Abstract
With the change of the seismic parameter zoning map of China (GB 18306-2015), the seismic grade of reservoir dams in some areas had changed. At the same time, the standard for seismic design of hydraulic structures (GB 51247-2018) also put forward new requirements for the seismic calculation of reservoir dams. In order to ensure the safe operation of reservoir dams, it is necessary to review the seismic safety based on the finite element numerical simulation technology. Taking the retaining dam section of a gravity dam as an example, a finite element model of the retaining dam section was established, the corresponding calculation and analysis of ground motion were carried out using the mode decomposition response spectrum method, and the seismic safety evaluation of the retaining dam section was carried out according to the calculation results. The results show the following: (1) after considering the effect of hydrodynamic pressure, the natural frequency of the dam body has been significantly reduced, and the first-order natural frequency has been reduced by about 10%; (2) in addition to the local tensile stress at the dam heel, the rest of the vertical stress on the foundation surface of the retaining dam section is compressive stress, and the length of the tensile stress is less than the distance from the dam heel to the curtain center line, which met the requirements; (3) the antisliding stability safety factor of the retaining dam section is greater than the design value in the specification and meets the safety requirements; and (4) the seismic safety of the retaining dam section meets the standard requirements, and the seismic grade is evaluated as grade A.
1. Introduction
Reservoir Dam Safety Appraisal Method (Shui Jianguan [2003] No. 271) stipulates that the periodic safety appraisal system should be carried out for dams. The first safety appraisal should be carried out within 5 years after completion and acceptance and every 6 to 10 years thereafter. Seismic safety review is an important part of safety evaluation of reservoir dams. China is a country with many earthquake disasters, especially in the southwest and northwest regions. These regions account for 80% of the national hydropower resources, but they are high earthquake intensity regions in China, and the earthquake frequency is very high. At the same time, gravity dams are widely used in China, and the gravity dam would be damaged under the action of earthquake load just like the Koyna gravity dam. So, it is necessary to study aseismic safety review of reservoir dams.
A concrete gravity dam project was completed in April 2013, completed and accepted in May 2014, and carried out the first safety appraisal in April 2019. The elevation of the dam is about 1320.6 m, the maximum dam height is 41.6 m, and the width of the dam is 8.4 m, which is a medium-sized III class project. Among them, the upstream dam slope of the dam retaining section is 1 : 0.1, and the downstream dam slope is 1 : 1, as shown in Figure 1. The designed seismic intensity of the gravity dam is VII degree. With the change of China Ground Motion Parameter Zoning Map (GB 18306-2015) [1], the seismic intensity of the engineering area is adjusted from VII degree to VIII degree. At the same time, seismic Design Standard for Hydraulic Structures (GB 5247-2018) [2] (hereinafter referred to as seismic Design Standard) also puts forward new requirements for seismic calculation of reservoirs and dams. Therefore, in order to ensure the safe operation of the gravity dam project, it is necessary to conduct seismic safety review study of the project.
At present, most scholars combine the material mechanics method and quasistatic method to conduct seismic review calculation of gravity dam. In fact, the material mechanics method can calculate the stress component and principal stress at any point in the dam, and the calculation results are relatively accurate, but due to the influence of foundation deformation, the calculated internal force results of the lower part of the dam have a large error [3]. In recent years, with the rapid development of computer technology, finite element numerical simulation technology has been widely used in various fields [4–14], and many scholars began to use the finite element numerical simulation technology to analyze the seismic problems of gravity dams [15, 16]. Therefore, it is necessary to adopt finite element numerical simulation technology to conduct seismic review study of gravity dams.
2. Research Technique
2.1. Basic Theory of Finite Element
The basic idea of the finite element method is to discretize the continuous solution region into a group of finite elements connected together in a certain way, and then the approximate function assumed in each element is used to represent the unknown field function to be solved in the whole solution domain. Once these unknowns are solved, the approximate value of the field function in each element can be calculated by interpolation function, and the approximate solution in the whole solution domain can be obtained.
The static equilibrium equation of the finite element is as follows: where is the stiffness matrix, is the displacement vector, and is the load matrix.
The finite element dynamic equilibrium equation is as follows: where is the damping matrix, is the overall mass matrix, is the velocity vector, is the acceleration vector, and is the dynamic load matrix.
where is the mass matrix, is the additional mass matrix, and can be simulated by using Westergard formula which is as follows:
where is the density of water body, is the dam surface area corresponding to this point, and is any water depth.
The free vibration equation of the structure can be obtained from the finite element dynamic equilibrium equation, which is as follows:
where is the structural circle frequency.
