | Bingol | 7142-xa | 01/05/2003 | 0.50 |

2309 | Bingol | 7142-ya | 01/05/2003 | 0.50 |

2142 | South Iceland (aftershock) | 6349-xa | 21/06/2000 | 0.72 |

2142 | South Iceland (aftershock) | 6332-ya | 21/06/2000 | 0.51 |

1635 | South Iceland | 6277-ya | 17/06/2000 | 0.35 |

- | Mean | | - | 0.46 |

**Table 3.**
Regular bridge—modal properties.

**Table 3.**
Regular bridge—modal properties.

Mode | Period | Participating Mass |
---|

N° | S | % |
---|

2 | 1.02 | 78.0 |

4 | 0.33 | 12.0 |

**Table 4.**
Irregular bridge—modal properties.

**Table 4.**
Irregular bridge—modal properties.

Mode | Period | Participating Mass |
---|

N° | S | % |
---|

1 | 0.65 | 16.9 |

3 | 0.53 | 71.3 |

4 | 0.13 | 4.5 |