# Influences on the Seismic Response of a Gravity Dam with Different Foundation and Reservoir Modeling Assumptions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hydrodynamic Pressure Modelling Approaches

#### 2.1. Westergaard Added Mass Method

_{i}is the effective area of i, H

_{i}is the total water depth of the vertical surface at which i is located, Z

_{i}is the height from i to the bottom of the structural surface subjected to hydrodynamic pressure, and λ

_{i}is the normal vector of i,${\lambda}_{i}=\left\{{\lambda}_{ix},{\lambda}_{iy},{\lambda}_{iz}\right\}$.

#### 2.2. Potential-Based Fluid Formulation

## 3. Viscoelastic Artificial Boundary and Earthquake Input Mechanisms

#### 3.1. Viscoelastic Artificial Boundary Condition

#### 3.2. Earthquake Input Mechanisms

_{p}, c

_{s}, λ are the foundation density, P-wave velocity, S-wave velocity, and Lame constant, respectively; H

_{s}is the vertical distance from the wave source to the bottom boundary; h is the vertical distance from q to the bottom boundary; $\Delta {t}_{1}$, $\Delta {t}_{2}$, $\Delta {t}_{3}$, and $\Delta {t}_{4}$ are the time delay of the incident P-wave at q, the reflected P-wave at the foundation surface, the incident S-wave at q, and the reflected S-wave at the foundation surface, respectively. The subscripts of the equivalent nodal loads represent the node number and component direction, and the superscripts represent the wave field for calculating the equivalent nodal loads and the outer normal direction of the boundary surface at which q is located, which is positive if the direction is the same as the coordinate axis and negative if the direction is opposite to the coordinate axis.

#### 3.3. Verification Test

^{10}N/m

^{2}, the mass density is 2777 kg/m

^{3}, the Poisson’s ratio is 0.23, the S-wave velocity is 1239.8 m/s, and the P-wave velocity is 2093.6 m/s. The dynamic time-history analysis is performed with a total calculation time of 1 s and a time step of 0.01 s. The input displacement, velocity, and acceleration time history are determined by Equations (11)–(13) respectively, and their time-history curves are shown in Figure 4.

## 4. General Description of the Numerical Example

#### 4.1. General Information

^{10}N/m

^{2}, mass density = 2400 kg/m

^{3}, Poisson’s ratio = 0.167 according to Design Code for Hydraulic Concrete Structures (SL 191-2008) [45], structural damping = 5%. Foundation: modulus of elasticity = 1.5 × 10

^{10}N/m

^{2}, mass density = 2700 kg/m

^{3}, Poisson’s ratio = 0.24. Reservoir: mass density of water = 1000 kg/m

^{3}, acoustic wave speed = 1440 m/s.

#### 4.2. Cases of the Numerical Analysis

## 5. Dynamic Characteristics

## 6. Dam-Foundation Interaction

#### 6.1. Simulation Methods of Foundation

#### 6.2. Sensitivity Analysis of Foundation Size

#### 6.3. The Radiation Damping Effect of Infinite Foundation

#### 6.3.1. Verification of the Foundation Model

^{2}in both downstream and vertical directions. Three observation points are set on the foundation surface: point M which is closest to the upstream side, point N which is the intermediate point of the upstream side, and point O where there is the wave source. The PGA values at these three points are listed in Table 5. It is seen that the amplitude of the wave is almost doubled when it reaches the free surface of the foundation, which is consistent with the theory and further verifies the applicability of the viscoelastic artificial boundary and corresponding earthquake input mechanism.

^{2}in the downstream direction and 1.528 m/s

^{2}in the vertical direction. The displacement and velocity time histories are given in Figure 12, and the PGA values at each observation point are shown in Table 6.

#### 6.3.2. The Radiation Damping Effect

## 7. Dam–Reservoir Interaction

#### 7.1. Simulation Methods of Reservoir

#### 7.2. Reservoir Water Length

## 8. Conclusions

- (1)
- The natural frequency of the dam decreases greatly in numerical analysis when the dam–foundation interaction is considered, and decreases slightly with the increase in the foundation size. The simulation methods of reservoir water have significant effects on the natural frequency of the dam, whereas the reservoir water lengths have no significant effect.
- (2)
- The dynamic interaction of the dam and the foundation cannot be ignored. The radiation damping effect should be considered in the dynamic numerical analysis. The viscoelastic artificial boundary foundation is more efficient than the massless foundation in simulating the radiation damping effect of the far-field foundation. It was found that a foundation range of 3 times the dam height in all directions, such as upstream, downstream, and depth, is the most reasonable range of the truncation boundary of the foundation.
- (3)
- The methods used for reservoir water simulation have no significant effects on the acceleration and displacement of the dam, but have a significant effect on the stress. Compared with the Westergaard added mass method, the potential-based fluid simulation method simultaneously takes into account the reservoir–dam and reservoir–foundation interactions. The static and dynamic water pressure was applied to the upstream foundation surface as the reservoir water, which may cause downward deformation of the foundation and consequently increase the normal tensile stress at the dam heel. It was found that a reservoir length of 3 times the dam height is feasible for the truncation boundary of the reservoirs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 10.**The principal stress envelopes of the dam for different foundation sizes obtained using the MLF model (Pa). (

