Shi et al 2023a - Revision history

Rimni at 12:23, 24 April 2023

2023-04-24T12:23:15Z

Rimni: /* 3.4 Resonance frequency */

2023-04-24T11:09:33Z

‎3.4 Resonance frequency

Rimni at 11:07, 24 April 2023

2023-04-24T11:07:37Z

Rimni: /* 2.2 Verification */

2023-04-24T10:51:45Z

‎2.2 Verification

Rimni at 10:39, 24 April 2023

2023-04-24T10:39:19Z

Show changes

Rimni: /* 4 Conclusions */

2023-04-24T10:01:16Z

‎4 Conclusions

Rimni: /* 3.5 Comparison with mode-superposition response spectrum method */

2023-04-24T09:58:41Z

‎3.5 Comparison with mode-superposition response spectrum method

Rimni at 09:51, 24 April 2023

2023-04-24T09:51:27Z

Rimni at 09:49, 24 April 2023

2023-04-24T09:49:23Z

Rimni at 09:37, 24 April 2023

2023-04-24T09:37:46Z

← Older revision		Revision as of 12:23, 24 April 2023
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−	where <math>a</math> is the maximum acceleration. The maximum amplitude is numerically equal to the maximum displacement, and the resonant frequency of the gate under the action of two seismic waves can be obtained by substituting ~~the~~ Eq. (8) as shown in [[#tab-3\|Table 3]]. Under the action of far-field wave, the resonance frequency of the gate is large because the maximum displacement value is small. The maximum frequency occurs at <math display="inline">M_1</math> point, reaching 65.24 Hz. The frequency of <math display="inline">M_3</math> point is also large, reaching 64.26 Hz. This is also the maximum dynamic stress response point of the gate, which indicates that the middle and lower part of the gate leaf is the weak point under the action of far-field seismic waves. However, the resonance frequency of EI wave is generally small, and the maximum frequency occurs at <math display="inline">M_3</math> point, reaching 49.13 Hz. The resonance frequency of all points is less than the resonance frequency under the action of far-field seismic wave.	+	where <math>a</math> is the maximum acceleration. The maximum amplitude is numerically equal to the maximum displacement, and the resonant frequency of the gate under the action of two seismic waves can be obtained by substituting Eq. (8) as shown in [[#tab-3\|Table 3]]. Under the action of far-field wave, the resonance frequency of the gate is large because the maximum displacement value is small. The maximum frequency occurs at <math display="inline">M_1</math> point, reaching 65.24 Hz. The frequency of <math display="inline">M_3</math> point is also large, reaching 64.26 Hz. This is also the maximum dynamic stress response point of the gate, which indicates that the middle and lower part of the gate leaf is the weak point under the action of far-field seismic waves. However, the resonance frequency of EI wave is generally small, and the maximum frequency occurs at <math display="inline">M_3</math> point, reaching 49.13 Hz. The resonance frequency of all points is less than the resonance frequency under the action of far-field seismic wave.

	<div class="center" style="font-size: 75%;">'''Table 3'''. Acceleration and resonance frequency at different positions of gate under different wave patterns</div>		<div class="center" style="font-size: 75%;">'''Table 3'''. Acceleration and resonance frequency at different positions of gate under different wave patterns</div>

@@ Line 434: / Line 434: @@
 |}
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 12'''. Acceleration of M2 monitoring point during earthquake. (a) EI wave. (b) Far-field wave
+| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 12'''. Acceleration of <math display="inline">M_2</math> monitoring point during earthquake. (a) EI wave. (b) Far-field wave
 |}

@@ Line 317: / Line 317: @@
 |}
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 9'''. Displacement curves of monitoring points gate structure under the action of EI wave. (a) M1 point. (b) M2 point. (c) M3 point. (d) M4 point. (e) M5 point. (f) M6 point
+| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 9'''. Displacement curves of monitoring points gate structure under the action of EI wave. (a) <math display="inline">M_1</math> point. (b) <math display="inline">M_2</math> point. (c) <math display="inline">M_3</math> point. (d) <math display="inline">M_4</math> point. (e) <math display="inline">M_5</math> point. (f) <math display="inline">M_6</math> point
 |}
@@ Line 338: / Line 338: @@
 |}
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 10'''. Displacement curves of monitoring points gate structure under the action of far field wave. (a) M1 point. (b) M2 point. (c) M3 point. (d) M4 point. (e) M5 point. (f) M6 point
+| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 10'''. Displacement curves of monitoring points gate structure under the action of far field wave. (a) <math display="inline">M_1</math> point. (b) <math display="inline">M_2</math> point. (c) <math display="inline">M_3</math> point. (d) <math display="inline">M_4</math> point. (e) <math display="inline">M_5</math> point. (f) <math display="inline">M_6</math> point
 |}

@@ Line 156: / Line 156: @@
-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 [[#img-2|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.
+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 [[#img-2|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.
 <div id='img-2'></div>
@@ Line 244: / Line 244: @@
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 4'''. Contour of displacement of the dam
 |}
 ==3. Seismic response of radial steel gate==

