# A Probabilistic Capacity Model and Seismic Vulnerability Analysis of Wall Pier Bridges

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Model of Wall Pier

## 3. Probabilistic Capability Model of Wall Piers

#### 3.1. Wall Pier Damage Indexes

#### 3.2. Probabilistic in-Plane Capability Model of Wall Piers

^{2}= 0.954, standard deviation: $\sigma =0.044$. We use the same method to obtain the probabilistic capability models of the yield point, peak point, and limit point in turn, as shown in Equations (6)–(8):

## 4. Seismic Vulnerability of Wall Pier Girder Bridges

#### 4.1. Bearing and Abutment Limit States

#### 4.2. Bridge Sample Establishment

#### 4.3. Time-History Analysis

#### 4.4. Seismic Vulnerability of Wall Pier Bridges

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Component Name | Section Size (cm) (Height × Width × Thickness) | Concrete Strength | Embedded Column Width (cm) | Axial Compression Ratio | Dark Column Reinforcement Ratio | Structural Reinforcement | Destruction Form | ||
---|---|---|---|---|---|---|---|---|---|

Vertical | Lateral | Vertical | Lateral | ||||||

SW1-1 | 200 × 100 × 12.5 | C30 | 20 | 0.1 | 1.84% | 0.57% | 0.38% | 0.36% | Bending |

SW2-1 | 100 × 100 × 12.5 | C40 | 20 | 0.3 | 1.84% | 0.57% | 0.38% | 0.36% | Shearing |

Component Name | Section Size (cm) (Height × Width × Thick) | Concrete Strength (MPa) | Axial Compression Ratio | Reinforcement Ratio | Destruction Form |
---|---|---|---|---|---|

Park and Paulay | 178 × 60 × 40 | 26.9 | 0.1 | Vertical 1.88% Transverse 2.2% | bending damage |

Wehbe | 234 × 61 × 38 | 27.2 | 0.098 | Vertical 2.22% Transverse 0.4% | bending damage |

Wight03 | 88 × 31 × 15 | 26.1 | 0.147 | Vertical 2.45% Transverse 0.5% | bending and shearing |

Wight04 | 88 × 31 × 15 | 26.1 | 0.147 | Vertical 2.45% Transverse 0.5% | bending and shearing |

Performance Level | Degree of Damage | Seismic Performance Index |
---|---|---|

level 1 | Intact | MDR ≤ 0.11% |

level 2 | Minor damage | 0.11% < MDR ≤ 0.38% |

level 3 | Medium damage | 0.38% < MDR ≤ 0.84% |

level 4 | Serious damage | 0.84% < MDR ≤ 2.23% |

level 5 | Complete destruction | MDR > 2.23% |

Level | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Aspect ratio | 3.2 | 3.8 | 4.4 | 5 |

Pier width (m) | 4 | 5.33 | 6.67 | 8 |

Vertical reinforcement ratio (%) | 0.5 | 0.63 | 0.77 | 0.9 |

Lateral reinforcement ratio (%) | 0.2 | 0.27 | 0.33 | 0.4 |

Axial pressure ratio | 0.02 | 0.04 | 0.06 | 0.08 |

Shear span ratio | 1.6 | 2.07 | 2.53 | 3 |

Reinforcement grade | HRB400 | HRB335 | — | — |

Concrete marking | C40 | C35 | — | — |

Performance Level | Damage State | Allowable Quantification of Shear Strain |
---|---|---|

level Ⅰ | intact | γ_{α} < 100% |

level Ⅱ | minor damage | 100% ≤ γ_{α} < 150% |

level Ⅲ | medium damage | 150% ≤ γ_{α} < 200% |

level Ⅳ | serious damage | 200% ≤ γ_{α} < 250% |

level Ⅴ | complete destruction | γ_{α} ≥ 250% |

Performance Level | Damage State | Abutment Limit Displacement Quantification (mm) |
---|---|---|

level Ⅰ | intact | Δ < 25 |

level Ⅱ | minor damage | 25 ≤ Δ < 50 |

level Ⅲ | medium damage | 50 ≤ Δ < 100 |

level Ⅳ | serious damage | 100 ≤ Δ < 150 |

level Ⅴ | complete destruction | Δ ≥ 150 |

Uncertainty Parameter | Distribution Type | Distribution Parameter | |
---|---|---|---|

