# Modelling and Characterizing a Concrete Gravity Dam for Fragility Analysis

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## Abstract

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## 1. Introduction

## 2. Finite Element Model of the Case Study Dam

#### 2.1. Model Configuration and Mesh Considerations

#### 2.1.1. Block-6

#### 2.1.2. Foundation

#### 2.1.3. Reservoir

#### 2.2. Boundary Conditions and Contact Interfaces

#### 2.3. Damping and Hourglassing Control

#### 2.4. Validation of the Numerical Model

#### Dynamic Behavior

## 3. Loads

#### 3.1. Static and Dynamic Loads

#### 3.2. Application of Static and Dynamic Loads

#### 3.3. Deconvolution of the Ground Motions

## 4. Uncertainty Modelling

#### 4.1. Material Properties

#### 4.1.1. Modulus of Elasticity

#### 4.1.2. Damping

#### 4.1.3. Shear and Tensile Strength

#### 4.2. Load Parameters

#### 4.2.1. Drain Efficiency

#### 4.2.2. Directionality Factor

#### 4.3. Ground Motions

#### 4.3.1. Levels of Seismic Intensity

#### 4.3.2. Selection and Scaling of Accelerograms

## 5. Fragility Analysis

#### 5.1. Limit States: Preliminary Analyses and Nonlinearities

#### 5.2. Fragility Curves

## 6. Expected Seismic Performance

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Comparison between response spectrum (before and after deconvolution) and target spectrum.

**Figure 6.**Spectra of selected horizontal accelerograms. (

**a**) Target spectrum; (

**b**) Selected horizontal accelerograms.

Soil foundation | Diorite rock, gneiss, granite, hybrid |

Max. height | 29.3 m |

Crest length | 192.6 m |

Max. width | 28.8 m |

Crest width | 12.2 m |

Drain | Drained with injection curtain |

Elasticity modulus | ${E}_{rock}=40$ GPa, ${E}_{concrete}=21.4$ GPa |

Damping | ${\xi}_{rock}=5\%$, ${\xi}_{concrete}=1.5\%$ |

Density | ${\rho}_{rock}=2600$ kg/m${}^{3}$, ${\rho}_{concrete}=2300$ kg/m${}^{3}$ |

Poisson coefficient | ${\nu}_{rock}=0.3$, ${\nu}_{concrete}=0.164$ |

Reservoir | Westergaard added mass |

Model | Fundamental Period (s) | Damping (%) |
---|---|---|

Reference | 0.096 | 5.50 |

LS-Dyna M1 | 0.106 | 2.88 |

LS-Dyna M0 | 0.110 | 3.19 |

Load | Details |
---|---|

Self-weight - Plot-6 | ${\rho}_{concrete}=2300\phantom{\rule{0.166667em}{0ex}}$ kg/m${}^{3}$ |

Reservoir weight | Reservoir level = MOL |

Upstream hydrostatic thrust | Reservoir level = MOL |

Uplift | Concrete–rock contact Reservoir level = MOL Drain efficiency: between 0% and 70% |

Hydrodynamic load | Automatic with fluid elements Reservoir level = MOL |

Horizontal seismic load | Representative accelerograms |

Vertical seismic load | Representative accelerograms |

Modelling Parameter | Probability Distribution | Distribution Parameter | Units | |
---|---|---|---|---|

Directionality factor | Uniform | $L=0.5$ | $U=0.8$ | - |

Concrete damping | Log-Normal | $\lambda =-2.99$ | $\zeta =0.35$ | % |

Concrete–concrete cohesion | Uniform | $L=0.5$ | $U=2.5$ | MPa |

Concrete–rock cohesion | Uniform | $L=0.1$ | $U=1.5$ | MPa |

Concrete–concrete angle of friction | Uniform | $L=45$ | $U=55$ | ${}^{\circ}$ |

Concrete–rock angle of friction | Uniform | $L=30$ | $U=45$ | ${}^{\circ}$ |

Concrete–concrete tensile strength | Uniform | $L=0.3$ | $U=2.3$ | MPa |

Concrete–rock tensile strength | Uniform | $L=0.05$ | $U=0.8$ | MPa |

Concrete elasticity modulus | Uniform | $L=18.5$ | $U=26.9$ | MPa |

Rock elasticity modulus | Uniform | $L=40$ | $U=50$ | MPa |

Drain efficiency | Uniform | $L=0$ | $U=70$ | % |

Material Properties | Constant Value |
---|---|

Concrete density | 2300 kg/m${}^{3}$ |

Rock density | 2600 kg/m${}^{3}$ |

Concrete Poisson’s coefficient | 0.164 |

Rock Poisson’s coefficient | 0.300 |

Foundation damping | 5% |

Limit State | Displacement (mm) |
---|---|

LS0—Minor damage | Incipient sliding |

LS1—Moderate damage | 25 |

LS2—Severe damage | 50 |

**Table 8.**Incremental dynamic analysis (IDA) results—Fraction of samples exceeding the limit state. PGA: peak ground acceleration.

Limit State | Seismic Intensity Levels—PGA (g) | ||||||
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.25 | 0.3 | 0.4 | 0.5 | 0.7 | |

LS0 | 0/22 | 2/22 | 5/22 | 10/22 | 15/22 | 17/22 | 20/22 |

LS1 | 0/22 | 0/22 | 1/22 | 3/22 | 6/22 | 10/22 | 19/22 |

LS2 | 0/22 | 0/22 | 0/22 | 0/22 | 3/22 | 5/22 | 11/22 |

Limit State | Log-Normal Parameters | |
---|---|---|

${\mathit{m}}_{\mathit{R}}$ (g) | ${\mathit{\beta}}_{\mathit{R}}$ | |

LS0 | 0.336 | 0.456 |

LS1 | 0.499 | 0.374 |

LS2 | 0.698 | 0.454 |

Limit State | PGA for HCLPF | PGA${}_{\mathbf{target}}$ |
---|---|---|

LS0 | 0.16g | 20% |

LS1 | 0.27g | 2% |

LS2 | 0.33g | 0.7% |

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**MDPI and ACS Style**

Segura, R.L.; Bernier, C.; Durand, C.; Paultre, P.
Modelling and Characterizing a Concrete Gravity Dam for Fragility Analysis. *Infrastructures* **2019**, *4*, 62.
https://doi.org/10.3390/infrastructures4040062

**AMA Style**

Segura RL, Bernier C, Durand C, Paultre P.
Modelling and Characterizing a Concrete Gravity Dam for Fragility Analysis. *Infrastructures*. 2019; 4(4):62.
https://doi.org/10.3390/infrastructures4040062

**Chicago/Turabian Style**

Segura, Rocio L., Carl Bernier, Capucine Durand, and Patrick Paultre.
2019. "Modelling and Characterizing a Concrete Gravity Dam for Fragility Analysis" *Infrastructures* 4, no. 4: 62.
https://doi.org/10.3390/infrastructures4040062