**a**) The distribution of σ

_{1}for B-1 (1H), (

**b**) the distribution of σ

_{3}for B-1 (1H), (

**c**) the distribution of σ

_{1}for B-2 (1.5H), (

**d**) the distribution of σ

_{3}for B-2 (1.5H), (

**e**) the distribution of σ

_{1}for B-3 (2H), (

**f**) the distribution of σ

_{3}for B-3 (2H), (

**g**) the distribution of σ

_{1}for B-4 (3H), (

**h**) the distribution of σ

_{3}for B-4 (3H).

**Figure 13.**The principal stress envelopes of the dam for different foundation sizes obtained using the VABF model (Pa). (

**a**) The distribution of σ

_{1}for C-1 (1H), (

**b**) the distribution of σ

_{3}for C-1 (1H), (

**c**) the distribution of σ

_{1}for C-2 (1.5H), (

**d**) the distribution of σ

_{3}for C-2 (1.5H), (

**e**) the distribution of σ

_{1}for C-3 (2H), (

**f**) the distribution of σ

_{3}for C-3 (2H), (

**g**) The distribution of σ1 for C-4 (3H), (

**h**) the distribution of σ

_{3}for C-4 (3H).

**Figure 14.**Hydrodynamic pressure distributions along the dam–reservoir interface. (

**a**) The VABF model, (

**b**) the MLF model.

Cases | Dam | Foundation | Reservoir | ||||
---|---|---|---|---|---|---|---|

Simulation Methods | Foundation Sizes (H = Dam Height) | Simulation Methods | Reservoir Lengths | ||||

Upstream | Downstream | Depth | |||||

A-1 | Linear | RF | / | / | / | WAMR | / |

B-1 | Linear | MLF | 1H | 1H | 1H | WAMR | / |

B-2 | Linear | MLF | 1.5H | 1.5H | 1.5H | WAMR | / |

B-3 | Linear | MLF | 2H | 2H | 2H | WAMR | / |

B-4 | Linear | MLF | 3H | 3H | 3H | WAMR | / |

C-1 | Linear | VABF | 1H | 1H | 1H | WAMR | / |

C-2 | Linear | VABF | 1.5H | 1.5H | 1.5H | WAMR | / |

C-3 | Linear | VABF | 2H | 2H | 2H | WAMR | / |

C-4 | Linear | VABF | 3H | 3H | 3H | WAMR | / |

D-1 | Linear | VABF | 3H | 3H | 3H | IPFR | 3H |

D-2 | Linear | VABF | 3H | 3H | 3H | CPFR | 3H |

D-3 | Linear | MLF | 3H | 1H | 1H | WAMR | 3H |

D-4 | Linear | MLF | 3H | 1H | 1H | IPFR | 3H |

D-5 | Linear | MLF | 3H | 1H | 1H | CPFR | 3H |

D-6 | Linear | MLF | 4H | 1H | 1H | CPFR | 4H |

D-7 | Linear | MLF | 5H | 1H | 1H | CPFR | 5H |

Case | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
---|---|---|---|---|---|---|---|---|---|---|

