A dynamic similarity design method for soil-structure super-scale reduced models

Abstract

A dynamic similarity design method for soil-structure super-scale reduced models. This method vectorizes the geometric dimensions. First, the geometric similarity constant is comprehensively determined based on the prototype mass of the soil and the structure and the equipment parameters of the centrifuge shaking table, so as to determine the centrifuge gravity field. Then, the physical quantities of the density and elastic modulus of the prototype structure and the soil are measured, the material of the reduced model structure is selected, and the physical quantities of the density and elastic modulus of the reduced model structure are measured. Using the dynamic equilibrium equation and based on the principle that the similarity of the structure frequency and the soil frequency is consistent, the geometric similarity constant in the direction of the seismic action is derived and determined. Finally, the similarity design of the test reduced model is completed according to the principle of dimensional coordination. The present invention is used for the centrifuge shaking table dynamic test of the soil-structure reduced model under unidirectional seismic action, can accurately restore the dynamic response characteristics of the prototype structure, and this method is simple and practical and easy to apply in the test.

Description

Technical Field

[0001] The present invention belongs to the technical field of civil engineering, and particularly relates to a dynamic similarity design method for a soil-structure overweight scaled model. Background Art

[0002] In the field of civil engineering, seismic analysis and checking calculations are often required through shaking table scaled model tests in the design of large and complex projects such as houses, bridges, and docks. In seismic analysis and checking calculations, the interaction between soil and structure needs to be considered. The scientific nature of the similarity design of the scaled model test considering the soil-structure interaction is directly related to the rationality and effectiveness of the test. Since the centrifuge test can increase the gravitational field acceleration of the scaled model and restore the original stress state of the soil mass, the centrifuge shaking table test is more suitable for studying the dynamic response of the soil-structure interaction compared to the conventional scaled model shaking table test in the 1g gravitational field.

[0003] Because the load-bearing capacity of the centrifuge shaking table is much smaller than that of the conventional shaking table and the structural model is smaller, a small scale ratio must be used for the similarity design of the test model. For reinforced concrete structures, conventional concrete and steel bars cannot be used for the design and production of small-scale scaled models. Usually, materials such as aluminum alloy, plexiglass, and micro-concrete are used instead. Among them, the physical parameters such as density and elastic modulus of micro-concrete are roughly the same as those of ordinary concrete. The scaled model steel bars are generally replaced by steel wires. Even so, due to the too small size of the scaled model, there are still great difficulties in the production of the scaled model. Therefore, most of the scaled models for centrifuge shaking table tests of elastic structures use materials such as aluminum alloy or plexiglass to simulate reinforced concrete materials, and there are differences in their physical parameters such as density and elastic modulus from those of reinforced concrete materials. The similarity relationship of the structural scaled model needs to be designed with emphasis. On the other hand, the centrifuge can change the gravity similarity relationship between the scaled model and the prototype by adjusting the gravitational field, and can better solve the problem of the similarity of the original stress of the soil scaled model. In the test, the prototype soil can be used as the soil material of the scaled model, and the density and elastic modulus and other physical parameters of the scaled model soil can be made similar to those of the prototype soil through consolidation. However, there are differences in the similarity constants of physical quantities such as the material density and elastic modulus of the structure and the soil, resulting in that the traditional design method cannot strictly satisfy the dynamic similarity relationship. Summary of the Invention

[0004] In order to solve the above problems, the purpose of the present invention is to provide a dynamic similarity design method for a soil-structure overweight scaled model, which is applicable to the similarity design of centrifuge shaking table tests of soil-structure scaled models under unidirectional seismic action, and the accuracy of its test scaled model in simulating the prototype is high.

[0005] In order to achieve the above purpose, the dynamic similarity design method for a soil-structure overweight scaled model provided by the present invention includes the following steps carried out in sequence:

[0006] Step 1: Determine the gravity similarity constant S according to the maximum effective load of the centrifuge shaking table test equipment and the mass of the prototype structure g ;

[0007] Step 2: Determine the apparent density and elastic modulus of the prototype structure and the soil through material tests

[0008] Step 3: Select the material of the scaled model structure for the test, and determine the apparent density and elastic modulus of the scaled model structure through material tests; select the same material for the scaled model soil as that of the prototype soil, with the density and elastic modulus the same as those of the prototype soil's apparent density and elastic modulus in Step 2 above. Combining the apparent density and elastic modulus of the prototype structure in Step 2 above, finally determine the density similarity constant S ρ and the elastic modulus similarity constant S E , and the density similarity constant S ρsoil and the elastic modulus similarity constant S Esoil of the scaled model soil;

