# Seismic Fragility Evaluation of Simply Supported Aqueduct Accounting for Water Stop’s Leakage Risk

^{*}

## Abstract

**:**

## 1. Introduction

^{3}, which will greatly alleviate the imbalance of water resources between the southern and western areas of China. The Middle Route of the South-to-North Water Diversion Project (MRP), part of the South-to-North Water Diversion Project, is the largest water conveyance project in China and 1432 km in length to transfer [2]. A total of 27 large aqueducts have been constructed in the MRP to cross rivers and valleys.

## 2. The Steps of Seismic Fragility Analysis

- (1)
- The appropriate numerical model was built.
- (2)
- According to the fortification intensity and site type of the selected project, the design response spectrum was obtained, which in this study is under Chinese Code GB_51247-2018 [26]. The first nine groups of ground motions were then selected according to the designed response spectrum from the PEER Ground Motion Database [27].
- (3)
- The Latin hypercube sampling (LHS) method was implemented to reduce the correlation between variables. This paper used LHS to extract six groups of three variables.
- (4)
- By running two groups of data analysis and making a comparison of the results, the most vulnerable ground motion direction was identified, and the transverse direction was found to be in the governing ground motion direction.
- (5)
- Each group of models was analyzed with nine groups of ground motion, and the obtained data were extracted and processed.
- (6)
- The damage index of the aqueduct pier and the water stop was proposed and determined.
- (7)
- The fragility curve was drawn, and then the possibility of the exceedance of a limit state under a specific intensity measure was discussed.

## 3. Numerical Modeling

#### 3.1. Introduction of the Aqueduct

#### 3.2. Methods of Modeling and Analysis

#### 3.2.1. Lumped Mass Method

_{0}and M

_{1}are H

_{0}and H

_{1}, respectively.

_{1}.

#### 3.2.2. Modeling and Analysis

## 4. Ground Motions and LHS

#### 4.1. Selection of Ground Motions

#### 4.2. Latin Hypercube Sampling

_{c}) in this study followed a normal distribution of mean = 20.1 MPa and standard deviation = 2.5 MPa [29]. The flow rate conformed to a normal distribution of flow, mean = 10.4 m

^{3}/s, and standard deviation = 2.6 m

^{3}/s [30]. The weight of water per segment was assumed to follow a normal distribution with a mean = 56.6 kN/m and a standard deviation = 14.1 kN/m. The friction coefficient of rubber bearing is a uniform distribution from 0.02 to 0.05 [31]. These variables are summarized in Table 2.

## 5. Seismic Fragility Evaluation of Aqueducts

#### 5.1. Definition of the Water Stop’s Limit State

#### 5.2. Definition of the Pier Limit State

#### 5.3. Seismic Fragility Evaluation

_{f}= P(a

_{i}, E

_{k}) is defined as the probability that the randomly selected aqueduct samples are in the limit state of E

_{k}under PGA = a

_{i}. The conditional probability of a structure exceeding a specific damage level for a given earthquake intensity can be generally expressed as Equation (7).

_{d}and m

_{c}are the demand and capacity of the aqueduct. IM is the PGA (g); β

_{d}is the logarithmic standard deviation of structural demand. β

_{c}is the logarithmic standard deviation of structural capacity. β

_{c}us 0.6 in this paper. Φ(x) is the standard normal cumulative distribution function. After solving and obtaining x, the corresponding transcendence probability P

_{f}can be obtained by checking the table according to the size of x.

_{0}and PGA.

## 6. Discussion

## 7. Conclusions

- An analytical model incorporating water variant, concrete strength, and bearing performance degradation is proposed, in which a water stop is considered a nonlinear gap element. This analytical model can be largely used for the vulnerability analysis of general aqueducts.
- The damage state of the rubber water stop is proposed based on the rubber pad theory. The damage limit state of the water stop is defined by the tangent of the shear angle. In the range of tanγ < 0.5, the damage status is defined as intact; in the range of 0.5–1, the damage status is defined as slight damage; in the range of 1–2, the damage status is defined as moderate damage; in the range of 2–3, the damage status is defined as extensive damage; and when tanγ exceeds 3, the damage status is defined as complete damage.
- With the increase in ground motion, the probability of the water stop’s leakage risk is significantly increased. However, the water stop has better ductility compared with the pier according to the vulnerability curve shown in Figure 11. It is necessary to evaluate the seismic fragility of old aqueducts with long service history, and seismic rehabilitation could be needed for these aqueducts based on the vulnerability curve.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**Fragility curves of aqueducts: (

