# IMPAβ: Incremental Modal Pushover Analysis for Bridges

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Static Analysis for Bridges

#### 2.1. General Considerations for Bridges

#### 2.2. The Incremental Modal Pushover Analysis for Bridges (IMPAβ)

_{n}for each mode:

_{n}is the nth modal participation factor; M is a diagonal mass matrix of order 2n, including the diagonal submatrices m, and 1: m is a diagonal matrix with m

_{jj}= m

_{j}, the lumped mass barycenter of the jth pier; φ

_{n}is the nth natural vibration mode of the structure consisting of three sub-vectors: r

_{n}, y

_{n}and θ

_{n}; and the n × 1 vector 1 is equal to unit.

- Compute the natural frequencies, w
_{n}and modes, n for the linear elastic vibration of the bridge. The modal properties of the bridge model are obtained from the linear dynamic modal analysis, and the relevant modes of the bridge are selected; - Define the seismic demand in terms of response spectra (RS) for a defined range of intensity levels;
- For the intensity level, i, represented by peak ground motion acceleration (PGA), the performance point (P.P.) for the selected (predominant) modes can be determined (Figure 1a);
- Using a combination rule to combine the P.P. corresponding to each mode for each intensity, i, the “multimodal performance point” (P.P.m,i) can be determined (Figure 1b). The P.P.m,i is expressed in terms of monitoring point displacement, u
_{rmmi}, and corresponding global base shear, V_{b,i}, for each intensity level considered: being u_{rni}the modal displacements of the monitoring point, in this paper, the transverse direction is considered.

_{r1i}, …, u

_{rni}) at the performance point (P.P., see Figure 2). In this paper, the P.Ps are determined with the Capacity Spectrum Method (ATC 40) [34], and the P.P.s obtained for each modal shape are combined by using the Square Root of the Sum of Squares rule (3), to obtain a multimodal performance point (P.P.m,i) for each specific seismic intensity level, as displayed in Figure 3.

_{i,IMPA}

_{β}defined in (4) and described in Figure 4.

_{i,IMPA}

_{β}= (max (u

_{r,i,MPA};u

_{r,i,UPA}); max (V

_{b,i,MPA};V

_{b,i,UPA}))

_{r,i,MPA;}V

_{b,i,MPA}) and (u

_{r,i,UPA;}V

_{b,i,UPA}) are the coordinates of the performance point for the intensity, i, obtained by performing MPA or UPA. The flowchart of this new procedure is displayed in Figure 5.

## 3. Case Studies

#### 3.1. Bridges

_{ck}= 20 MPa), while B450C steel (characteristic yield strength f

_{yk}= 450 MPa) reinforcement was used throughout the structure. Both bridges were designed according to Eurocode 8, using a design peak ground acceleration of 0.35 g and a behavior factor (coefficient q) of 3.5 for the regular one and 3.0 for the irregular one. The design loads are summarized in Table 1.

#### 3.2. Seismic Input

_{s30}≤ 800), and (c) the return period considered was Tr = 949 years (life safety limit state: P

_{VR}= 10%; a

_{g}= 0.35 g; T

_{B}= 0.172 s, T

_{C}= 0.516 s, T

_{D}= 3.035 s; F

_{0}= 2.464; S

_{T}= 1.0). Using Rexel [42], we selected a set of seven unscaled records, compatible on average with the target spectrum, and with the minimum dispersion of individual spectra: The average response spectrum matched the target spectrum at a specified period range that included all the periods considered relevant (participating mass >1% along the transverse direction of the bridge).

#### 3.3. Numerical Models

#### 3.4. Modal Properties

## 4. Nonlinear Analyses

_{bn}− u

_{rn}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):

_{rn}is the value of f

_{n}at the monitoring point; M

^{*}

_{n}= L

_{n}G

_{n}is the effective modal mass; L

_{n}= f

^{T}

_{n}m l, G

_{n}= L

_{n}/M

_{n}and M

_{n}= f

^{T}

_{n}mf

_{n}were the generalized mass for the natural mode, n.

_{n}·f

_{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.

_{rni}, of the bridge gives a different value, not only because of the deviation of the elastic mode shape, f

_{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. Figure 12 and Figure 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).