[0009] Step 4: Determine the geometric similarity constant S according to the relationship between the gravity similarity constant S g determined in Step 1 above and the geometric similarity constant l ;

[0010] Step 5: Based on the principle of the similarity of the structure frequency and the soil frequency being consistent, and based on the results of Steps 3 to 4 above, use the dynamic equilibrium equation to determine the geometric similarity constant S in the direction of the seismic action lx ;

[0011] Step 6: Based on the results of Steps 3 to 5 above, use the dimensional coordination principle to determine the similarity constants of each physical quantity of the scaled model structure

[0012] In Step 1, the method for determining the gravity similarity constant S g is as follows:

[0013] The gravity similarity constant S g should satisfy the following equation with the maximum effective load N g of the centrifuge shaking table test equipment and the mass m of the prototype structure:

[0014]

[0015] And the gravity similarity constant S g should not exceed 2 / 3 of the maximum design gravity similarity constant that the centrifuge shaking table test equipment can withstand

[0016] In Step 3, the material of the scaled model structure is selected as steel, aluminum alloy or plexiglass; the density similarity constant S of the scaled model structureρ = Apparent density of the scaled - down model structure / Apparent density of the prototype structure, Elastic modulus similarity constant \(S\) of the scaled - down model structure E = Elastic modulus of the scaled - down model structure / Elastic modulus of the prototype structure; Since the apparent density and elastic modulus of the scaled - down model soil are the same as those of the prototype soil, the density similarity constant \(S\) of the scaled - down model soil ρsoil = 1, Elastic modulus similarity constant \(S\) Esoil = 1.

[0017] In step four, the gravity similarity constant \(S\) g and the geometric similarity constant \(S\) l have the relationship:

[0018]

[0019] In step five, assuming that the frequency similarity constant \(S\) of the structure f is equal to the frequency similarity constant \(S\) of the soil fsoil , then the expression of the geometric similarity constant \(S\) in the direction of seismic action lx is:

[0020]

[0021] Where: \(S\) Esoil is the elastic modulus similarity constant of the scaled - down model soil; \(S\) ρsoil is the density similarity constant of the scaled - down model soil; \(S\) E is the elastic modulus similarity constant of the scaled - down model structure; \(S\) ρ is the density similarity constant of the scaled - down model structure; \(S\) l is the geometric similarity constant.

[0022] In step six, using the principle of dimensional harmony, the similarity constants of various physical quantities of the scaled - down model structure are determined as:

[0023] The mass similarity constant \(S\) of the scaled - down model structure m = \(S\) ρ \(S\) lx \(S\) l 2 , The acceleration similarity constant \(S\) in the direction of seismic action ax = \(S\) lx \(S\) Esoil / (\(S\) ρsoil \(S\) l 2 ), The concentrated force similarity constant \(S\) in the direction of seismic action F = \(S\) ρ \(S\) lx \(S\) l 2 \(S\) axand the moment similarity constant S M = S F S l .

[0024] The advantages and positive effects of the present invention are as follows: According to the prototype mass of the soil body and the structure and the parameters of the centrifuge shaking table equipment, the geometric similarity constant is comprehensively determined to determine the centrifuge gravity field. Then, the physical quantities of the density and elastic modulus of the prototype structure and the soil body are measured, the material of the scaled-down model structure is selected, and the physical quantities of the density and elastic modulus of the scaled-down model structure are measured. Using the dynamic equilibrium equation and based on the principle of the similarity of the structure frequency and the soil body frequency, the geometric similarity constant in the direction of the seismic action is deduced and determined. Finally, according to the principle of dimensional coordination, the similarity design of the test scaled-down model is completed. This similarity method is simple and practical, and has a higher accuracy in simulating the prototype dynamic response compared with the traditional method, which can provide a new method for the design of the scaled-down model of the soil-structure interaction centrifuge shaking table test under unidirectional seismic action. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 (a) and (b) are respectively the front view and side view of the prototype of the pile foundation pier adopted in the present invention;

[0026] Figure 2 is a schematic diagram of the soil layer distribution and the direction of the seismic action of the pile foundation pier adopted in the present invention;

[0027] Figure 3 is a comparison diagram of the displacements at the top of the pier using the method of the present invention and the traditional method;

[0028] Figure 4 is a comparison diagram of the bending moments at the bottom of the pier using the method of the present invention and the traditional method;

[0029] Figure 5 is a comparison diagram of the shear forces at the bottom of the pier using the method of the present invention and the traditional method;

