# Modeling Variability in Seismic Analysis of Concrete Gravity Dams: A Parametric Analysis of Koyna and Pine Flat Dams

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Seismic Analysis of Dams

#### 2.1. Linear vs. Nonlinear Analysis

#### 2.2. Horizontal vs. Combined Horizontal and Vertical Earthquake Components

#### 2.3. System Configuration

## 3. Modeling Variabilities in Seismic Analysis

#### 3.1. EAGD-84

#### 3.2. ADRFS v1

#### 3.3. Abaqus

## 4. Case Study: Koyna and Pine Flat Dams

#### 4.1. Location and Geometry Description

#### 4.2. Static and Dynamic Loading

#### 4.3. Material Properties

_{cu}is considered based on the work of Rahman Raju et al. [50], which is consistent with the resistance factor for concrete used in design in different codes and standards. Other relevant input parameters such as E, σ

_{co}, and σ

_{to}are estimated. A comparison of crest displacement histories for both CDP and CDPM2 is shown in Figure 11, where it can be seen that the original CDP model slightly overpredicts the crest displacement as compared to the CDPM2 model, the observed difference is expected as the CDPM2 model is deemed more adaptive. The traditional CDP model produces a slightly conservative response, i.e., 5% to 6% more compared to the CDPM2 model. So, the original CDP model was adopted for all cases of nonlinear analysis and implemented using the implicit solver in Abaqus.

## 5. Results and Discussion

#### 5.1. Modal Analysis

#### 5.1.1. Comparison of the Modal Period

#### 5.1.2. Impact of Model Complexity

#### 5.1.3. Impact of the Solution Procedure

#### 5.1.4. Variation in the Higher Modes

#### 5.2. Crest Displacement

#### 5.2.1. Mean and Standard Deviation for S1 and S2

#### 5.2.2. Displacement Histories for S1 and S2

#### 5.2.3. Displacement Histories for S3 and S4

#### 5.3. Comparison among Software Tools

#### 5.4. Verification with ICOLD Benchmark Study

_{cu}, and σ

_{to}, between the current article (22.41 MPa and 2.24 MPa, respectively) and those used in [7] (28.0 MPa and 2.0 MPa, respectively). Similarly, variations in foundation geometry are evident, with the current article using dimensions of 412 m by 206 m, contrasting with [7] where dimensions of 700 m by 122 m were used. The reservoir level considered in the current study is 290.02 m vs. 278.57 m in Case A2 [8]. Lastly, regarding differences in Rayleigh viscous damping parameters for the dam, the current article employs α = 0 and β = 0.004333, while [7] uses α = 0.751 and β = 0.0005.

## 6. Conclusions

- (i)
- For both dams, in the case of the dam-only model, the modal parameters and crest displacement histories match quite well across all three software systems in scenarios S1 and S2.
- (ii)
- The observed discrepancy in the crest displacement values between the Westergaard-added mass approach and the reservoir modeled using acoustic elements in the DFR models for the Koyna and Pine Flat dams highlights the impact of using simplified approaches such as the added mass approach. Specifically, in the case of Koyna, the Westergaard-added mass approach yields values of the crest displacement approximately 40% lower than those obtained using the acoustic elements, for both scenarios S1 and S2. Conversely, for Pine Flat, the same approach reports 33% higher values.
- (iii)
- The reservoir, when modeled using acoustic elements, captures fluid–structure interaction more accurately than the Westergaard-added mass approach, which considers only a fraction of reservoir participation. This distinction led to an observed increase in the vibration period when acoustic elements were employed. This indicates that the model with acoustic elements attracts lower seismic force compared to the added mass model. Thus, careful selection of reservoir models for fluid–structure interaction is essential.
- (iv)
- Although isotropic and homogeneous material properties and boundary conditions were adopted for the foundation, a variation in results across the DF models for scenarios S1 and S2 in all three software tools was observed. This variation can be attributed to the way each tool implements the soil–structure interaction.
- (v)
- The maximum crest displacement showed increasing variation with increasing modeling complexity, emphasizing the importance of progressive simulation with increasing model complexity for a proper understanding of the system’s behavior.
- (vi)
- In the case of scenarios S3 and S4 for both the D and DF models, with the reservoir empty condition, the effect of nonlinear analysis was more pronounced in the Koyna dam. An empty reservoir condition is often more critical during seismic events as it lacks the damping effects of water, potentially leading to more significant structural stress and displacement in concrete gravity dams.
- (vii)
- Similarly, in the DR and DFR models, the effect of nonlinear analysis was more pronounced for the reservoir-filled condition in the case of Pine Flat Dam; this behavior can be attributed to its dynamic properties.
- (viii)
- The reservoir modeled using acoustic elements in the case of DR and DFR models provided the most consistent results in all four scenarios across both the dams.
- (ix)
- Nonlinear effects were more pronounced for Koyna than for Pine Flat Dam, highlighting that while providing a more adequate response, the use depends on the level of seismic safety evaluation desired.
- (x)
- EAGD-84 and ADRFS v1 are easy to use, and quicker-turnaround tools can be helpful for a preliminary safety assessment. For comprehensive safety assessments, however, Abaqus or similar software remains a suitable choice for a detailed analysis.
- (xi)
- Both EAGD-84 and ADRFS v1 utilize a pressure-based formulation for reservoir modeling, akin to Abaqus acoustic elements. Therefore, for linear assessments, these tools can be a suitable alternative to Abaqus.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Taxonomy of uncertainty in risk analysis adapted from [1].