2.2. Revitalizing Decomposition Reaction Spectroscopy
This time, the standard design response spectrum in the seismic Design Standard for Hydraulic Structures (GB 5247-2018) [2] is used to conduct seismic recheck calculation of the gate chamber structure, as shown in Figure 2, where the damping ratio is 10% and the representative value of maximum value of standard design response spectrum is 2.0.
When the mode decomposition response spectrum method is used to calculate the seismic effect, the seismic effect of each mode can be combined according to the square sum square root. When the ratio of the absolute value of the frequency difference between two vibration modes to one of the smaller frequencies is less than 0.1, the seismic action effect should adopt the complete quadratic root combination which is as follows:
where is the seismic effect, and are seismic effects of the -th and -th modes, respectively, is the number of mode shapes used for calculation, is the mode shape correlation coefficients for -th and -th modes, and are damping ratios of the -th and -th modes, respectively, and is the circular frequency ratio.
In consideration of the retroactivity of earthquakes, the response indexes such as displacement and stress obtained by the response spectrum method can be positive or negative. Therefore, when carrying out the static and static force superposition of structural response, the superposition principle adopted in this calculation is as follows: ① the static calculation results are directly added to the reaction spectrum calculation results (static + dynamic); ② the static calculation results are directly subtracted from the reaction spectrum calculation results (referred to as static-dynamic).
2.3. Calculation of Antisliding Stability
According to the Code for Design of Concrete Gravity DAMS (SL319-2018) [17], the shear strength formula or shear strength formula is adopted to calculate the sliding stability safety factor of the foundation surface. The specific shear strength formula is as follows:
where is the sliding stability safety factor calculated for shear strength, is the shear friction coefficient of the contact surface between concrete and dam foundation, is the shear cohesion of the interface between concrete of dam body and dam foundation, Pa, is the cross-sectional area of the contact surface between concrete of dam body and dam foundation, m2, is the normal score of all loads acting on the dam body on the sliding surface, N, and is the tangential score of all the loads acting on the dam body on the sliding surface, N.
3. Finite Element Calculation Model
3.1. Finite Element Model
According to the actual size of dam retaining section of a gravity dam, a two-dimensional finite element model of dam retaining section including foundation, curtain grouting, and corridor is established, as shown in Figure 3. In order to ensure the calculation accuracy, quadrilateral mesh is used for discrete mesh, and triangular element is used for partial transition, in which the number of cells is 18008 and the number of nodes is 18271. The Cartesian coordinate system is adopted, with the -direction in the direction of the water and -direction in the direction of the vertical direction. Three-way fixed constraints are imposed on the bottom of the foundation and normal constraints on both sides during calculation. In addition, due to the different mechanical properties of concrete materials and bedrock materials, thin layer units are set in the part where the dam body, impermeable curtain, and foundation contact, as shown in Figure 4.
(a) Oblique drawing
(b) Front view
3.2. Calculation Parameters
The material partition of the dam body is shown in Figure 1, and the material used in this calculation material parameters is shown in Table 1.
3.3. Calculated
The calculation condition is normal storage water level operation period plus VIII degree earthquake action. The water depth in front of the dam is 37.0 m, and there is no water behind the dam when the normal water level is running.
3.4. Load Calculation
In this calculation, the main consideration is dead weight, water load, silt pressure, wave pressure, uplift pressure, and earthquake load. The peak acceleration of ground motion at the dam site is 0.20 g, and the characteristic period of foundation response spectrum is 0.40s.
4. Finite Element Calculation Results and Analysis
4.1. Calculation Results and Analysis of Natural Vibration Characteristics
The natural vibration analysis of concrete gravity dam with water and without water is carried out by using the structural natural vibration characteristic analysis method. The influence of reservoir water on dam body is simulated by using Westergard formula to calculate hydrodynamic pressure by the additional mass method.
The first five natural vibration frequencies and periods of dam retaining section under normal storage level are shown in Table 2. It can be seen from the table that the natural vibration frequency of dam retaining section is larger in empty storage condition, and the first frequency of structure natural vibration is 8.37 Hz. Considering the hydrodynamic pressure, the natural vibration frequency of the dam decreases obviously, and the first order frequency of the dam is 7.53 Hz under normal storage level condition. Compared with empty reservoir, considering the additional mass of water body, the first natural vibration frequency of dam body is reduced by about 9.7%.
4.2. Displacement Calculation Results and Analysis
Figures 5 and 6, respectively, give the -direction and -direction displacement cloud maps of the lower retaining dam section with static and dynamic superposition. It can be seen from the figure that, under the earthquake action of normal storage level, the maximum downstream displacement of the dam section under different stacking modes is 3.69 mm and 3.23 mm, respectively, and both positions appear at the top of the dam body. The maximum vertical displacement of the dam retaining section is 1.35 mm and 2.95 mm, respectively, and the positions appear at the top of the downstream dam and the top of the upstream dam.