[0030] Figure 6 is a comparison diagram of the soil pressures in the middle of the pile using the method of the present invention and the traditional method. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0031] In order to further understand the content, features and effects of the present invention, the following embodiments are exemplified and described in detail in conjunction with the drawings and tables as follows:

[0032] The dynamic similarity design method of the soil-structure overweight scaled-down model provided by the present invention includes the following steps carried out in sequence:

[0033] Step 1: Determine the gravity similarity constant S according to the maximum effective load of the centrifuge shaking table test equipment and the mass of the prototype structure g ;

[0034] The method for determining the gravity similarity constant S g is as follows:

[0035] The gravity similarity constant S g should satisfy the following equation with the maximum effective load N of the centrifuge shaking table test equipment g and the mass m of the prototype structure:

[0036]

[0037] And the gravity similarity constant S g should not exceed 2 / 3 of the maximum designed gravity similarity constant that the centrifuge shaking table test equipment can withstand.

[0038] Step 2: Determine the apparent density and elastic modulus of the prototype structure and the soil through material tests;

[0039] Step 3: Select the material for the scaled model structure for the test, and determine the apparent density and elastic modulus of the scaled model structure through material tests; select the same material for the scaled model soil as that of the prototype soil, with the density and elastic modulus the same as those of the prototype soil's apparent density and elastic modulus in Step 2 above. Combine with the apparent density and elastic modulus of the prototype structure in Step 2 above to finally determine the density similarity constant S ρ and the elastic modulus similarity constant S E of the scaled model structure, as well as the density similarity constant S ρsoil and the elastic modulus similarity constant S Esoil of the scaled model soil;

[0040] The material of the scaled model structure is selected from steel, aluminum alloy or plexiglass; the density similarity constant S ρ of the scaled model structure = apparent density of the scaled model structure / apparent density of the prototype structure, and the elastic modulus similarity constant S E of the scaled model structure = elastic modulus of the scaled model structure / elastic modulus of the prototype structure; since the apparent density and elastic modulus of the scaled model soil are the same as those of the prototype soil, the density similarity constant S ρsoil of the scaled model soil = 1, and the elastic modulus similarity constant S Esoil = 1.

[0041] Step 4: Determine the geometric similarity constant S g using the relationship between the gravity similarity constant S l determined in Step 1 above and the geometric similarity constant;

[0042] The relationship between the gravity similarity constant S g and the geometric similarity constant S l is as follows:

[0043]

[0044] Step 5: Based on the results of the above Steps 3 to 4 and with the principle that the similarity between the structural frequency and the soil body frequency is consistent, use the dynamic equilibrium equation to determine the geometric similarity constant S in the direction of the seismic action. lx ;

[0045] Let the frequency similarity constant S of the structure f be equal to the frequency similarity constant S of the soil body. fsoil Then the expression for the geometric similarity constant S in the direction of the seismic action lx is:

[0046]

[0047] In the formula: S Esoil is the elastic modulus similarity constant of the soil body of the scaled model; S ρsoil is the density similarity constant of the soil body of the scaled model; S E is the elastic modulus similarity constant of the structure of the scaled model; S ρ is the density similarity constant of the structure of the scaled model; S l is the geometric similarity constant.

[0048] Step 6: Based on the results of the above Steps 3 to 5, use the principle of dimensional consistency to determine the similarity constants of each physical quantity of the scaled model structure.

[0049] The similarity constants of each physical quantity of the scaled model structure determined by using the principle of dimensional consistency are:

[0050] The mass similarity constant S of the scaled model structure m = S ρ S lx S l 2 The acceleration similarity constant S in the direction of the seismic action ax = S lx S Esoil / (S ρsoil S l 2 ) The concentrated force similarity constant S in the direction of the seismic action F = S ρ S lx S l 2 S ax And the moment similarity constant S M = S F S l .

[0051] By vectorizing the geometric similarity constant, the present invention designs and determines the geometric similarity constant for the direction of seismic action, realizes the coordinated similarity of the dynamic characteristics of soil and structure, and achieves the purpose that the soil-structure overweight scaled model can accurately simulate the dynamic response of the prototype.

[0052] The superiority of the present invention is illustrated by an application example below.

[0053] In this embodiment, the maximum gravitational acceleration of the centrifuge shaking table test equipment selected is 100g, and the maximum effective load is 500 kg.

