Model similarity design method suitable for wave-underwater vibration table test

Abstract

The invention discloses a model similarity design method suitable for a wave-underwater vibration table test. The method comprises the steps of determining a geometric similarity ratio of a reduced scale model; determining a structural density similarity ratio, a structural elastic modulus similarity ratio and a water density similarity ratio of the reduced scale model; determining a vertical geometric similarity ratio; determining an x-direction geometric similarity ratio under the action of a horizontal earthquake; determining a geometric similarity ratio in a horizontal y direction; the wavelength similarity ratio in the x direction is determined; and determining a wave height similarity ratio and the like. The method has the advantages that the reduced scale model in the design test is adopted, coordination similarity between water and the structure is achieved by changing the cross section size similarity ratio of the structure without changing the structure density similarity ratio, the method is simple and practical, model manufacturing is easy and convenient, and the method is suitable for large-scale production. And a feasible method can be provided for similar design of a wave-underwater vibration table test model under the earthquake-wave action.

Description

Technical Field

[0001] This invention belongs to the field of civil engineering technology, and in particular relates to a model similarity design method suitable for wave-underwater shaking table tests. Background Technology

[0002] With the advancement of China's maritime power construction, large-scale projects such as cross-sea bridges and offshore wind power are increasingly being built. These structures are not only massive and complex, but also face threats from earthquakes and extreme wave damage, thus requiring specialized analysis and research in their design. Among these, wave-underwater shaking table model tests are a key experimental method for analyzing the dynamic response of structures under earthquake-wave conditions. Reasonable model similarity design is a core step in conducting scaled-down model tests, ensuring that the scaled-down model can reproduce the dynamic response of the prototype structure.

[0003] In wave-underwater shaking table model tests, two media materials are involved: water and the structure. Ensuring a reasonable design for the water-structure scaled-down model requires maintaining a consistent density similarity ratio between the two materials. Since the density of water cannot be changed during the test, the density similarity ratio of water is 1.0, therefore, the density similarity ratio of the designed structure should also be 1.0. However, in traditional shaking table model similarity design, to ensure the dynamic response of the scaled-down model is similar to the prototype structure, additional mass is added to change the structural density. This design method leads to an inconsistency in the density similarity ratio between the structure and water, distorting the water-structure interaction test. Therefore, current model similarity design for wave-underwater shaking table tests presents significant challenges. Summary of the Invention

[0004] To address the aforementioned problems, the present invention aims to provide a model similarity design method suitable for wave-underwater shaking table tests. The scaled-down model produced by this method accurately reproduces the dynamic response of the prototype structure and is simple and convenient to manufacture.

[0005] To achieve the above objectives, the model similarity design method suitable for wave-underwater shaking table tests provided by the present invention includes the following steps performed in sequence:

[0006] Step 1: Based on the dimensions of the water tank and the parameters of the underwater vibration table test equipment, and in conjunction with the dimensions of the prototype structure, determine the geometric similarity ratio λ of the scaled-down model. l ;

[0007] Step 2: Select the same materials as the prototype structure and water as the materials for the scaled-down model, and determine the structural density similarity ratio λ of the scaled-down model. ρ , structural elastic modulus similarity ratio λ E Similarity ratio to water density λ ρw ;

[0008] Step 3: Utilize the geometric similarity ratio λ of the scaled model obtained in Step 1 l Determine the vertical geometric similarity ratio λ lz ;

[0009] Step 4: Using the equality of the structural inertial force similarity ratio and the structural elastic force similarity ratio as a constraint, based on the structural elastic modulus similarity ratio λ obtained in Step 2... E Similarity ratio to structural density λ ρ Step 3 obtains the vertical geometric similarity ratio λ lz Determine the x-direction geometric similarity ratio λ under horizontal seismic action. lx ;

[0010] Step 5: Using the equality of the structural inertial force similarity ratio and the seismic hydraulic similarity ratio as a constraint, based on the x-direction geometric similarity ratio λ obtained in Step 4 under horizontal seismic action... lx Determine the geometric similarity ratio λ in the horizontal y-direction. ly ;