α | β | ||

C35 concrete compressive strength (MPa) | normal distribution | 35 | 4.5 |

HRB335 steel yield strength (MPa) | logarithmic normal distribution | 5.81 | 0.1 |

abutment initial stiffness (kN/mm/m) | uniform distribution | 11.5 | 28.5 |

Scale factor of horizontal resistance coefficient (kN/m4) | uniform distribution | 60000 | 100000 |

damping ratio | normal distribution | 0.045 | 0.0125 |

vertical reinforcement ratio of piers (%) | uniform distribution | 0.55 | 0.85 |

transverse reinforcement ratio (%) | uniform distribution | 0.2 | 0.4 |

pier height (m) | uniform distribution | 11 | 16 |

expansion joint width (cm) | normal distribution | 8 | 0.5 |

shear elastic modulus (MPa) | normal distribution | 1.18 | 0.16 |

Type of Ground Motions | Types | Fault Distance (R) | Intensity Magnitude | PGV/PGA |
---|---|---|---|---|

near field | pulse type | 0~20km | 6~8 | >0.15 |

far field | non-pulse type | 20~100km | 6~8 | ≤0.15 |

**Table 9.**The median and logarithmic standard deviation in the fragility models of bridge components and bridge systems to different damage states.

Component | Minor Damage | Medium Damage | Severe Damage | Complete Destruction | ||||
---|---|---|---|---|---|---|---|---|

Median Value (g) | Logarithmic Standard Deviation | Median Value (g) | Logarithmic Standard Deviation | Median Value (g) | Logarithmic Standard Deviation | Median value (g) | Logarithmic Standard Deviation | |

Pier (out of plane) | 0.2541 | 0.8003 | 0.7618 | 0.4712 | 1.3108 | 0.7875 | 3.1159 | 1.1027 |

Pier (in plane) | 0.316 | 1.5414 | 1.0808 | 0.6534 | 3.3743 | 0.7875 | 3.3743 | 0.7875 |

Pier (overall) | 0.1892 | 0.9701 | 0.695 | 0.463 | 1.3108 | 0.7875 | 3.1159 | 1.1027 |

Bearing | 0.2312 | 0.8775 | 0.3966 | 0.6135 | 0.5306 | 0.5739 | 0.6435 | 0.4278 |

Abutment | 0.5759 | 0.8253 | 0.6877 | 0.4962 | 0.8648 | 0.3866 | 1.4911 | 0.4057 |

Bridge system | 0.1767 | 0.8758 | 0.3682 | 0.5241 | 0.5031 | 0.5449 | 0.6435 | 0.4278 |

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**MDPI and ACS Style**

Chen, L.; Tu, Y.; He, L.
A Probabilistic Capacity Model and Seismic Vulnerability Analysis of Wall Pier Bridges. *Appl. Sci.* **2020**, *10*, 926.
https://doi.org/10.3390/app10030926

**AMA Style**

Chen L, Tu Y, He L.
A Probabilistic Capacity Model and Seismic Vulnerability Analysis of Wall Pier Bridges. *Applied Sciences*. 2020; 10(3):926.
https://doi.org/10.3390/app10030926

**Chicago/Turabian Style**

Chen, Libo, Yi Tu, and Leqia He.
2020. "A Probabilistic Capacity Model and Seismic Vulnerability Analysis of Wall Pier Bridges" *Applied Sciences* 10, no. 3: 926.
https://doi.org/10.3390/app10030926