**Figure 2.**Progressive seismic safety evaluation adapted from [21].

**Figure 4.**Dam–foundation–reservoir system [11].

**Figure 13.**Displacement variation among software tools for different model complexities: (

**a**) Koyna Dam and (

**b**) Pine Flat Dam.

**Figure 14.**Koyna Dam horizontal time displacement for S1 scenario: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR.

**Figure 15.**Pine Flat horizontal time displacement for S1 scenario: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR.

**Figure 16.**Koyna Dam horizontal time displacement for S2 scenario: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR.

**Figure 17.**Pine Flat Dam horizontal time displacement for S2 scenario: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR.

**Figure 18.**Koyna Dam horizontal time displacement: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR, linear (S1) vs. nonlinear analysis (S3) considering horizontal ground motion component only.

**Figure 19.**Pine Flat Dam horizontal time displacement: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR, linear (S1) vs. nonlinear analysis (S3) considering the horizontal ground motion component.

**Figure 20.**Koyna Dam horizontal time displacement: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR, linear (S2) vs. nonlinear analysis (S4) considering horizontal and vertical ground motion components.

**Figure 21.**Pine Flat Dam horizontal time displacement: (

**a**) D, (

**b**) DF, (

**c**) DR, and (

**d**) DFR, linear (S2) vs. nonlinear analysis (S4) considering horizontal and vertical ground motion components.

Software, z = | ||||
---|---|---|---|---|

Scenario, x = | 1 | 2 | 3 | 4 |

1 | ✓ | ✓ | ✓ | ✓ |

2 | ✓ | ✘ | ✓ | ✓ |

3 | ✘ | ✘ | ✓ | ✓ |

4 | ✘ | ✘ | ✓ | ✓ |

**Table 2.**Elastic material properties of dam, foundation, and reservoir for Koyna Dam [45].

Material Properties | Dam | Foundation | Reservoir |
---|---|---|---|

General and Specific to Abaqus | |||

Density (ρ) | 2643 kg/m^{3} | 2643 kg/m^{3} | 1000 kg/m^{3} |

Modulus of elasticity (E) | 31,027 MPa | 27,580 MPa | - |

Bulk modulus (K) | - | - | 2070 MPa |

Poisson’s ratio (ν) | 0.15 | 0.333 | - |

Rayleigh damping Alpha (α) | - | 1.64 | - |

Rayleigh damping Beta (β) | 0.00323 | 0.0012 | - |

Specific to EAGD-84 | |||

Wave reflection coefficient (α) | - | - | 0.75 |

Hysteric damping for dam | 0.07 | 0.04 | - |

Specific to ADRFS v1 | |||

Wave reflection coefficient | - | - | 0.75 |

Wave speed | - | - | 1440.00 m/s |

Damping ratio (ζ) | 0.03 | 0.02 | - |

**Table 3.**Elastic material properties of dam, foundation, and reservoir for Pine Flat dam [46].

Material Properties | Concrete | Foundation | Reservoir |
---|---|---|---|

General and Specific to Abaqus | |||

Density (ρ) | 2482 kg/m^{3} | 2640 kg/m^{3} | 1000 kg/m^{3} |

Modulus of elasticity (E) | 22,407 MPa | 22,407 MPa | - |

Bulk modulus (K) | - | - | 2070 MPa |

Poisson’s ratio | 0.2 | 0.333 | - |

Rayleigh damping Alpha (α) | - | 1.64 | |

Rayleigh damping Beta (β) | 0.004333 | 0.00668 | |

Specific to EAGD-84 | |||

Wave reflection coefficient (α) | - | - | 0.75 |

Hysteric damping for dam | 0.1 | 0.1 | - |

Specific to ADRFS v1 | |||

Wave reflection coefficient | - | - | 0.75 |

Wave speed | - | - | 1440.00 m/s |

Damping ratio (ζ) | 0.04 | 0.07 | - |

ψc * | σ_{co} (MPa) | σ_{cu} (MPa) | σ_{to} (MPa) | e | R | |
---|---|---|---|---|---|---|

Koyna | 36.31° | 13.0 | 24.1 | 2.90 | 0.1 | 1.16 |

Pine Flat | 36.31° | 12.08 | 22.41 | 2.24 | 0.1 | 1.16 |

_{c}: dilatation angle; σ

_{co}: compressive initial yield stress; σ

_{cu}: compressive ultimate yield stress; σ

_{to}: tensile failure stress; e: flow potential eccentricity; R: ratio of the initial equibiaxial to the uniaxial compressive yield stress.