[0011] Step Six: Based on the vertical geometric similarity ratio λ obtained in Step Three lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Using the water depth and wavelength in the prototype, the wavelength similarity ratio λ in the x-direction is determined using the dispersion formula. L ;

[0012] Step 7: Using the equality of the structural inertial force similarity ratio and the wave force similarity ratio as a constraint, based on the structural density similarity ratio λ obtained in Step 2... ρ Similarity ratio to water density λ ρw Step 3 obtains the vertical geometric similarity ratio λ lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Step 5: Obtain the horizontal y-axis geometric similarity ratio λ ly And the x-axis wavelength similarity ratio λ obtained in step six L Determine the wave height similarity ratio λ H .

[0013] In step one, the geometric similarity ratio λ l = Scaled-down model size / Prototype structure size, determined by the following method:

[0014] According to the geometric similarity ratio λ l The weight of the scaled-down model must not exceed the maximum design weight that the underwater vibration table test equipment can withstand, and must be based on the geometric similarity ratio λ. l The water depth after scaling down must not exceed the designed water depth of the pool.

[0015] In step two, the structural density similarity ratio λ ρ=Structural density of scaled model / structural density of prototype, structural elastic modulus similarity ratio λ E = Elastic modulus of the scaled-down model structure / Elastic modulus of the prototype structure, water density similarity ratio λ ρw =Water density in the scaled-down test / Water density in the prototype; Since the scaled-down model test uses the same materials as the prototype structure and water, the structural density similarity ratio λ ρ =1, structural elastic modulus similarity ratio λ E =1, water density similarity ratio λ ρw =1.

[0016] In step three, the vertical geometric similarity ratio λ lz The expression is:

[0017] .

[0018] In step four, the constraint condition is that the structural inertial force similarity ratio and the structural elastic force similarity ratio are equal, based on the structural elastic modulus similarity ratio λ obtained in step two. E Similarity ratio to structural density λ ρ Step 3 obtains the vertical geometric similarity ratio λ lz Determine the x-direction geometric similarity ratio λ under horizontal seismic action. lx The method is:

[0019] Assuming the structural inertial force similarity ratio is equal to the structural elastic force similarity ratio, then the x-direction geometric similarity ratio λ under horizontal seismic loading is... lx The expression is:

[0020] ;

[0021] In the formula, λ g The similarity ratio for gravitational acceleration is 1.0.

[0022] In step five, the constraint condition is that the structural inertial force similarity ratio and the seismic hydraulic similarity ratio are equal, based on the x-direction geometric similarity ratio λ obtained in step four under horizontal seismic action. lx Determine the horizontal y-axis geometric similarity ratio λ ly The method is:

[0023] Assuming the structural inertial force similarity ratio is equal to the seismic hydraulic similarity ratio, then the horizontal y-direction geometric similarity ratio λ ly The expression is:

[0024] .

[0025] In step six, the vertical geometric similarity ratio λ obtained in step three is... lz Step 4: Obtain the x-axis geometric similarity ratio λlx Using the water depth and wavelength in the prototype, the wavelength similarity ratio λ in the x-direction is determined using the dispersion formula. L The method is:

[0026] The wavelength similarity λ in the x-direction L We can obtain the following by solving the following formula:

[0027] ;

[0028] In the formula, h is the water depth in the prototype, and L is the wavelength in the prototype.

[0029] In step seven, the constraint condition of the structural inertial force similarity ratio being equal to the wave force similarity ratio is used, based on the structural density similarity ratio λ obtained in step two. ρ Similarity ratio to water density λ ρw Step 3 obtains the vertical geometric similarity ratio λ lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Step 5: Obtain the horizontal y-axis geometric similarity ratio λ ly And the x-axis wavelength similarity ratio λ obtained in step six T Determine the wave height similarity ratio λ H The method is:

[0030] Assuming the structural inertial force similarity ratio is equal to the wave force similarity ratio, then the wave height similarity ratio λ H The expression is:

[0031] ;

[0032] In the formula, R is the cross-sectional radius of the prototype structure, which is taken as half the length of the water-facing surface. , , and These are the first-order Bessel functions of the first kind and the second-order Bessel functions of the second kind, respectively.