**a**) slight damage; (

**b**) moderate damage; (

**c**) extensive damage; (

**d**) complete damage.

No. | Earthquake | Station | Year | Magnitude | Horizontal Acc. |
---|---|---|---|---|---|

1 | Imperial Valley-02 | El Centro Array #9 | 1940 | 6.95 | RSN6_IMPVALL.I_I-ELC180 |

2 | Kern County | Santa Barbara Courthouse | 1952 | 7.36 | RSN14_KERN_SBA132 |

3 | Kern County | Taft Lincoln School | 1952 | 7.36 | RSN15_KERN_TAF021 |

4 | Northern Calif-03 | Ferndale City Hall | 1954 | 6.5 | RSN20_NCALIF.FH_H-FRN044 |

5 | San Fernando | LA–Hollywood Stor FF | 1971 | 6.61 | RSN68_SFERN_PEL090 |

6 | San Fernando | Lake Hughes #1 | 1971 | 6.61 | RSN70_SFERN_L01021 |

7 | San Fernando | Palmdale Fire Station | 1971 | 6.61 | RSN78_SFERN_PDL120 |

8 | San Fernando | Pasadena–CIT Athenaeum | 1971 | 6.61 | RSN79_SFERN_PAS090 |

9 | Managua_Nicaragua-02 | Managua_ ESSO | 1972 | 5.2 | RSN96_MANAGUA_B-ESO180 |

Variable | Probability Distribution | μ | σ | Range of Variations |
---|---|---|---|---|

Water weight | Normal distribution | 56.6 kN/m | 14.1 kN/m | 0.2–113 kN |

ƒ′_{c} | Normal distribution | 20.1 MPa | 2.5 MPa | 15–25 MPa |

Bearing friction coefficient | Uniform distribution | 0.02–0.05 |

Group No. | Water Weight (kN) | ƒ′_{c} (MPa) | Friction Coefficient of Bearing |
---|---|---|---|

1 | 58.97 | 20.7 | 0.0468 |

2 | 22.1 | 22.3 | 0.0492 |

3 | 83.9 | 16.5 | 0.0395 |

4 | 48.9 | 23.9 | 0.0300 |

5 | 39.9 | 18.4 | 0.0356 |

6 | 86.2 | 20.3 | 0.0221 |

Ground Motion No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Response acceleration, g_{max} | 0.80 | 0.35 | 0.60 | 0.48 | 0.41 | 0.41 | 0.38 | 0.41 | 0.80 |

Maximum acceleration, g | 0.28 | 0.13 | 0.16 | 0.16 | 0.22 | 0.15 | 0.11 | 0.11 | 0.22 |

AM acceleration IDA, g | 0.39 | 0.32 | 0.48 | 0.66 | 0.75 | 0.77 |

Damage Limit State | Damage Index |
---|---|

Intact | tanγ < 0.5 |

Slight damage | 0.5 ≤ tanγ < 1 |

Moderate damage | 1 ≤ tanγ < 2 |

Extensive damage | 2 ≤ tanγ < 3 |

Complete damage | tanγ ≥ 3 |

Damage Limit State | Damage Index |
---|---|

Intact | d_{0} < 0.0246% |

Slight damage | 0.0246% ≤ d_{0} < 0.084% |

Moderate damage | 0.084% ≤ d_{0} < 0.09% |

Extensive damage | 0.09% ≤ d_{0} < 0.113% |

Complete damage | d_{0} ≥ 0.113% |

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**MDPI and ACS Style**

Xiong, Z.; Liu, C.; Zhang, A.; Zhu, H.; Li, J. Seismic Fragility Evaluation of Simply Supported Aqueduct Accounting for Water Stop’s Leakage Risk. *Water* **2021**, *13*, 1404.
https://doi.org/10.3390/w13101404

**AMA Style**

Xiong Z, Liu C, Zhang A, Zhu H, Li J. Seismic Fragility Evaluation of Simply Supported Aqueduct Accounting for Water Stop’s Leakage Risk. *Water*. 2021; 13(10):1404.
https://doi.org/10.3390/w13101404

**Chicago/Turabian Style**

Xiong, Zhihua, Chen Liu, Aijun Zhang, Houda Zhu, and Jiawen Li. 2021. "Seismic Fragility Evaluation of Simply Supported Aqueduct Accounting for Water Stop’s Leakage Risk" *Water* 13, no. 10: 1404.
https://doi.org/10.3390/w13101404