The paper focuses on the failure process and mechanism of the concrete gravity dam considering different nonlinear models under strong earthquakes. By taking a typical monolith of a concrete gravity dam as a case study, a comparative analysis of the failure process and mechanism of the dam considering the plastic damage model and the dynamic contact model, respectively, is performed using the seismic overload method. Moreover, the ultimate seismic capacity of the dam is evaluated for both of the nonlinear models. It is found that the ultimate seismic capacity of the dam is slightly different, but the failure process has significant distinctions in each model. And, the damage model is recommended when the conditions permit.

Seismic safety of high concrete dams is extremely important. Once a reservoir dam holding hundreds of millions of tons of lake water fails, it will cause unimaginable catastrophic consequences [

The damage, cracking, or even collapse of dams could happen leading to the stress redistribution when the stress exceeds the allowable strength under strong earthquakes. Therefore, the assumption of linear behavior is no longer applicable, and the nonlinear situation of the material should be considered to conform to the actual situation better [

Besides, the tensile strength and shear strength of the RCC gravity dam with the rolled surface are usually lower than that of normal concrete, which may lead to the decrease of stability and safety of the dam [

At present, a more commonly used model for simulating material nonlinearity is the plastic damage model proposed by Lubliner et al. [

The concrete plastic damage model [

When the stress is inside the yield surface, the concrete material is in an elastic state. When the stress is on the yield surface, the concrete material begins to enter the plastic state. The yield function can be expressed as follows:

The plastic damage model assumes the nonassociated potential flow:

The flow potential

When the concrete is subjected to the uniaxial tension and uniaxial compression, the plastic strain can be expressed as

The stress-strain relations under uniaxial tension and compression loading are, respectively,

Under the uniaxial cyclic loading conditions, the stiffness degradation mechanisms of the material can be expressed as

Under the action of multiaxial cyclic loading, the elastic stiffness degradation is assumed to be isotropic, which can be expressed by a single scalar variable

The equivalent plastic strain rates are evaluated according to the expressions:

In the dynamic contact model, the contact problem established by adding a supplementary equation constructed by the contact constraints to the dynamic equation is solved in the form of point-to-point contact. The explicit integration method of the central differential method combined with the Newmark constant average acceleration method is adopted, and the detailed procedures are shown in Reference [

For the convenience of derivation, formula (

To get the normal contact force, taking the contact point pairs

When the normal contact between

When the static friction force exceeds

From the above calculation,

Taking a typical nonoverflow monolith of a concrete dam as a case study, the dam bottom and crest elevation are 1970 m and 2155 m, respectively. The elevation of the upstream breakpoint is 2040 m, the width of the dam crest is 16 m, and the width of the dam bottom is 165.5 m. The geometric dam section and concrete partition are shown in Figure

The geometric dam section and concrete partition (unit: m).

Finite element model of the dam.

Material properties of dam concrete and foundation rock.

Number | Concrete strength grade | The standard value of dynamic tensile strength (MPa) | Dynamic elastic modulus (GPa) | Poisson’s ratio | Density |
---|---|---|---|---|---|

2.00 | 42.00 | 0.167 | 2448.98 | ||

2.00 | 42.00 | 0.167 | 2448.98 | ||

2.00 | 42.00 | 0.167 | 2448.98 | ||

1.60 | 38.25 | 0.167 | 2448.98 | ||

2.00 | 42.00 | 0.167 | 2448.98 | ||

1.60 | 38.25 | 0.167 | 2448.98 | ||

Rock | — | — | 10.50 | 0.230 | 2777.00 |

The static loads mainly include the self-weight, upstream and downstream hydrostatic pressure, and silting pressure. Westergaard’s added mass without considering the compressibility of the reservoir water is used to consider the hydrodynamic interaction between the dam and the reservoir. The normal upstream water level is 2150 m, and the corresponding downstream water level is 2019.25 m. The elevation of silt is 2023.7 m, and its floating bulk density is 8

Time histories of normalized acceleration. (a) Vertical. (b) Along the river.

