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[[#img-9|Figures 9]] and [[#img-10|10]] show the displacement curves of monitoring points <math display="inline">M_1</math>-<math display="inline">M_6</math> of gate structure under the action of EI wave and far-field wave, respectively. Under the action of EI wave, the maximum displacements of monitoring points <math display="inline">M_1</math> to <math display="inline">M_6</math> are 10.2 mm, 38.2 mm, 21.2 mm, 5.5 mm, 34.8 mm and 25.0 mm, respectively. The displacements of monitoring points <math display="inline">M_2</math>, <math display="inline">M_3</math>, <math display="inline">M_5</math> and <math display="inline">M_6</math> exceed the allowable value. The maximum dynamic displacement occurs in the middle and upper part of the panel (<math display="inline">M_2</math>), and the occurrence time is within the sub-step time range of 0.02s before and after 2s. Under the action of far-field wave, the maximum displacements of <math display="inline">M_1</math> to <math display="inline">M_6</math> monitoring points are 6.4 mm, 22.4 mm, 12.7 mm, 3.3 mm, 20.6 mm and 14.7 mm respectively, which are less than the displacement values of corresponding monitoring points under the action of EI wave, and only the displacements of <math display="inline">M_2</math> and <math display="inline">M_5</math> monitoring points exceed the allowable values. The maximum displacement value also occurs in the middle and upper part of the panel (<math display="inline">M_2</math>), and the occurrence time is within the sub-step time range of 0.02s before and after 1.3s. It can be seen that the dynamic displacement counter of the structure after the earthquake shown in [[#img-8|Figure 8]] can reflect the deformation of the gate to a certain extent, but it cannot fully show the true deformation of the structure during the dynamic response duration. The extreme deformation of the structure may occur at a certain time during the response duration. | [[#img-9|Figures 9]] and [[#img-10|10]] show the displacement curves of monitoring points <math display="inline">M_1</math>-<math display="inline">M_6</math> of gate structure under the action of EI wave and far-field wave, respectively. Under the action of EI wave, the maximum displacements of monitoring points <math display="inline">M_1</math> to <math display="inline">M_6</math> are 10.2 mm, 38.2 mm, 21.2 mm, 5.5 mm, 34.8 mm and 25.0 mm, respectively. The displacements of monitoring points <math display="inline">M_2</math>, <math display="inline">M_3</math>, <math display="inline">M_5</math> and <math display="inline">M_6</math> exceed the allowable value. The maximum dynamic displacement occurs in the middle and upper part of the panel (<math display="inline">M_2</math>), and the occurrence time is within the sub-step time range of 0.02s before and after 2s. Under the action of far-field wave, the maximum displacements of <math display="inline">M_1</math> to <math display="inline">M_6</math> monitoring points are 6.4 mm, 22.4 mm, 12.7 mm, 3.3 mm, 20.6 mm and 14.7 mm respectively, which are less than the displacement values of corresponding monitoring points under the action of EI wave, and only the displacements of <math display="inline">M_2</math> and <math display="inline">M_5</math> monitoring points exceed the allowable values. The maximum displacement value also occurs in the middle and upper part of the panel (<math display="inline">M_2</math>), and the occurrence time is within the sub-step time range of 0.02s before and after 1.3s. It can be seen that the dynamic displacement counter of the structure after the earthquake shown in [[#img-8|Figure 8]] can reflect the deformation of the gate to a certain extent, but it cannot fully show the true deformation of the structure during the dynamic response duration. The extreme deformation of the structure may occur at a certain time during the response duration. | ||
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− | | | + | | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 9'''. |
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The water resources in southwest China is abundant and the seismicity is strong, so it is necessary to study the dynamic response and safety of hydraulic structures under rare earthquake. Taking a typical radial steel gate as an example, a three-dimensional numerical model considering the interaction between water and gate during the earthquake is established. The accuracy and applicability of the model are verified by comparing with the measured results of the dynamic response of Zipingpu dam during the Wenchuan earthquake. Thereafter, the dynamic displacement and stress, and resonance frequency of the radial gate under the rare earthquake of two wave types are analyzed. The water-structure coupling effect has a great influence on the seismic dynamic response of the radial steel gate. The calculated result of the dynamic response of the gate considering the fluid-structure coupling effect is significantly larger than that of the specification, and the maximum ratio of the two is more than 2.27 times. Under the action of EI wave, the peak value of dynamic stress response is at the bottom of the panel, and the maximum value of resonance frequency (about49.13 Hz) is located in the middle and lower part of the panel. Under the action of far-field wave, the peak area of dynamic displacement response of the gate is basically the same as that under the action of EI wave, while the maximum value of some measuring points is only half of the maximum value under the action of EI wave. However, the resonance frequency is significantly greater than that of EI wave, the maximum value reaches 65.24 Hz, which appears at the top of the gate. The dynamic response of the gate structure caused by two different wave types of earthquakes is not completely consistent. The comprehensive consideration of different wave types is of significance for the structural design and safety evaluation of the radial steel gate in the earthquake-prone areas.
