# Eccentricity-Induced Seismic Behavior of Curved Bridges Based on Controllability

## Abstract

**:**

## 1. Motivation

## 2. Introduction

## 3. Analytical Model of Curved Bridges Based in Rigid-Deck Assumption

_{si}is the azimuth angle of the i

^{th}pier, and $\mathsf{\beta}$ is the intersection angle. ${\mathrm{f}}_{\mathrm{t}}$ and ${\mathrm{f}}_{\mathrm{n}}$ are friction and contact pounding forces, respectively.

## 4. Controllability Measures

## 5. The Prototype Bridge

^{6}kg and is evenly distributed along the line.

## 6. Eccentricity Conceptual Analysis

## 7. Eccentricity Controllability Analysis

_{r}represents radial HSVs, H

_{t}represents tangential HSVs, H

_{rot}represents rotational HSVs. Same legend applies all plots.

_{r}and H

_{t}decrease when GMs change from the bridge’s longitudinal (x-axis) direction to transverse (y-axis) direction, but when GM directions are between 50° and 130° (where #2 and #4 supports located), the values remain at low level; H

_{rot}and case 2 are opposite. The results show that cases 1 and 3 are more “controllable” by the x-components than by the y-components of GMs; case 2 is more “controllable” by the y-components than the x-components of GMs, and radius difference does not influence the HSVs too much for case 2. These results agree with the previous conceptual analysis and provide visual and deep insight into the eccentricity essence.

## 8. Earthquake Time History Analysis

#### 8.1. Earthquake Records to Use

^{2}), corresponding to the intensity 8 design earthquake level.

#### 8.2. Simulations

- Examine the radius influence. So, comparison was made for the case 3 bridges with different radiuses and four GMs inputting at 90°. The time history responses (5~15–20 s) of Pt. #5 are shown in Figure 9. Clearly, peak responses are in order of red lines (R = 110 m) < green lines (R = 100 m) < blue lines (R = 90 m), for both radius responses (U
_{r}) and tangential responses (U_{t}), and for all earthquakes. Furthermore, U_{r}and U_{t}are nearly of the same order for all cases. The results show that smaller radius bridges have greater earthquake response, which agrees with the Grammian analyses in Figure 7 (case 3) and validates the fact that a smaller radius worsens the eccentricity. - Examine the bearing influence and the earthquake superpose effects. Comparison was made on the 3 bearing cases of the R = 100 m bridge, with Kobe and Jiji GMs inputting at 0° and 45°, respectively. The two translational motions (radius U
_{r}, and tangential U_{t}) are plotted in Figure 10.

_{r}and U

_{t}are nearly of same order under 0° and 45° GMs for all cases, indicating the coupling is strong.

_{r}and U

_{t}responses to 50° GM are not the smallest among the three angles as Figure 6 indicates. The reason might lie in the large rotational responses U

_{rot}(Figure 6) and strongly coupled U

_{rot}with U

_{r}and U

_{t}(Figure 8).

## 9. Discussions

- This study applied the controllability-based concepts on the evaluation of curved bridges. To make it work, a rigid-deck assumption was made to obtain the analytical equations of motion that can account for coupling effects, and six eccentricity cases were designed for the study of different sources of eccentricities.
- HSV–GM direction and controllability Grammian–GM direction curves show the potential structural responses to earthquakes from different directions, and response variances for different stiffness cases and radius cases. Analyses confirmed that HSV and Grammian could reflect the link of structural geometric and physical properties to excitations. Their values could provide useful information for positions on the structure that are sensitive to excitations and measurements, and thus, factors that make the structure “irregular” or “asymmetrical” could be figured out.
- A smaller radius indicates stronger eccentricity for curved bridges, which makes the bridge more “controllable” to ground motions in the chord’s direction, i.e., curved bridges illustrate strong irregularity to such ground motions. This phenomenon is shown in the typical y-symmetric case 1.
- The support restriction (or stiffness) distribution is a more important factor for eccentricity control than radius for curved bridges, because it determines the relative position of CS and CM. After all, the CM location could not be adjusted once the line of the curved bridge is determined. The space between the curves on HSV and Grammian figures confirmed it.
- Stiffness eccentricity literally aggravates the irregularity; however, its action could either amplify or counteract the other eccentricity source, ground motion directions. Comparison of case 2 and case 3 found that, case 2 “deteriorated” the eccentric condition, while case 3 “neutralized” the eccentric condition, and thus performed better than the y-symmetric case 2. So, proper design of stiffness eccentricity could achieve “symmetric” effects to some unfavorable ground motions, decrease the coupling effects, and result in a more robust bridge.
- HSV is more comprehensive and general and better used in feedback control, while Grammian seems to be the right index for this control-only case. However, the indices still need to be studied further for easier implementation.

## 10. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematics for the analytical model deduction of the arc bridge. (

**a**) Plan of the rigid-body arc bridge—deck and box beam. (

**b**) Stiffness deduction of the beam support (k

_{s}) from series of bearing and pier.

**Figure 2.**The prototype bridge. (

**a**) Plan of six spans. (

**b**) Picture: bird view. (

**c**) Picture: the vase-shape pier.

**Figure 7.**Controllability Grammians to single ground motion inputting from 0° to 180°, where radius–radius and tangent–tangent are auto-Grammians, and radius–tangent is cross-Grammians.

**Figure 10.**Displacement responses of Pt. #1 of the R = 100 m bridge to Kobe and Jiji inputting at 0° and 45°.

Pier No. | #1 | #2 | #3 | #4 | #5 |
---|---|---|---|---|---|

Radius of curve (m) | 102.05 | 100 | 100 | 100 | 102.27 |

Azimuth angle φ (°) | 157.37 | 132.76 | 90 | 47.24 | 22.63 |

Pier height (m) | 14.2 | 19.81 | 21.92 | 18.82 | 13.6 |

Restraint Parameters | LNR700 | LRB700-140 |
---|---|---|

Stiffness K_{i}, (kN/m) | 1922 | 2843 |

Post-yield stiffness ratio (α_{ri}, α_{φi}) | 1.0 | 0.114 |

Damping ratio % | 5 | 26.4 |

R = 90 m | R = 100 m | R = 110 m | ||||
---|---|---|---|---|---|---|

CM | 0 | 66.5372 | 0 | 78.5388 | 0 | 90.2549 |

Case 1 | 0.0056 | 87.3306 | 0.0059 | 97.546 | 0.0062 | 107.7388 |

Case 2 | 64.1551 | 60.3207 | 62.5925 | 72.9837 | 69.6676 | 85.2347 |

Case 3 | −38.102 | 57.0166 | −38.978 | 69.8754 | −39.633 | 82.3367 |

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**MDPI and ACS Style**

Wang, Y.
Eccentricity-Induced Seismic Behavior of Curved Bridges Based on Controllability. *Symmetry* **2020**, *12*, 1633.
https://doi.org/10.3390/sym12101633

**AMA Style**

Wang Y.
Eccentricity-Induced Seismic Behavior of Curved Bridges Based on Controllability. *Symmetry*. 2020; 12(10):1633.
https://doi.org/10.3390/sym12101633

**Chicago/Turabian Style**

Wang, Yumei.
2020. "Eccentricity-Induced Seismic Behavior of Curved Bridges Based on Controllability" *Symmetry* 12, no. 10: 1633.
https://doi.org/10.3390/sym12101633