**Table 5.**Maximum crest displacement in (m) for scenarios S1 and S2 across different solution procedures and model complexities.

Scenario (x =) | Koyna Dam | Pine Flat Dam | |||||||
---|---|---|---|---|---|---|---|---|---|

Model Complexity (y =) | Software (z =) | ||||||||

1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | ||

S1 | 1 | 0.042 | 0.042 | 0.039 | - | 0.027 | 0.030 | 0.033 | - |

S1 | 2 | 0.037 | 0.039 | 0.035 | - | 0.026 | 0.038 | 0.034 | - |

S1 | 3 | 0.045 | 0.051 | 0.049 | 0.050 | 0.041 | 0.043 | 0.057 | 0.042 |

S1 | 4 | 0.056 | 0.046 | 0.029 | 0.050 | 0.031 | 0.043 | 0.056 | 0.042 |

S2 | 1 | 0.046 | - | 0.042 | - | 0.029 | - | 0.036 | - |

S2 | 2 | 0.035 | - | 0.039 | - | 0.028 | - | 0.036 | - |

S2 | 3 | 0.047 | - | 0.046 | 0.048 | 0.031 | - | 0.058 | 0.044 |

S2 | 4 | 0.059 | - | 0.027 | 0.048 | 0.045 | - | 0.058 | 0.044 |

**Table 6.**Maximum crest displacement in (m) for scenarios S3 and S4 across different solution procedures and model complexities.

Scenario (x =) | Model Complexity (y =) | Koyna | Pine Flat | ||
---|---|---|---|---|---|

Software (z =) | |||||

3 | 4 | 3 | 4 | ||

S3 | 1 | 0.040 | - | 0.035 | - |

S3 | 2 | 0.042 | - | 0.035 | - |

S3 | 3 | 0.035 | 0.043 | 0.066 | 0.043 |

S3 | 4 | 0.037 | 0.044 | 0.077 | 0.044 |

S4 | 1 | 0.051 | - | 0.043 | - |

S4 | 2 | 0.047 | - | 0.043 | - |

S4 | 3 | 0.035 | 0.040 | 0.054 | 0.044 |

S4 | 4 | 0.037 | 0.040 | 0.077 | 0.046 |

**Table 7.**Comparison of modal parameters with the 15th ICOLD International Benchmark Workshop [8].

Mode | Natural Frequency (Hz) | St. DeviationICOLD Benchmark ^{3} | ||
---|---|---|---|---|

Present Study ^{1} | Present Study ^{2} | ICOLD Benchmark ^{3} | ||

1 | 1.90 | 1.81 | 2.15 | 0.34 |

2 | 3.67 | 3.04 | 3.28 | 0.63 |

3 | 3.76 | 3.32 | 3.91 | 0.79 |

4 | 4.86 | 3.54 | 4.51 | 0.88 |

^{1}scenario, y = 4, z = 3 and

^{2}scenario, y = 4, z = 4 and

^{3}Case A2 of 15th ICOLD International Benchmark Workshop.

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**MDPI and ACS Style**

Patra, B.K.; Segura, R.L.; Bagchi, A.
Modeling Variability in Seismic Analysis of Concrete Gravity Dams: A Parametric Analysis of Koyna and Pine Flat Dams. *Infrastructures* **2024**, *9*, 10.
https://doi.org/10.3390/infrastructures9010010

**AMA Style**

Patra BK, Segura RL, Bagchi A.
Modeling Variability in Seismic Analysis of Concrete Gravity Dams: A Parametric Analysis of Koyna and Pine Flat Dams. *Infrastructures*. 2024; 9(1):10.
https://doi.org/10.3390/infrastructures9010010

**Chicago/Turabian Style**

Patra, Bikram Kesharee, Rocio L. Segura, and Ashutosh Bagchi.
2024. "Modeling Variability in Seismic Analysis of Concrete Gravity Dams: A Parametric Analysis of Koyna and Pine Flat Dams" *Infrastructures* 9, no. 1: 10.
https://doi.org/10.3390/infrastructures9010010