Keywords: Radial steel gate, rare earthquake, fluid-structure coupling, seismic dynamic response
Southwest China is located at the junction of the first and second steps of the terrain, with large river drop, so it is rich in hydropower resources. A large number of high dam hydropower stations, such as Ertan Hydropower Station and Jinping Hydropower Station, have been built here [1-4]. Radial steel gates are widely used in these projects due to their large orifice area, simple pier structure, good drainage conditions, convenient opening and closing, and few embedded parts [5]. However, as the regulating throat of hydraulic hub, its structural response characteristics under dynamic load need to be further studied [6]. Southwest China is an earthquake prone area, many gates of hydraulic structures were seriously damaged during the M_s8.0 Wenchuan earthquake in 2008 [7]. The damage and deformation of the gate structure not only affect the normal use function of the gate, but also lead to the loss of the function of regulating and storing water flow of the dam in serious cases, leading to dam break and other secondary disasters. Therefore, it is urgent to study the dynamic response of the radial steel gate under the action of rare earthquake.
Generally, the dynamic response analysis of radial steel gates can be carried out by means of filed monitoring, model test and numerical simulation. Field monitoring can directly record the deformation and stress of the gate structure, but it is difficult to predict the safety of the structure under earthquake. In the aspect of model test, its manufacturing process is complex and costly, and it has obvious size effect. With the breakthrough of computer in the bottleneck of numerical calculation rate, numerical simulation has become the main method to analyze the dynamic stress response of radial steel gate. Zhang et al. [6] used the finite element ANSYS to improve the structure of radial steel gate. The stress and displacement of each component of the improved radial steel gate are significantly lower than those of the traditional gate. Liu et al. [8] carried out a three-dimensional nonlinear analysis of a radial steel gate of a reservoir and investigated the influence of stiffeners on the panel. Wu and Xie [9] used the Galerkin method to derive the system finite element equations of fluid and structure, and analyzed the dynamic characteristics of a radial gate.
Currently, there are mainly three methods can be used to simulate the dynamic response of the gate under earthquake load: (1) the mode-superposition response spectrum method [10], in which only the vibration of the gate is considered, (2) the additional mass method considering the superposition of fluid and vibration [11], and (3) the fluid-solid coupling method considering the coupling effect of fluid and structure [12]. In the earthquake, the vibration movement of the gate structure acts on the water body, making its flow field change. The water body after the change of the flow field also affects the dynamic characteristics of the gate structure, such as damping force, elastic force, inertial force, etc., thus affecting the dynamic response characteristics of the gate structure. Therefore, many studies have pointed out that the coupling process of the water body on the upstream of the gate in the gate movement under earthquake cannot be ignored. Faridmehr et al. [12] established a three-dimensional radial gate model in ABAQUS/Explicit, and found that the results of structural dynamic responses considering fluid-solid coupling are quite different from those of static analysis. Buldgen et al. [11] investigated the classical Westergaard seismic hydrodynamic pressure solution formula through numerical simulation and model test. The results show that the calculated value is conservative than considering the fluid-solid coupling numerical solution because the formula assumes that the structure is completely rigid.