A-1 | 1.926 | 4.261 | 6.951 | 7.251 | 10.147 | 12.751 | 14.612 | 15.029 | 16.732 | 17.064 |

B-1/C-1 | 1.286 | 2.943 | 3.306 | 5.082 | 7.949 | 10.771 | 11.046 | 12.752 | 13.752 | 14.344 |

B-2/C-2 | 1.247 | 2.838 | 3.032 | 4.994 | 7.906 | 10.681 | 11.023 | 12.726 | 13.730 | 14.310 |

B-3/C-3 | 1.223 | 2.747 | 2.892 | 4.941 | 7.881 | 10.633 | 11.011 | 12.713 | 13.719 | 14.295 |

B-4/C-4 | 1.194 | 2.582 | 2.775 | 4.877 | 7.853 | 10.583 | 10.997 | 12.701 | 13.709 | 14.280 |

D-1 | 1.294 | 1.968 | 2.405 | 3.183 | 3.574 | 5.263 | 5.710 | 7.394 | 9.151 | 10.025 |

D-2 | 1.243 | 1.474 | 1.921 | 2.607 | 3.022 | 3.227 | 3.912 | 4.510 | 4.843 | 4.997 |

D-3 | 1.282 | 2.919 | 3.303 | 5.047 | 7.927 | 10.770 | 11.032 | 12.750 | 13.747 | 14.344 |

D-4 | 1.387 | 2.976 | 3.492 | 3.499 | 4.304 | 5.580 | 5.993 | 7.548 | 9.222 | 10.116 |

D-5 | 1.322 | 1.751 | 2.152 | 2.810 | 3.392 | 3.484 | 3.986 | 4.903 | 4.929 | 5.150 |

D-6 | 1.322 | 1.708 | 1.970 | 2.415 | 2.962 | 3.419 | 3.484 | 3.907 | 4.595 | 4.903 |

D-7 | 1.322 | 1.681 | 1.871 | 2.185 | 2.604 | 3.060 | 3.440 | 3.484 | 3.856 | 4.407 |

Case | A-1 | B-1 | C-1 | |
---|---|---|---|---|

Simulation Methods of Foundation | RF | MLF | VABF | |

Downstream direction | Acceleration magnification factor | 5.894 | 4.680 | 2.112 |

Relative displacement (m) | 0.081 | 0.148 | 0.100 | |

Vertical direction | Acceleration magnification | 5.483 | 3.447 | 1.781 |

Relative displacement (m) | −0.031 | −0.037 | −0.022 | |

Dam stress (MPa) | Vertical normal tensile stress (σ_{zz}) | 7.369 | 9.227 | 3.042 |

Vertical normal compressive stress (σ_{zz}) | −8.255 | −20.05 | −15.08 | |

Principal tensile stress (σ_{1}) | 7.969 | 16.64 | 10.56 | |

Principal compressive stress (σ_{3}) | −10.03 | −28.00 | −24.89 |

Case | B-1 | B-2 | B-3 | B-4 | |
---|---|---|---|---|---|

Foundation Size | 1H | 1.5H | 2H | 3H | |

Downstream direction | Acceleration magnification factor | 4.680 | 5.057 | 5.789 | 3.477 |

Relative displacement (m) | 0.148 | 0.150 | 0.154 | 0.141 | |

Vertical direction | Acceleration magnification | 3.447 | 3.056 | 3.856 | 4.220 |

Relative displacement (m) | −0.037 | −0.038 | −0.048 | −0.042 | |

Dam stress (MPa) | Vertical normal tensile stress (σ_{zz}) | 9.227 | 9.903 | 8.770 | 6.754 |

Vertical normal compressive stress (σ_{zz}) | −20.05 | −20.79 | −21.15 | −18.72 | |

Principal tensile stress (σ_{1}) | 16.64 | 16.37 | 13.70 | 10.66 | |

Principal compressive stress (σ_{3}) | −28.00 | −29.52 | −27.21 | −28.01 |

Foundation Size | Downstream | Vertical | ||||
---|---|---|---|---|---|---|

M | N | O | M | N | O | |

1H | 25.185 (2.008) | 26.065 (2.078) | 24.103 (1.922) | 25.712 (2.050) | −27.683 (2.207) | 23.948 (1.909) |

1.5H | −25.773 (2.055) | 26.833 (2.139) | 23.757 (1.894) | −25.197 (2.009) | −26.204 (2.089) | −24.580 (1.960) |

2H | 25.141 (2.005) | −27.651 (2.205) | 23.269 (1.855) | 25.277 (2.015) | 26.125 (2.083) | 25.152 (2.005) |

3H | 25.451 (2.029) | −27.316 (2.178) | 23.394 (1.865) | 25.234 (2.012) | 26.589 (2.120) | 25.749 (2.053) |

Foundation Size | Downstream | Vertical | ||||
---|---|---|---|---|---|---|

M | N | O | M | N | O | |

1H | 3.506 (−24.6%) | 4.133 (−11.1%) | 4.121 (−11.3%) | 2.807 (−8.1%) | 3.410 (+11.6%) | 3.125 (+2.3%) |

1.5H | −4.259 (−8.4%) | −4.572 (−1.6%) | −3.819 (−17.8%) | 3.541 (+15.9%) | 2.898 (−5.2%) | 3.178 (+4.0%) |

2H | −4.200 (−9.6%) | −4.716 (+1.5%) | −5.092 (+9.6%) | 3.027 (−0.9%) | 3.031 (−0.8%) | 3.888 (+27.2%) |

3H | 4.493 (−3.3%) | −4.472 (−3.8%) | 4.509 (−3.0%) | 2.792 (−8.6%) | −3.267 (+6.9%) | 3.405 (+11.4%) |

Case | C-1 | C-2 | C-3 | C-4 | |
---|---|---|---|---|---|

Foundation Size | 1H | 1.5H | 2H | 3H | |

Downstream direction | Acceleration magnification factor | 2.112 | 2.052 | 1.992 | 1.910 |