Figures

Dynamic tensile damage evolution curve of

Dynamic tensile damage evolution curve of

Dynamic compressive damage evolution curve of

Dynamic compressive damage evolution curve of

The damage evolution process of the concrete gravity dam is discussed in detail through the earthquake overload analysis [

Only the tension damage of the dam concrete is considered, and the earthquake overload analysis is performed with increasing overload coefficients of 1.2, 1.4, 1.52, 1.53, and 2.0. The results are shown in Figure

Damage rupture process only considering the tension damage under different earthquake overload coefficients. (a) 1.0. (b) 1.2. (c) 1.4. (d) 1.52. (e) 1.53. (f) 2.0.

Figure

Number of damage elements of the dam head under different overload coefficients. (a) 1.2. (b) 1.53. (c) 2.0.

Time histories of relevant variables at the element 3143 for the earthquake overload coefficient of 1.2. (a) Time history of tensile damage variables. (b) The relative horizontal displacement is 0.39 mm after the earthquake. (c) The relative vertical displacement is 0.34 mm after the earthquake.

Time histories of relevant variables at the element 3143 for the earthquake overload coefficient of 1.53. (a) Time history of tensile damage variables. (b) The relative horizontal displacement is 3.20 mm after the earthquake. (c) The relative vertical displacement is 4.15 mm after the earthquake.

Time histories of relevant variables at the element 3400 for the earthquake overload coefficient of 1.53. (a) Time history of tensile damage variables. (b) The relative horizontal displacement is 0.28 mm after the earthquake. (c) The relative vertical displacement is 0.14 mm after the earthquake.

Time histories of relevant variables at the element 3143 for the earthquake overload coefficient of 2.0. (a) Time history of tensile damage variables. (b) The relative horizontal displacement is 8.97 mm after the earthquake. (c) The relative vertical displacement is 17.38 mm after the earthquake.

Time histories of relevant variables at the element 3401 for the earthquake overload coefficient of 2.0. (a) Time history of tensile damage variables. (b) The relative horizontal displacement is 0.95 mm after the earthquake. (c) The relative vertical displacement is 8.95 mm after the earthquake.

Figure

The tension damage-rupture process considering both the tension and compression damage. (a) 1.0. (b) 1.2. (c) 1.4. (d) 1.54. (e) 1.55. (f) 2.0.

The compression damage-rupture process considering both the tension and compression damage. (a) 1.0. (b) 1.2. (c) 1.4. (d) 1.54. (e) 1.55. (f) 2.0.

It can be obtained that the compression damage near the dam toe expands slightly with the increasing of the earthquake overload coefficients (Figure

To explain this phenomenon, the results of the overload coefficient of 1.0 are taken as an example of a detailed discussion. The concrete strength grade at the dam heel is

The compression damage at the integration point of the dam heel for the overload coefficient of 1.0.

The relevant variables at the integral point of the dam heel for the overload coefficient of 1.0 at 5.82 s.

Maximum principal stress (MPa) | Minimum principal stress (MPa) | Tensile damage variable | Compressive damage variable |
---|---|---|---|

1.925 | −0.839 | 0.63228 | 0.00246 |

Tensile equivalent plastic strain | Compressive equivalent plastic strain | Maximum plastic strain | Minimum plastic strain |

− |

Figure

The compressive stress-strain of

The related variable values of the element 1327 near the dam toe in the time history under different overload coefficients.

Overload coefficients | Maximum compressive strain | Compressive equivalent plastic strain | Compressive damage variable |
---|---|---|---|

1.00 | 270.545 | 34.529 | 0.02452 |

1.20 | 297.189 | 47.455 | 0.03371 |

1.40 | 330.269 | 60.951 | 0.04330 |

1.54 | 353.277 | 70.733 | 0.05024 |

1.55 | 355.124 | 71.484 | 0.05077 |

2.00 | 422.743 | 114.483 | 0.08107 |

3.00 | 791.907 | 333.332 | 0.23424 |

From Figure

The compression damage-rupture distribution considering both the tension and compression damage for the overload coefficient of 3.0.

Moreover, the tension damage begins to appear at the downstream slope of the dam head for the overload coefficient of 1.2. Penetrating macrocracks through the upstream and downstream appear when the overload coefficient is 1.55. Therefore, it is suggested that 1.54 times the design earthquake can be considered as the ultimate seismic capacity, which is similar to that only considering the tensile damage.

As shown in Figure

Preset joints.

Results show that the preset contact joints did not open until the overload coefficient reaches 1.49, indicating that no damage and cracking occurs. From Figures

Time histories of displacement difference of joint pairs of 16. (a) Horizontal. (b) Vertical.