The studies mentioned above have demonstrated that the fluid-structure coupling method is more suitable for simulating the dynamic response of radial steel gates under earthquake. However, most of the previous studies were carried out under the conditions of frequent earthquakes, and the input seismic waves are mostly single EI waves. Hence, study on the dynamic response of the radial steel gates under different seismic waves of rare earthquakes is insufficient. The answer to this question is of special significance for the safety evaluation of radial steel gates under earthquake. Based on the three-dimensional coupled numerical model of gate and water, this paper revises the seismic acceleration curve according to the characteristics of rare earthquakes, and the dynamic response and spatial-temporal distribution characteristics of arc steel gate are studied by using the external excitation of EI wave and far-field seismic wave. The significance of considering water-gate interaction under earthquake has been revealed.
In the finite element analysis of fluid-solid coupling under linear small deformation, the fluid is assumed to be a uniform, inviscid and vortex-free ideal fluid [13-14]. According to Euler equation, the dynamic balance equations of fluid can be deduced as follows:
|
(1) |
or
|
(2) |
where, is the density of the fluid, is the pressure of fluid particle, , and are the displacement components of the fluid particle in the , and directions, and , and are the velocity components of the fluid particle in the , and directions, is time. respectively. And the continuity equation of compressible fluid is
|
(3) |
where is the compressive modulus of fluid.
Using Eqs. (1)-(3), the governing equations of fluid motion can be obtained the following equation:
|
(4) |
where is the Laplace operator.
Obviously, it is difficult to solve Eq. (4) under three-dimensional conditions using conventional finite element methods such as Galerkin's method [15], especially for the fluid-solid coupling interface. Therefore, it is necessary to divide the fluid domain into fluid elements, and then aggregate the discrete motion equations of the whole fluid domain. The Gauss numerical integration of the solution results of each fluid element is weighted and aggregated to obtain the coefficient matrix of the governing equation, and then the motion equation of the whole fluid domain can be expressed as:
|
(5) |
where , , and are the coefficients, is the displacement (unit: m), is the excitation vector (unit: N). It is also necessary to discretize the solid structure by finite element method, and the motion equation:
|
(6) |
where, , and are the mass matrix, damping matrix and stiffness matrix of the structure, respectively; is the fluid force at the interface (unit: N), and by weighted aggregation through Gauss numerical integration; and is the external excitation vector other than the fluid force at the interface (unit: N). Then Eqs. (5) and (6) can be combined as:
|
(7) |
Based on Eq. (7), the structural model and flow field model are first generated in the geometric model software. The external flow field is generated in Workbench, and the finite element model is generated by importing the calculation platform. Then the contact surface between the structure and the fluid is set as the fluid-structure coupling interface, and the external excitation load is input to drive the coupling iteration of the model. In FLUENT, the fluid domain generates pressure under the action of external excitation and transmits it to the structure. The structure then reacts the pressure on the fluid domain with boundary deformation. Such iterative process until the convergence condition is reached. This dynamic mesh technology updates and modifies the fluid domain mesh to ensure that the model mesh does not distort.
The accuracy of the above method is analyzed by analyzing the interaction characteristics of Zipingpu dam and reservoir water under the Wenchuan earthquake. Zipingpu Water Control Project [16] is located in the upper reaches of the Minjiang River, more than 60 kilometers northwest of Chengdu, Sichuan Province, and 9 kilometers away from Dujiangyan City. On May 12, 2008, a strong earthquake of magnitude 8.0 occurred in Wenchuan, about 17 km west of Zipingpu Dam, with the maximum intensity of the epicenter as high as XI. This earthquake is characterized by large magnitude, shallow focus (about 14 km), long fault (nearly 300 km) and long duration (about 90 s of main shock) [17]. The strong earthquake caused obvious damage to Zipingpu Dam as shown in Figura 1.