[0054] Please refer to Figure 1 , in this embodiment, the pier 1 in the prototype structure has a rectangular cross-section, with a length of 4.0 m, a width of 1.6 m, and a height of 17.0 m. The cuboid-shaped cap 2 has a height of 2.0 m and a width of 2.2 m. There are four piles 3 in a row under the cap 2, and the length of the piles 3 to the bearing stratum is 10.0 m. The load transferred by the main beam to the top of the pier 1 is equivalent to a mass of 100 t. The pier 1, the cap 2, and the piles 3 are all reinforced concrete structures, with a concrete strength grade of C40 and steel bars of HRB400. Through tests, the apparent density of the prototype reinforced concrete structure is measured as 2600 kg / m 3 and the elastic modulus is 35.0 GPa.

[0055] Please refer to Figure 2 , in this embodiment, considering the unidirectional seismic action, the soil where the piles 3 are located is divided into three layers, and the soil layer distribution parameters are shown in Table 1.

[0056] Table 1 Soil layer parameters

[0057]

[0058]

[0059] In this embodiment, the material of the scaled model structure selected is aluminum alloy. Through tests, the apparent density of the aluminum alloy is measured as 2600 kg / m 3 , and the elastic modulus is 70.0 GPa. Therefore, the density similarity constant S ρ = 1, and the elastic modulus similarity constant S E = 2. Considering the parameters of the test equipment and the mass of the prototype structure, the gravitational similarity constant S g = 50 is determined, and thus the geometric similarity constant S l = 1 / 50 is determined. Therefore, the geometric dimension similarity constant S lx for the direction of seismic action = 1 / 70.71. The similarity constants of the scaled model structure designed by the method of the present invention are shown in Table 2, and the similarity constants of the scaled model structure designed by the traditional method are shown in Table 3.

[0060] Table 2 Similarity Constants of the Scaled Model Structure of the Present Invention

[0061]

[0062] Table 3 Similarity Constants of the Conventional Scaled Model Structure

[0063]

[0064] The scaled model structures of the above-mentioned bridge piers 1, pile caps 2 and pile foundations 3 are designed by using the method of the present invention and the conventional method respectively. The numerical calculations of the dynamic time-history responses of the prototype structures of the bridge piers 1, pile caps 2 and pile foundations 3, the scaled model structures designed by using the method of the present invention and the scaled model structures designed by using the conventional method under the action of El-Cenreo ground motion are carried out by using finite element software. The numerical calculation results of the displacement at the top of the pier, the bending moment at the bottom of the pier, the shear force at the bottom of the pier and the soil pressure in the middle of the soil layer are extracted. Then, the seismic response results of the prototype structures are obtained by back-calculating the seismic response numerical calculation results of the scaled model structures designed by using the method of the present invention and the scaled model structures designed by using the conventional method according to the similarity relationships in Table 2 and Table 3 respectively, and are compared with the seismic response results of the prototype structures obtained by direct numerical calculation.

[0065] Please refer to Figure 3 , under the action of El-Centro earthquake, the maximum value of the seismic response of the displacement at the top of the prototype pier is 17.79 mm. The maximum values of the seismic response of the displacement at the top of the scaled model structures designed by using the method of the present invention and the scaled model structures designed by using the conventional method after back-calculation are 18.22 mm and 20.82 mm respectively, and the relative errors are 2.4% and 15.8% respectively.

[0066] Please refer to Figure 4 , under the action of El-Centro earthquake, the maximum value of the seismic response of the bending moment at the bottom of the prototype pier is 2.12 MN·m. The maximum values of the seismic response of the bending moment at the bottom of the scaled model structures designed by using the method of the present invention and the scaled model structures designed by using the conventional method after back-calculation are 2.44 MN·m and 1.54 MN·m respectively, and the relative errors are 15.2% and 27.5% respectively.

[0067] Please refer to Figure 5 , under the action of El-Centro earthquake, the maximum value of the seismic response of the shear force at the bottom of the prototype pier is 81.95 kN. The maximum values of the seismic response of the shear force at the bottom of the scaled model structures designed by using the method of the present invention and the scaled model structures designed by using the conventional method after back-calculation are 99.22 kN and 115.71 kN respectively, and the relative errors are 21.1% and 41.2% respectively.

[0068] Please refer to Figure 6, under the El-Centro earthquake action, the maximum value of the seismic response of the earth pressure of the prototype structure is 25.49 kPa. The maximum values of the seismic response of the earth pressure back-calculated for the scaled model structure designed by the method of the present invention and the scaled model structure designed by the traditional method are 26.43 kPa and 28.12 kPa respectively, and the relative errors are 3.7% and 10.3%.