Time histories of displacement difference of joint pairs of 1. (a) Horizontal. (b) Vertical.

According to the results of the plastic damage model, the damage of the downstream surface of the dam head first appears when the overload coefficient is 1.2. With the increase of the earthquake overload coefficient, the damage gradually extends to the upstream surface. The penetrating cracks of the dam appear when the overload coefficients are 1.53 and 1.55 for the damage model only considering the tensile damage and that considering the tension and compression damage, respectively. However, for the contact model, when the overload coefficient is 1.49, the preset contact joint causes tensile cracks and completely breaks after the earthquake.

To sum up, the earthquake overload coefficient of the cracking at the dam head in the damage model is lower than that in the contact model, but the failure process of the contact model is faster than that of the damage model. The main reasons lie in that the damage cracking of concrete in the damage model is mainly judged by the state of the maximum principal stress at the element Gauss point, and there is no restriction on the damage propagation direction. However, for the contact model, the contact status changes among the separation, adhesion, and contact states. The failure of the contact surface is mainly judged by the normal and tangential force, and the influence area of the node is also larger, so the earthquake overload coefficient of the cracking at the dam head of the damage model is lower than that of the contact model. Besides, in the damage model, the dam concrete will enter the softening stage when the maximum principal stress exceeds the tensile strength. In the softening stage, the damage develops and the stiffness gradually degenerates, while the bearing capacity will not be completely lost immediately. Moreover, the elements near the penetrating part of the dam head are also damaged, which plays a role in energy dissipation. However, for the contact model, once the cracks appear in the normal direction, the contact joints cannot bear the normal tensile stress, and the tensile bearing capacity will be completely lost, so the damage and cracking process are faster than those of the damage model.

If the penetrating crack of the dam is taken as the failure criteria, the ultimate seismic capacities of the damage model only considering tensile damage and considering tensile and compression damage are 1.52 and 1.54 times the design earthquake, respectively. For the contact model, it is suggested as 1.48 times the design earthquake. Therefore, the difference in the ultimate seismic capacities of the concrete gravity dam obtained by the two different models is slight.

Compared with the contact model, the damage model is more consistent with the actual situation without presetting the contact joints. It only needs to consider the damage softening process of the material. However, the damage model requires a refined finite element mesh leading to a large number of computing efforts. The contact model is more suitable for RCC gravity dams with clear weak interfaces. The coarse finite element mesh is sufficient for the analysis, and computing efforts are reduced. Moreover, it can also give a reasonable result to evaluate the ultimate seismic capacity of concrete dams, while, for the dams without clear weak interfaces, the damage model is recommended when the calculation condition permits.

In this paper, seismic overload analysis is performed by taking a typical nonoverflow monolith of a concrete gravity dam considering the damage model and contact model. The tension and compression damage development process and failure mechanism of the concrete gravity dam under the strong earthquake including its ultimate seismic capacity are of comparative analysis. The main conclusions are as follows:

For the concrete dam with the material damage model, with the increase of the overload coefficients, the tension and compression damage may occur simultaneously in multiaxial stress conditions, but the damage of the concrete gravity dam is mainly controlled by the tensile damage. A slight difference in the development pattern of the damage is found between the damage model considering only the tension damage and that considering the tension and compression damage.

Compared with the contact model, the earthquake overload coefficient of the cracking of the damage model is earlier, but the damage development process is slower. Based on the failure criteria of the penetrating crack of the dam, the ultimate seismic capacity of the gravity dam with the damage model considering only tensile damage is 1.52 times the design earthquake, and that, with the damage model considering the tension and compression damage, it is 1.54 times the design earthquake. However, the ultimate seismic capacity based on the contact model is 1.48 times the design earthquake. In summary, the ultimate seismic capacities of the gravity dam under the two models are similar, and the contact model is slightly lower.

It is suggested that the contact model should be used for the RCC gravity dam with clear weak interfaces and coarse meshes, while for the dams without clear weak interfaces, the damage model is recommended when the calculation conditions permit.

All data generated or analyzed during this study are included within this article.

The authors declare that they have no conflicts of interest.

This study was supported by the National Key Research and Development Program (Grant no. 2016YFC0401807) and National Key Research and Development Program (Grant no. 2017YFC0404903) and Research Project of China Three Gorges Corporation (Grant no. XLD/2115). The authors are grateful for these supports.