# Numerical Evaluation of Natural Periods and Mode Shapes of Earth Dams for Probabilistic Seismic Hazard Analysis Applications

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology and Data

^{−9}.

_{S}(and of the shear modulus, G), of soil with depth. Over the last few decades, several authors investigated this effect for different boundary conditions and soil types [8,18,19,20,21,22,23,24,25]. The scheme used by Dakoulas and Gazetas [8] considers the dependency of the variability with depth of the shear modulus in dams and embankments on an inhomogeneity factor, m, varying from 0 to 1 (Equation (4)):

_{b}is the shear modulus at the base of the dam, z is the depth starting from the top of the model, and H is the total height of the dam. Similarly, it is possible to calculate the variation of V

_{S}with depth using the well-known Equation (5):

_{H}is the shear wave velocity at the base of the model, n is the inhomogeneity factor that accounts for the variation of V

_{S}with depth (notice that n differs than the inhomogeneity factor used in Equation (4), m—the latter is defined as the inhomogeneity factor relative to the shear modulus profile), and b is defined by Equation (6):

_{0}is the initial shear wave velocity at the top of the model (crest of the dam), as shown in the geometrical scheme presented in Figure 3.

_{H}= 550 m/s and V

_{0}= 250 m/s, for the shells of Farneto del Principe dam. For the core, we decided to use a constant, V

_{S}= 250 m/s (inhomogeneity factor n = 0, according to previous studies for overconsolidated clays [28,29]). These values are in good agreement with available measurements at the Farneto del Principe, San Valentino, Camastra, and Bilancino dam cores [2,30,31]. Input parameters for models 1–5 are summarized in Table 2.

_{i}is a parameter that depends on the geometry of the dam (provided by the authors for several modes). For the case of i = 1 (fundamental period of the dam, T

_{1}), Equation (7) becomes the well-known Equation (8):

^{th}mode, m is the inhomogeneity factor (m = 0 corresponds to the homogeneous case—i.e., a

_{i}≡ β

_{i}—in this case, Equation (9) can be reduced to Equation (7)) that is provided by the authors for different dam geometries (m = 0.57 for the Farneto del Principe dam), a

_{i}is dependent on the inhomogeneity factor m and soil properties (stiffness, geometry, etc.), and $\overline{{V}_{S}}$ is the average shear wave velocity of the dam, calculated using Equation (10):

_{b}is the shear wave velocity at the base of the dam and λ is the so-called truncation ratio of the dam, equal to 0.05 for the case study, that can be calculated using Equation (11):

## 3. Results

_{1}) is equal to 22%. Such elongation is also evident, looking at the second modal period, where ΔT

_{2}= 29%. These results highlight the importance of considering the dam foundation when performing modal analyses. SSI effects due to the flexibility of the foundation have a strong impact on all important modes of earth dams. This finding is consistent with other studies on the vibrational characteristics of earth dams [33]. Such effect is much more important than that of the degree of saturation within the dam and the variability of shear modulus values in the dam core.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of the Farneto del Principe dam (Southern Italy), shallow crustal seismogenic faults, and Calabrian Arc subduction zone.

**Figure 2.**Representative cross-section and location of the monitoring instrumentation of the Farneto del Principe dam (adapted from [2]).

**Figure 4.**Finite element method (FEM) Model 1: (

**a**) undeformed mesh, (

**b**) first mode shape, and (

**c**) second mode shape.

Model # | Rigid Base | Flexible Base | Presence of Water | V_{S} Variable with Depth |
---|---|---|---|---|

1 | ✓ | ✗ | ✗ | ✗ |

2 | ✓ | ✗ | ✓ | ✗ |

3 | ✓ | ✗ | ✗ | ✓ |

4 | ✓ | ✗ | ✓ | ✓ |

5 | ✗ | ✓ | ✓ | ✓ |

^{1}✓: modelling assumption is present in the model; ✗: modelling assumption is not present in the model.

Parameter | Above the Phreatic Surface | Below the Phreatic Surface | Foundation | ||
---|---|---|---|---|---|

Core | Shells | Core | Shells | ||

γ (kN/m^{3}) | 18 | 24 | 21.3 | 25.1 | 24.1 |

Poisson’s ratio | 0.35 | 0.33 | 0.49 | 0.49 | 0.33 |

V_{S} (m/s) | 250 | Variable with depth (models 4 and 5) | 250 | Variable with depth models 4 and 5) | 650 |

Mode | Model 1 | Model 2 | Model 3 | Model 4 | ||||
---|---|---|---|---|---|---|---|---|

Period (s) | MPMR (%) | Period (s) | MPMR (%) | Period (s) | MPMR (%) | Period (s) | MPMR (%) | |

1 | 0.213 | 63 | 0.205 | 65 | 0.205 | 60 | 0.197 | 62 |

2 | 0.102 | 14 | 0.099 | 10 | 0.099 | 16 | 0.097 | 12 |

3 | 0.059 | 2 | 0.055 | 2 | 0.062 | 2 | 0.063 | 2 |

4 | 0.045 | 1 | 0.040 | 2 | 0.004 | 1 | 0.038 | 2 |

Model | Period (s) | Period Elongation (%) | ||
---|---|---|---|---|

T1 | T2 | ΔT_{1} | ΔT_{2} | |

4 (fixed base) | 0.197 | 0.097 | - | - |

5 (flexible base) | 0.240 | 0.126 | 22 | 29 |

**Table 5.**Modal periods for three analytical solutions used in this study and complete fixed-base FEM solution (Model 4).

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**MDPI and ACS Style**

Zimmaro, P.; Ausilio, E.
Numerical Evaluation of Natural Periods and Mode Shapes of Earth Dams for Probabilistic Seismic Hazard Analysis Applications. *Geosciences* **2020**, *10*, 499.
https://doi.org/10.3390/geosciences10120499

**AMA Style**

Zimmaro P, Ausilio E.
Numerical Evaluation of Natural Periods and Mode Shapes of Earth Dams for Probabilistic Seismic Hazard Analysis Applications. *Geosciences*. 2020; 10(12):499.
https://doi.org/10.3390/geosciences10120499

**Chicago/Turabian Style**

Zimmaro, Paolo, and Ernesto Ausilio.
2020. "Numerical Evaluation of Natural Periods and Mode Shapes of Earth Dams for Probabilistic Seismic Hazard Analysis Applications" *Geosciences* 10, no. 12: 499.
https://doi.org/10.3390/geosciences10120499