@@ Line 616: / Line 616: @@
 |}
-==4 Conclusions==
+==4. Conclusions==
 (1) The difference between the maximum displacement of Zipingpu dam calculated by the established fluid-structure coupling model and the measured results is not more than 0.9%, which shows that the fluid-structure coupling model in this paper can achieve high accuracy and can be applied to the dynamic response study of radial gate structure.
@@ Line 622: / Line 622: @@
 (2) The maximum dynamic displacement of the gate structure under EI wave occurs at the center of the middle and upper part of the gate leaf, which is 40 mm, which is far more than the allowable value of 20 mm in the current code, and is in danger of damage. The maximum dynamic displacement calculated by far-field seismic wave is also located at the center of the middle and upper part of the gate panel, which is 23 mm, which exceeds the allowable value of the specification, but is significantly less than the displacement under the action of EI wave.
-<span id='_Hlk132043224'></span>(3) Lower part of the gate leaf and the support arm are the areas with weak dynamic response under the rare earthquake, and effective structural measures should be taken for reinforcement to enhance local strength.
+(3) Lower part of the gate leaf and the support arm are the areas with weak dynamic response under the rare earthquake, and effective structural measures should be taken for reinforcement to enhance local strength.
 (4) The resonance frequency of the gate under the action of far-field wave is significantly larger, with the maximum frequency of 65.24 Hz, which is located at the top of the gate leaf. The resonance frequency under the action of EI wave is small, with the maximum value of 49.13 Hz, which is located in the middle and lower part of the gate leaf. Under the same basic acceleration value of rare earthquake, different seismic wave modes have great differences in the damage area and damage situation of the gate structure, indicating that the impact of different seismic wave modes on the structure should be considered during the gate design.
-<span id='_Hlk132041680'></span>(5) The coupling action has obvious influence on the structure. The calculated displacement and stress of the gate ignoring the coupling effect is far less than those considering, which may lead to insecurity. The model proposed in this paper can also be used for seismic response analysis under other liquid solid interactions. However, in some special cases, seismic effects may cause large swells in the liquid. Further research will be conducted on the interaction between large swells caused by earthquakes and existing structures.
+(5) The coupling action has obvious influence on the structure. The calculated displacement and stress of the gate ignoring the coupling effect is far less than those considering, which may lead to insecurity. The model proposed in this paper can also be used for seismic response analysis under other liquid solid interactions. However, in some special cases, seismic effects may cause large swells in the liquid. Further research will be conducted on the interaction between large swells caused by earthquakes and existing structures.
 ==References==

@@ Line 534: / Line 534: @@
 |}
-where <math>S</math> is horizontal seismic effect, <math>S_j</math> is the effect of horizontal seismic action of <math>j</math> mode, including internal force and deformation.
+where <math>S</math> is horizontal seismic effect, and <math>S_j</math> is the effect of horizontal seismic action of <math>j</math> mode, including internal force and deformation.
 [[#tab-4|Table 4]] shows the maximum values of dynamic displacement and stress of each monitoring point of the gate under EI wave and far-field seismic wave calculated by the mode decomposition response spectrum method. And the ratio of these values to the maximum values of corresponding monitoring points under the consideration of fluid-structure coupling method in this paper is also given. Since the coupling between the water and gate is not considered by the mode decomposition response spectrum method, both the maximum displacement and the maximum stress are less than those calculated by the fluid-structure coupling method. The maximum displacement and stress of the monitoring points under EI wave are only 0.64 and 0.83 of those under the consideration of fluid-structure coupling, and the maximum displacement and stress under far-field wave are 0.79 and 0.68 of those under the consideration of fluid-structure coupling. Especially for <math display="inline">M_2</math> monitoring point, the maximum stress under far-field wave is only 0.44 of that considering the coupling effect.
@@ Line 549: / Line 549: @@
 !  Value !! Ratio !! Value !! Ratio !! Value !! Ratio !! Value !! Ratio
 |-style="text-align:center"
-|  M1
+|  <math display="inline">M_1</math>
 |  Top of gate leaf
 |  6.6
@@ Line 560: / Line 560: @@
 |  0.65
 |-style="text-align:center"
-|  M2
+|  <math display="inline">M_2</math>
 |  Middle and upper part of gate leaf
 |  25.9
@@ Line 571: / Line 571: @@
 |  0.44
 |-style="text-align:center"
-|  M3
+|  <math display="inline">M_3</math>
 |  Middle and lower part of gate leaf
 |  13.1
@@ Line 582: / Line 582: @@
 |  0.83
 |-style="text-align:center"
-|  M4
+|  <math display="inline">M_4</math>
 |  Bottom of the gate leaf
 | 4.2
@@ Line 593: / Line 593: @@
 |  0.84
 |-style="text-align:center"
-| M5
+| <math display="inline">M_5</math>
 | Upper part of arm
 | 20.1
@@ Line 604: / Line 604: @@
 |  0.69
 |-style="text-align:center"
-|  M6
+|  <math display="inline">M_6</math>
 |  Lower part of arm
 |  15.4

@@ Line 22: / Line 22: @@
 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.
+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. [5] 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.