[0033] The advantages and positive effects of this invention are as follows: In wave-underwater shaking table tests, in order to meet the coordination requirements of the water-structure scaled-down model similarity design, it is necessary to ensure that the density similarity ratio of the water is equal to the density similarity ratio of the structure. However, traditional shaking table model similarity design methods cannot meet this condition. By using the scaled-down model designed in this invention, the coordination similarity between water and structure is achieved by changing the cross-sectional size similarity ratio of the structure without changing the density similarity ratio of the structure. This method is simple and practical, and the model is easy to make. It can provide a feasible method for the similarity design of wave-underwater shaking table test models under earthquake-wave action. Attached Figure Description

[0034] Figure 1A comparison diagram of the displacement of the top of the bridge pier under seismic loading between the scaled model and the prototype structure designed using the method of this invention.

[0035] Figure 2 A comparison diagram of the acceleration at the top of the bridge pier under seismic loading between the scaled-down model and the prototype structure designed using the method of this invention.

[0036] Figure 3 A comparison diagram of the strain at the bottom of the bridge pier under seismic loading between a scaled-down model and a prototype structure designed using the method of this invention.

[0037] Figure 4 A comparison diagram of the dynamic water pressure at the midpoint of the bridge pier under seismic loading between the scaled-down model and the prototype structure designed using the method of this invention.

[0038] Figure 5 A comparison diagram of the dynamic water pressure at the midpoint of the bridge pier under wave action between the scaled-down model and the prototype structure designed using the method of this invention. Detailed Implementation

[0039] To further understand the invention's content, features, and effects, the following embodiments are provided, along with detailed descriptions in conjunction with the accompanying drawings and tables:

[0040] The model similarity design method suitable for wave-underwater shaking table tests provided by this invention includes the following steps performed in sequence:

[0041] Step 1: Based on the dimensions of the water tank and the parameters of the underwater vibration table test equipment, and in conjunction with the dimensions of the prototype structure, determine the geometric similarity ratio λ of the scaled-down model. l ;

[0042] The geometric similarity ratio λ l = Scaled-down model size / Prototype structure size, determined by the following method:

[0043] According to the geometric similarity ratio λ l The weight of the scaled-down model must not exceed the maximum design weight that the underwater vibration table test equipment can withstand, and must be based on the geometric similarity ratio λ. l The water depth after scaling down must not exceed the designed water depth of the pool.

[0044] Step 2: Select the same materials as the prototype structure and water as the materials for the scaled-down model, and determine the structural density similarity ratio λ of the scaled-down model. ρ , structural elastic modulus similarity ratio λ E Similarity ratio to water density λ ρw ;

[0045] The structural density similarity ratio λ ρ =Structural density of scaled model / structural density of prototype, structural elastic modulus similarity ratio λ E= Elastic modulus of the scaled-down model structure / Elastic modulus of the prototype structure, water density similarity ratio λ ρw =Water density in the scaled-down test / Water density in the prototype; Since the scaled-down model test uses the same materials as the prototype structure and water, the structural density similarity ratio λ ρ =1, structural elastic modulus similarity ratio λ E =1, water density similarity ratio λ ρw =1.