[0069] In summary, under the El-Centro earthquake action, the relative errors of the seismic responses of the scaled model structures designed by the method of the present invention are between 2.4% and 15.2%, and the relative errors of the seismic responses of the scaled model structures designed by the traditional method are between 10.3% and 41.2%. Therefore, the relative error of the seismic response of the scaled model structure designed by the method of the present invention is smaller than that of the scaled model structure designed by the traditional method.

[0070] Although the preferred embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the above specific embodiments. The above specific embodiments are merely illustrative and not restrictive. Under the inspiration of the present invention, those of ordinary skill in the art can also make many forms without departing from the spirit of the present invention and the scope protected by the claims. All of these fall within the protection scope of the present invention.

Claims

1. A dynamic similarity design method for soil-structure overweight reduced-scale models, characterized in that: The dynamic similarity design method for soil-structure overweight reduced-scale models includes the following steps carried out in sequence: Step 1: Determine the gravity similarity constant S according to the maximum effective load of the centrifuge shaking table test equipment and the mass of the prototype structure g ; Step 2: Determine the apparent density and elastic modulus of the prototype structure and soil through material tests; Step 3: Select the material for the scaled model structure for the test, and determine the apparent density and elastic modulus of the scaled model structure through material tests; select the same material for the scaled model soil as that of the prototype soil, with the density and elastic modulus being the same as the apparent density and elastic modulus of the prototype soil in Step 2 above. Combine the apparent density and elastic modulus of the prototype structure in Step 2 above to finally determine the density similarity constant S ρ and the elastic modulus similarity constant S E for the scaled model structure, as well as the density similarity constant S ρsoil and the elastic modulus similarity constant S Esoil for the scaled model soil; Step 4: Determine the geometric similarity constant S by using the relationship between the gravity similarity constant S determined in Step 1 above g and the geometric similarity constant l ; Step 5: Based on the results of the above Steps 3 to 4, with the principle that the similarity between the structural frequency and the soil body frequency is consistent, use the dynamic equilibrium equation to determine the geometric similarity constant S in the direction of seismic action lx ; Step 6: Based on the results of Steps 3 to 5 above, use the principle of dimensional coordination to determine the similarity constants of various physical quantities of the reduced-scale model structure; In Step 5, let the frequency similarity constant S of the structure f be equal to the frequency similarity constant S of the soil mass fsoil , then the geometric similarity constant in the direction of seismic action is S lx and its expression is: Where: S Esoil is the similarity constant of the elastic modulus of the soil in the scaled model; S ρsoil is the similarity constant of the density of the soil in the scaled model; S E is the similarity constant of the elastic modulus of the structure in the scaled model; S ρ is the similarity constant of the density of the structure in the scaled model; S l is the geometric similarity constant; In Step 6, the similarity constants of various physical quantities of the reduced-scale model structure determined by using the principle of dimensional coordination are: Mass similarity constant \(S\) of the scaled model structure m = \(S\) ρ \(S\) lx \(S\) l 2 , Acceleration similarity constant \(S\) in the direction of seismic action ax = \(S\) lx \(S\) Esoil / (\(S\) ρsoil \(S\) l 2 ), Concentrated force similarity constant \(S\) in the direction of seismic action F = \(S\) ρ \(S\) lx \(S\) l 2 \(S\) ax And moment similarity constant \(S\) M = \(S\) F \(S\) l .

2. The dynamic similarity design method of soil-structure overweight reduced-scale model according to claim 1, wherein: In step one, the method for determining the gravity similarity constant S g is as follows: Gravity similarity constant S g And the maximum effective load N of the centrifuge shaking table test equipment g And the mass m of the prototype structure should satisfy the following formula: and the gravity similarity constant S g shall not exceed 2 / 3 of the maximum designed gravity similarity constant that the centrifuge shaking table test equipment can withstand.

3. The dynamic similarity design method of soil-structure overweight reduced-scale model according to claim 1, characterized in that: In Step 3, the material of the scaled model structure is selected from steel, aluminum alloy or plexiglass; the density similarity constant S ρ of the scaled model structure = apparent density of the scaled model structure / apparent density of the prototype structure, and the elastic modulus similarity constant S E of the scaled model structure = elastic modulus of the scaled model structure / elastic modulus of the prototype structure; since the apparent density and elastic modulus of the scaled model soil are the same as those of the prototype soil, the density similarity constant S ρsoil of the scaled model soil = 1, and the elastic modulus similarity constant S Esoil = 1.

4. The dynamic similarity design method of soil-structure overweight reduced-scale model according to claim 1, characterized in that: In Step 4, the gravity similarity constant S g and the geometric similarity constant S l are related as follows:

Images