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Figure 1. Damage of the dam caused by the Wenchuan earthquake [17] |
The dam is a concrete face rockfill dam. The length and elevation of the dam crest are 663.77 m and 884 m, respectively. The normal and dead water level are 877 m and 817 m, respectively. The slope ratio of the upstream dam is 1:1.4, the lowest foundation elevation is 728 m, and the dam is constructed in three phases, of which the top elevation in the first and second phases are 796 m and 845 m respectively, and the reservoir water level during the earthquake is 828.65 m. Based on this, the established three-dimensional model is shown in Figure 2. For the boundary conditions, the upper surface of the fluid is free and incompressible. The contact surface between the dam body and the water body is set as the fluid-solid coupling boundary, and the rest of the fluid surfaces are fixed.
![]() |
Figure 2. Model of Zipingpu Dam and water |
Considering that hydraulic structures are mainly affected by horizontal earthquakes, seismic load input is only considered along the river. The central station of the National Strong Motion Network of China has obtained a large number of main earthquake records with complete seismic phases. Table 1 shows the information collected by the regional station at Zipingpu dam site during the Wenchuan earthquake. Although 7 stations have been set up in this area, the difference of peak acceleration of each station is relatively large, so the measured ground motion cannot be directly used for the simulation calculation of ground motion input. Chen et al. [17] estimated that the peak ground motion of the dam bedrock in the Wenchuan earthquake was more than 0.5 g (g is the acceleration of gravity) according to the measured peak acceleration at the dam crest of Zipingpu Dam. According to the method proposed by Yu et al. [18], the attenuation relationship between the bedrock acceleration and the distance from the fault zone is calculated, and the horizontal peak value of the bedrock of Zipingpu Dam during the Wenchuan earthquake is 0.52 g. Based on these research results and station monitoring data, the peak acceleration used in the final simulation is 0.56 g. In view of this, the seismic wave in the simulation calculation adopts the EI wave as shown in Figure 3, which is generated manually according to the Chinese relevant provisions of the Standard for seismic design of hydraulic structures [19]. The sub-step of seismic wave action time is 0.02 s, and the total action time is 10 s, and the peak acceleration is 0.56 g. And according to the test results [16], the density and friction angle of the dam material are 21.6 g/cm2 and 30°, respectively.
Station | Latitude (°) | Longitude (°) | Site type | Peak acceleration along the river (g) |
---|---|---|---|---|
Wolong, Wenchuan | 31.0N | 103.2E | Soil | 0.96 |
Bajiao, Shifang | 33.3N | 104.0E | Soil | 0.56 |
Qingping, Mianzhu | 31.5N | 104.1E | Soil | 0.82 |
Nanxin, Maoxian | 31.6N | 103.7E | Soil | 0.42 |
Diban, Maoxian | 31.7N | 103.9E | Rock | 0.31 |
Zoushixian, Pixian | 30.9N | 103.8E | Rock | 0.12 |
Zhonghe, Chengdu | 30.6N | 104.1E | Rock | 0.0 |
![]() |
Figure 3. EI seismic wave acceleration curve |
The calculated displacement of the dam considering the coupling effect of water and dam is shown in Figure 4. The dam displacement gradually increases with the elevation, which is the same as the actual damage of Zipingpu dam. The maximum displacement occurring at the dam crest is 273.4 mm. According to the measured data of Chen et al. [17], the measured value of the maximum displacement of the dam is 270.8 mm. The calculated value is close to the measured value, and the error is only 0.9%, indicating that the calculation method in this paper has high accuracy. Meanwhile, calculations were also conducted without considering the coupling effect of water and dam, which showed that the maximum displacement of the dam (198.4 mm) is far less than the actual displacement.