[0046] Step 3: Utilize the geometric similarity ratio λ of the scaled model obtained in Step 1 l Determine the vertical geometric similarity ratio λ lz ;

[0047] The vertical geometric similarity ratio λ lz The expression is:

[0048] ;

[0049] Step 4: Using the equality of the structural inertial force similarity ratio and the structural elastic force similarity ratio as a constraint, based on the structural elastic modulus similarity ratio λ obtained in Step 2... E Similarity ratio to structural density λ ρ Step 3 obtains the vertical geometric similarity ratio λ lz Determine the x-direction geometric similarity ratio λ under horizontal seismic action. lx ;

[0050] Assuming the structural inertial force similarity ratio is equal to the structural elastic force similarity ratio, then the x-direction geometric similarity ratio λ under horizontal seismic loading is... lx The expression is:

[0051] ;

[0052] In the formula, λ g The similarity ratio for gravitational acceleration is 1.0.

[0053] Step 5: Using the equality of the structural inertial force similarity ratio and the seismic hydraulic similarity ratio as a constraint, based on the x-direction geometric similarity ratio λ obtained in Step 4 under horizontal seismic action... lx Determine the geometric similarity ratio λ in the horizontal y-direction. ly ;

[0054] Assuming the structural inertial force similarity ratio is equal to the seismic hydraulic similarity ratio, then the horizontal y-direction geometric similarity ratio λ ly The expression is:

[0055] ;

[0056] Step Six: Based on the vertical geometric similarity ratio λ obtained in Step Three lzStep 4: Obtain the x-axis geometric similarity ratio λ lx Using the water depth and wavelength in the prototype, the wavelength similarity ratio λ in the x-direction is determined using the dispersion formula. L ;

[0057] The wavelength similarity λ in the x-direction L We can obtain the following by solving the following formula:

[0058] ;

[0059] In the formula, h is the water depth in the prototype, and L is the wavelength in the prototype.

[0060] Step 7: Using the equality of the structural inertial force similarity ratio and the wave force similarity ratio as a constraint, based on the structural density similarity ratio λ obtained in Step 2... ρ Similarity ratio to water density λ ρw Step 3 obtains the vertical geometric similarity ratio λ lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Step 5: Obtain the horizontal y-axis geometric similarity ratio λ ly And the x-axis wavelength similarity ratio λ obtained in step six L Determine the wave height similarity ratio λ H .

[0061] Assuming the structural inertial force similarity ratio is equal to the wave force similarity ratio, then the wave height similarity ratio λ H The expression is:

[0062] ;

[0063] In the formula, R is the cross-sectional radius of the prototype structure, which is taken as half the length of the water-facing surface. , , and These are the first-order Bessel functions of the first kind and the second-order Bessel functions of the second kind, respectively.

[0064] This invention achieves coordinated similarity between water and structure by changing the cross-sectional size similarity ratio, thereby enabling accurate reproduction of the dynamic response of the prototype structure under earthquake-wave action.

[0065] The advantages of this invention will be illustrated below through an application example.

[0066] In this embodiment, the maximum water depth of the selected pool is 2.0m, and the maximum load-bearing capacity of the underwater shaking table test equipment is 26t. To verify the accuracy of the dynamic response of the scaled-down model in restoring the prototype structure, a rectangular prototype pier and a geometric scaling ratio λ were also designed. lThe prototype pier is a scaled-down version of the original pier, and both the original and scaled-down piers meet the maximum load-bearing requirements of the underwater shaking table test equipment. The original pier has a rectangular cross-section, 0.4m long, 0.16m wide, and 2.4m high, with a top mass of 64kg, made of weighted rubber material with a density of 2382kg / m³. 3 The elastic modulus is 2.4 GPa, the water depth in the prototype is 2.0 m, the wavelength is 8.725 m, and the wave height is 0.38 m.

[0067] In this embodiment, the scaled-down pier material is the same as the prototype pier material; therefore, the structural density similarity ratio λ is used in the experimental design. ρ =1.0, structural elastic modulus similarity ratio λ E =1.0, water density similarity ratio λ ρw =1.0. Therefore, the geometric similarity ratio λ in the x-direction under horizontal seismic loading can be obtained. lx =0.397, horizontal y-axis geometric similarity ratio λ ly =0.397, vertical geometric similarity ratio λ lz =0.5, wavelength similarity ratio λ L The wave height similarity ratio λ is 0.417. H It is 0.476.