(a) -direction displacement cloud map
(b) -direction displacement cloud map
(a) -direction displacement cloud map
(b) -direction displacement cloud map
4.3. Stress Calculation Results and Analysis
Figure 7(a) shows the vertical stress cloud diagram of the dam body in the retaining dam section under the static action of normal storage level. It can be seen that, under the action of normal storage water level, the vertical stress of the dam body is basically compressive stress, only the tensile stress zone appears around the downstream corridor, and the maximum tensile stress value does not exceed the static tensile strength of C15 concrete (0.91 MPa), which meets the safety requirements. Figure 7(b) shows the cloud diagram of vertical stress calculation results of dam body under dynamic and static superposition of dam retaining section under earthquake action. As can be seen from the figure, under the earthquake action of normal storage water level, tensile stress concentration occurs at geometric changes such as the heel break angle of the dam section and the slope break angle of the downstream dam, and the maximum vertical tensile stress value reaches 0.99 MPa. The maximum tensile stress value does not exceed the dynamic tensile strength of C20 concrete (2.22 MPa), which meets the safety requirements.
(a) Normal water level static action
(b) Normal storage level seismic action (static+dynamic)
Figure 8 shows the vertical stress distribution on the foundation surface of the dam body of the retaining dam section under the static and static-static action of normal storage water level. It can be seen that under the static action of normal storage water level, the vertical stress of foundation surface is compressive stress, which meets the safety requirement. Under the seismic action of normal storage water level, the vertical stress of foundation surface is compressive stress except the partial tensile stress at the bottom of dam, and the length of tensile stress is less than the bottom-line of dam to the center line of curtain, which meets the safety requirements.
(a) Vertical stress distribution on foundation surface of retaining dam section under static action of normal storage water level situation
(b) Vertical stress distribution on foundation surface of retaining dam section under static and dynamic superposition (static + dynamic)
4.4. Analysis of Acceleration Calculation Results
Figure 9 shows the acceleration distribution cloud diagram of typical section of concrete gravity dam retaining section under earthquake action. As can be seen from the figure, the downstream acceleration of the retaining dam section increases gradually with the increase of dam height under earthquake action and reaches the maximum at the dam top, with a maximum value of 8.93 m/s2 and an amplification coefficient of 4.55. Similar to the calculation law of downstream acceleration, the vertical acceleration of dam retaining section increases gradually with the increase of dam height and reaches the maximum at the top of the upstream dam, with the maximum value of 4.00 m/s2 and the amplification coefficient of 2.04.
(a) -direction displacement cloud map
(b) -direction displacement cloud map
4.5. Stability Calculation Results and Analysis
Table 3 shows the calculation results of antisliding stability of foundation surface of retaining dam section under static and dynamic superposition. As can be seen from the table, according to the dynamic calculation results of finite element, the horizontal seismic load acting on the dam section is 3377.61 kN. Since the seismic action is random and reciprocating, when the horizontal seismic inertia force is downstream, combined with the static calculation results, the antisliding stability safety factor of the foundation surface of the dam section is 3.05, which meets the safety requirements. When the horizontal seismic inertia force is upstream, combined with the static calculation results, the antisliding stability safety factor of the foundation surface of the dam retaining section is 9.88, which meets the safety requirements.
4.6. Seismic Safety Review
Based on the above calculation results, according to the Guidelines for Reservoir Dam Safety Evaluation [18], (SL258-2017), the seismic safety of dam retaining section meets the standard requirements, and its seismic grade is grade A.
5. Conclusion
Based on the finite element numerical simulation technology, the seismic recheck calculation of the retaining section of a gravity dam is carried out. According to the calculation results, the following conclusions are drawn: (1)Considering the hydrodynamic pressure, the natural vibration frequency of the dam body decreases obviously. Compared with empty reservoir(2)Under the seismic action of normal storage water level, concrete stress of retaining dam section meets safety requirements(3)The safety coefficient of antisliding stability of foundation surface of retaining dam section meets the safety requirements. The seismic safety of dam retaining section meets the standard requirements, and its seismic grade is grade A
Data Availability
The data that support the findings of this study are available from the corresponding author, Bowen Guo, upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The present work is supported by the National Natural Science Foundation of China (51709115), the Key Research and Promotion Project of Henan Province (182102210066), the Natural Science Foundation of Henan Province (202300410545), and the Open Research Fund of Jiangxi Hydraulic Safety Engineering Technology Research Center (2020GGCZX03).