Relative displacement (m) | 0.100 | 0.088 | 0.081 | 0.074 | |

Vertical direction | Acceleration magnification | 1.781 | 1.529 | 1.654 | 1.654 |

Relative displacement (m) | −0.022 | −0.023 | −0.024 | −0.022 | |

Dam stress (MPa) | Vertical normal tensile stress (σ_{zz}) | 3.042 | 2.524 | 2.368 | 2.403 |

Vertical normal compressive stress (σ_{zz}) | −15.08 | −13.78 | −12.97 | −12.03 | |

Principal tensile stress (σ_{1}) | 10.56 | 7.719 | 6.361 | 4.698 | |

Principal compressive stress (σ_{3}) | −24.89 | −23.45 | −22.39 | −21.19 |

Case | C-4 | D-1 | D-2 | D-3 | D-4 | D-5 | |
---|---|---|---|---|---|---|---|

Simulation Method of Foundation | VABF | MLF | |||||

Simulation Method of Reservoir Water | WAMR | IPFR | CPFR | WAMR | IPFR | CPFR | |

Downstream direction | Acceleration magnification factor | 1.910 | 2.429 | 2.219 | 4.575 | 4.833 | 5.249 |

Relative displacement (m) | 0.074 | 0.069 | 0.068 | 0.149 | 0.112 | 0.172 | |

Vertical direction | Acceleration magnification | 1.654 | 1.822 | 1.715 | 3.433 | 4.085 | 2.660 |

Relative displacement (m) | −0.022 | −0.017 | −0.018 | −0.037 | −0.032 | −0.027 | |

Dam stress (MPa) | Vertical normal tensile stress (σ_{zz}) | 2.403 | 5.477 | 6.374 | 8.122 | 9.161 | 11.224 |

Vertical normal compressive stress (σ_{zz}) | −12.03 | −8.797 | −9.121 | −20.27 | −11.84 | −13.65 | |

Principal tensile stress (σ_{1}) | 4.698 | 12.11 | 13.98 | 14.48 | 15.07 | 17.91 | |

Principal compressive stress (σ_{3}) | −21.19 | −19.09 | −19.50 | −27.99 | −22.55 | −24.76 | |

Hydrodynamic pressure of dam heel (KN/m^{2}) | 716.8 | 429.8 | 371.4 | 1188 | 1127 | 994.3 |

Case | D-5 | D-6 | D-7 | |
---|---|---|---|---|

Reservoir Water Length | 3H | 4H | 5H | |

Downstream direction | Acceleration magnification factor | 5.249 | 4.275 | 4.303 |

Relative displacement (m) | 0.172 | 0.129 | 0.140 | |

Vertical direction | Acceleration magnification | 2.660 | 2.235 | 2.172 |

Relative displacement (m) | −0.027 | −0.027 | −0.024 | |

Dam stress (MPa) | Vertical normal tensile stress (σ_{zz}) | 11.224 | 9.52 | 10.15 |

Vertical normal compressive stress (σ_{zz}) | −13.65 | −12.25 | −11.96 | |

Principal tensile stress (σ_{1}) | 17.91 | 15.23 | 15.62 | |

Principal compressive stress (σ_{3}) | −24.76 | −24.11 | −23.35 | |

Hydrodynamic pressure of dam heel (KN/m^{2}) | 994.3 | 704.0 | 615.9 |

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**MDPI and ACS Style**

Wang, C.; Zhang, H.; Zhang, Y.; Guo, L.; Wang, Y.; Thira Htun, T.T.
Influences on the Seismic Response of a Gravity Dam with Different Foundation and Reservoir Modeling Assumptions. *Water* **2021**, *13*, 3072.
https://doi.org/10.3390/w13213072

**AMA Style**

Wang C, Zhang H, Zhang Y, Guo L, Wang Y, Thira Htun TT.
Influences on the Seismic Response of a Gravity Dam with Different Foundation and Reservoir Modeling Assumptions. *Water*. 2021; 13(21):3072.
https://doi.org/10.3390/w13213072

**Chicago/Turabian Style**

Wang, Chen, Hanyun Zhang, Yunjuan Zhang, Lina Guo, Yingjie Wang, and Thiri Thon Thira Htun.
2021. "Influences on the Seismic Response of a Gravity Dam with Different Foundation and Reservoir Modeling Assumptions" *Water* 13, no. 21: 3072.
https://doi.org/10.3390/w13213072