[0068] The method of this invention was used to design scaled-down models of the aforementioned bridge piers and water. Dynamic response tests of the prototype and scaled-down piers under El-Centro earthquake and wave action were conducted using a wave-underwater shaking table. Test data on the top displacement, acceleration, bottom strain, and hydrodynamic pressure at the mid-section of the prototype and scaled-down piers were collected. Then, the response results of the scaled-down piers were used to back-calculate the response results of the prototype piers according to the corresponding similarity ratios, and these results were compared with the corresponding response results of the prototype piers.

[0069] Please see Figure 1 Under the El-Centro earthquake, the maximum displacement of the top of the prototype pier was 9.83 mm, while the maximum displacement of the top of the scaled-down pier designed using the method of this invention was 9.48 mm, with a relative error of 3.5%.

[0070] Please see Figure 2 Under the El-Centro earthquake, the maximum acceleration at the top of the prototype bridge pier was 5.09 m / s². 2 The maximum acceleration at the top of the scaled-down bridge pier designed using the method of this invention is 5.38 m / s². 2 The relative error is 5.7%.

[0071] Please see Figure 3Under the El-Centro earthquake, the maximum strain at the bottom of the prototype pier was 601 με, while the maximum strain at the bottom of the scaled-down pier designed using the method of this invention was 572 με, with a relative error of 4.7%.

[0072] Please see Figure 4 Under the El-Centro earthquake, the maximum dynamic water pressure in the middle of the prototype pier was 734 Pa, while the maximum dynamic water pressure in the middle of the scaled-down pier designed using the method of this invention was 794 Pa, with a relative error of 10.8%.

[0073] Please see Figure 5 Under wave action, the maximum dynamic water pressure in the middle of the prototype bridge pier is 1424.3 Pa, while the maximum dynamic water pressure in the middle of the scaled-down bridge pier designed using the method of this invention is 1423.6 Pa, with a relative error of 0.1%.

[0074] In summary, the relative errors of the responses of the scaled models designed using the method of this invention under El-Centro earthquake and wave action are between 3.5% and 10.8%, indicating that the scaled models of the wave-underwater shaking table test designed in this invention accurately reproduce the dynamic response of the prototype structure.

[0075] Although preferred embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other modifications under the guidance of the present invention without departing from the spirit and scope of the claims, and all of these modifications are within the scope of protection of the present invention.

Claims

1. A model similarity design method suitable for wave-underwater shaking table tests, characterized in that: The model similarity design method suitable for wave-underwater shaking table tests includes the following steps performed in sequence: Step 1: Based on the dimensions of the water tank and the parameters of the underwater vibration table test equipment, and in conjunction with the dimensions of the prototype structure, determine the geometric similarity ratio λ of the scaled-down model. l ; Step 2: Select the same materials as the prototype structure and water as the materials for the scaled-down model, and determine the structural density similarity ratio λ of the scaled-down model. ρ , structural elastic modulus similarity ratio λ E Similarity ratio to water density λ ρw ; Step 3: Utilize the geometric similarity ratio λ of the scaled model obtained in Step 1 l Determine the vertical geometric similarity ratio λ lz ; Step 4: Using the equality of the structural inertial force similarity ratio and the structural elastic force similarity ratio as a constraint, based on the structural elastic modulus similarity ratio λ obtained in Step 2... E Similarity ratio to structural density λ ρ Step 3 obtains the vertical geometric similarity ratio λ lz Determine the x-direction geometric similarity ratio λ under horizontal seismic action. lx ; Step 5: Using the equality of the structural inertial force similarity ratio and the seismic hydraulic similarity ratio as a constraint, based on the x-direction geometric similarity ratio λ obtained in Step 4 under horizontal seismic action... lx Determine the geometric similarity ratio λ in the horizontal y-direction. ly ; Step Six: Based on the vertical geometric similarity ratio λ obtained in Step Three lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Using the water depth and wavelength in the prototype, the wavelength similarity ratio λ in the x-direction is determined using the dispersion formula. L ; Step 7: Using the equality of the structural inertial force similarity ratio and the wave force similarity ratio as a constraint, based on the structural density similarity ratio λ obtained in Step 2... ρ Similarity ratio to water density λ ρw Step 3 obtains the vertical geometric similarity ratio λ lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Step 5: Obtain the horizontal y-axis geometric similarity ratio λ ly And the x-axis wavelength similarity ratio λ obtained in step six L Determine the wave height similarity ratio λ H .

2. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step one, the geometric similarity ratio λ l = Scaled-down model size / Prototype structure size, determined by the following method: According to the geometric similarity ratio λ l The weight of the scaled-down model must not exceed the maximum design weight that the underwater vibration table test equipment can withstand, and must be based on the geometric similarity ratio λ. l The water depth after scaling down must not exceed the designed water depth of the pool.

3. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step two, the structural density similarity ratio λ ρ =Structural density of scaled model / structural density of prototype, structural elastic modulus similarity ratio λ E = Elastic modulus of the scaled-down model structure / Elastic modulus of the prototype structure, water density similarity ratio λ ρw =Water density in the scaled-down test / Water density in the prototype; Since the scaled-down model test uses the same materials as the prototype structure and water, the structural density similarity ratio λ ρ =1, structural elastic modulus similarity ratio λ E =1, water density similarity ratio λ ρw =1.

4. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step three, the vertical geometric similarity ratio λ lz The expression is: 。 5. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step four, the constraint condition is that the structural inertial force similarity ratio and the structural elastic force similarity ratio are equal, based on the structural elastic modulus similarity ratio λ obtained in step two. E Similarity ratio to structural density λ ρ Step 3 obtains the vertical geometric similarity ratio λ lz Determine the x-direction geometric similarity ratio λ under horizontal seismic action. lx The method is: Assuming the structural inertial force similarity ratio is equal to the structural elastic force similarity ratio, then the x-direction geometric similarity ratio λ under horizontal seismic loading is... lx The expression is: ； In the formula, λ g The similarity ratio for gravitational acceleration is 1.

0.

6. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step five, the constraint condition is that the structural inertial force similarity ratio and the seismic hydraulic similarity ratio are equal, based on the x-direction geometric similarity ratio λ obtained in step four under horizontal seismic action. lx Determine the horizontal y-axis geometric similarity ratio λ ly The method is: Assuming the structural inertial force similarity ratio is equal to the seismic hydraulic similarity ratio, then the horizontal y-direction geometric similarity ratio λ ly The expression is: 。 7. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step six, the vertical geometric similarity ratio λ obtained in step three is... lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Using the water depth and wavelength in the prototype, the wavelength similarity ratio λ in the x-direction is determined using the dispersion formula. L The method is: The wavelength similarity λ in the x-direction L We can obtain the following by solving the following formula: ； In the formula, h is the water depth in the prototype, and L is the wavelength in the prototype.

8. The model similarity design method suitable for wave-underwater shaking table tests according to claim 1, characterized in that: In step seven, the constraint condition of the structural inertial force similarity ratio being equal to the wave force similarity ratio is used, based on the structural density similarity ratio λ obtained in step two. ρ Similarity ratio to water density λ ρw Step 3 obtains the vertical geometric similarity ratio λ lz Step 4: Obtain the x-axis geometric similarity ratio λ lx Step 5: Obtain the horizontal y-axis geometric similarity ratio λ ly And the x-axis wavelength similarity ratio λ obtained in step six T Determine the wave height similarity ratio λ H The method is: Assuming the structural inertial force similarity ratio is equal to the wave force similarity ratio, then the wave height similarity ratio λ H The expression is: ； In the formula, R is the cross-sectional radius of the prototype structure, which is taken as half the length of the water-facing surface. , , and These are the first-order Bessel functions of the first kind and the second-order Bessel functions of the second kind, respectively.

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