| ||
Figure 4. Contour of displacement of the dam |
Radial steel gate shown in Figure 5 is selected for the seismic dynamic response analysis considering the fluid-structure coupling effect. The radius of the gate is 12 m, the vertical distance between the support hinge and the gate bottom is 10.9 m, and the panel width is 12 m. Radial gate is a spatial thin-walled structure system mainly composed of gate leaf and arm. The gate leaf structure adopts shell element shell93 which can reflect the spatial stress, and the material is Q235B. The steel material of the support arm is Q345B, and the section is rectangular. In total, there are 14,922 elements of the radial steel gate model. Figure 6 shows the gate - water coupling model. According to the design conditions, the design water level is 8.5 m, and the calculated water length is selected to be more than 3 times the height of the water, which is 30 m. The water-gate interface is set as the fluid-solid coupling surface, the top of the water is free, and the rest of the surfaces are fixed. Six key parts of the gate are selected for monitoring, including at the top of the gate leaf, at the middle and upper part of the gate leaf, at the middle and lower part of the gate leaf, at the bottom of the gate leaf, at the, and at the lower part of the arm.
![]() |
Figure 5. Model of radial steel gate |
![]() |
Figure 6. Water-gate coupling model |
According to the relevant regulations of the basic acceleration value of rare earthquake of the Standard for seismic design of hydraulic structures [19], the seismic acceleration value in this simulation is 0.30 g, and the corresponding seismic fortification intensity is 8 degrees. The sub-step length of seismic wave action time is 0.02 s, the action time is 10 s in total, and the seismic action load direction is downstream. In addition, EI wave (Figure 3) and far-field seismic wave T1-Ⅱ-1 (Figure 7) are respectively considered for analysis.
![]() |
Figure 7. Far field seismic wave T1-Ⅱ-1 acceleration curve |
According to the Design specification for steel gates of water and hydropower projects [20], the maximum allowable displacement of the gate is 20 mm. Figure 8 shows the displacement of the gate considering the fluid-structure coupling effect after earthquake. The displacement distribution of EI wave and far-field seismic wave is quite different. Under the action of EI wave, the maximum displacement of the radial gate is 33.7 mm, which exceeds the allowable displacement of the specification. The displacement mainly occurs in the middle of the gate leaf and the middle of the support arm. The maximum displacement is concentrated in the middle and upper part of the gate leaf, and the constrained support hinge and the upper and lower end of the panel have the minimum displacement. Under the action of far-field seismic wave, the maximum displacement of the gate is 18.3 mm, which meets the rigidity requirements of the specification. The displacement mainly occurs in the upper part of the gate leaf, the maximum displacement is concentrated in the upper part of the gate leaf, and the displacement at the connection between the arm and the panel and the arm is small.
| ||
Figure 8. Dynamic displacement of gate after earthquake. (a) EI wave. (b) Far-field wave |
Figures 9 and 10 show the displacement curves of monitoring points - of gate structure under the action of EI wave and far-field wave, respectively. Under the action of EI wave, the maximum displacements of monitoring points to are 10.2 mm, 38.2 mm, 21.2 mm, 5.5 mm, 34.8 mm and 25.0 mm, respectively. The displacements of monitoring points , , and exceed the allowable value. The maximum dynamic displacement occurs in the middle and upper part of the panel (), and the occurrence time is within the sub-step time range of 0.02s before and after 2s. Under the action of far-field wave, the maximum displacements of to monitoring points are 6.4 mm, 22.4 mm, 12.7 mm, 3.3 mm, 20.6 mm and 14.7 mm respectively, which are less than the displacement values of corresponding monitoring points under the action of EI wave, and only the displacements of and monitoring points exceed the allowable values. The maximum displacement value also occurs in the middle and upper part of the panel (), and the occurrence time is within the sub-step time range of 0.02s before and after 1.3s. It can be seen that the dynamic displacement counter of the structure after the earthquake shown in Figure 8 can reflect the deformation of the gate to a certain extent, but it cannot fully show the true deformation of the structure during the dynamic response duration. The extreme deformation of the structure may occur at a certain time during the response duration.
Published on 24/04/23
Accepted on 20/04/23
Submitted on 10/04/23
Volume 39, Issue 2, 2023
DOI: 10.23967/j.rimni.2023.04.004
Licence: CC BY-NC-SA license
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