#### 4.1. Analysis of the Regular Bridge (RB)

_{r}-Sa curve showed how, for an intensity level (Sa) beyond the design value (0.35 g), the displacement estimate with IMPAβ differs from IDA, but it is conservative, whereas the u

_{r}-V

_{b,x}curve (V

_{b,x}is the base shear evaluated for all the piers) better matched the curve derived from IDA since, beyond the design intensity (V

_{b,x}~ 9200 kN), the higher displacements derived with IMPAβ are related to a base shear value that are very similar, since the bridge (hinges of the piers) is in a plastic stage.

#### 4.2. Analysis of the Irregular Bridge (IB)

## 5. Discussion

## 6. Conclusions and Future Developments

- The procedure keeps the simplicity and low computational effort typical of a standard pushover analysis. In fact, IMPAβ implies the execution of one nonlinear static analysis for each loading pattern, while performing IDA, several nonlinear response history analyses have to be performed, i.e., one for each of the seven ground motions, and it has to be repeated for every intensity level considered.
- If the capacity curve is obtained by using a multimodal load pattern only, it well matches the IDA curve, for both the regular and irregular bridge, up to the seismic design intensity only, while the present proposal allows us to analyze even higher intensities.
- IMPAβ produces a capacity curve very similar to what derived with IDA also beyond the seismic design intensity (0.35 g), and, for very high seismic intensities (beyond 0.6 g), the capacity curve of IMPAβ is conservative.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

C.S.M. | capacity spectrum method |

GM | ground motion |

IDA | incremental dynamic analysis |

IMPA | incremental modal pushover analysis |

IMPAβ | incremental modal pushover analysis for bridges |

ISPA | incremental standard pushover |

IUPA | incremental uniform pushover analysis |

NSA | nonlinear static analysis |

NRHA | nonlinear response history analysis |

M.P. | monitoring point |

MPA | modal pushover analysis |

P.P. | performance point |

RS | response spectrum |

RSm | mean response spectrum |

SPA | standard pushover analysis—loading pattern proportional to 1st modal shape |

UPA | uniform pushover analysis—loading pattern proportional to masses |

u_{r} | monitored displacement |

V_{b} | base shear |

## Appendix A

**Figure A1.**Regular bridge—bending moment (M

_{b}) and curvatures (q

_{b}) at the hinge location, derived by performing nonlinear dynamic analysis and modal pushover according to each single mode (PGA from 0.175 to 0.7 g).

**Figure A2.**Regular bridge—bending moment (M

_{b}) and curvatures (q

_{b}) at the hinge location, derived by performing nonlinear dynamic analysis and pushover analysis (PGA from 0.175 to 0.7 g).

**Figure A3.**Irregular bridge—bending moment (M

_{b}) and curvatures (q

_{b}) at the hinge location, derived by performing nonlinear dynamic analysis and modal pushover according to each single mode (PGA from 0.175 to 0.7 g).

**Figure A4.**Irregular bridge—bending moment (M

_{b}) and curvatures (q

_{b}) at the hinge location, derived by performing nonlinear dynamic analysis and pushover analysis (PGA from 0.175 to 0.7 g).

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**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 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 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 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 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 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 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).

**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).

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

Dead | Self-weight | 200 | - |

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

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

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 |

Mode | Period | Participating Mass |
---|---|---|

N° | S | % |

2 | 1.02 | 78.0 |

4 | 0.33 | 12.0 |

Mode | Period | Participating Mass |
---|---|---|

N° | S | % |

1 | 0.65 | 16.9 |

3 | 0.53 | 71.3 |

4 | 0.13 | 4.5 |

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## Share and Cite

**MDPI and ACS Style**

Bergami, A.V.; Nuti, C.; Lavorato, D.; Fiorentino, G.; Briseghella, B.
IMPAβ: Incremental Modal Pushover Analysis for Bridges. *Appl. Sci.* **2020**, *10*, 4287.
https://doi.org/10.3390/app10124287

**AMA Style**

Bergami AV, Nuti C, Lavorato D, Fiorentino G, Briseghella B.
IMPAβ: Incremental Modal Pushover Analysis for Bridges. *Applied Sciences*. 2020; 10(12):4287.
https://doi.org/10.3390/app10124287

**Chicago/Turabian Style**

Bergami, Alessandro Vittorio, Camillo Nuti, Davide Lavorato, Gabriele Fiorentino, and Bruno Briseghella.
2020. "IMPAβ: Incremental Modal Pushover Analysis for Bridges" *Applied Sciences* 10, no. 12: 4287.
https://doi.org/10.3390/app10124287