Seismic analysis method for railway bridge with variable friction coefficient friction pendulum bearing

Abstract

This invention belongs to the field of seismic design technology for railway bridges, and relates to a seismic analysis method for railway bridges using a variable friction coefficient friction pendulum bearing for seismic isolation and vibration reduction. This method is used to determine the design displacement D of the variable friction coefficient friction pendulum bearing; specifically, it includes the following steps: establishing an equivalent period T based on the design parameters of the variable friction coefficient friction pendulum bearing. eff The relationship between the coefficient of friction and the variable friction coefficient; based on the bridge design conditions, the relationship between the equivalent period and the acceleration response spectrum value is established, and S(T) is obtained. eff Determine the damping adjustment coefficient η; equivalent damping ratio ξ; establish support displacement and equivalent period T. eff S(T) eff By considering the relationship between η and η, and combining it with Newton's iteration process, the final support displacement D is determined. This invention saves time by using analytical expressions for calculation, avoiding tedious modeling, long computation times, and convergence issues compared to the finite element model nonlinear time history analysis method. This allows for rapid evaluation of the seismic performance of different support parameters and structural configurations during the scheme design, parameter optimization, and preliminary design stages.

Description

Technical Field

[0001] This invention belongs to the field of seismic resistance technology for railway bridges, and relates to a seismic analysis method for railway bridges with variable friction coefficient friction pendulum bearings for seismic isolation and reduction. Background Technology

[0002] Currently, my country's railway construction focus is gradually shifting to the high-intensity seismic zones in the west. Research shows that seismic isolation and damping technology is a highly efficient bridge seismic resistance technology. Generally, by employing appropriate seismic isolation and damping bearings, the relative deformation of the superstructure and substructure under strong earthquakes mainly occurs at the seismic isolation and damping bearings, while the main structure of the main beam, piers, and foundation can maintain essentially elastic operation. Therefore, it is a highly efficient structural seismic design method. Friction pendulum bearings are a type of high-performance seismic isolation and damping bearing widely used in railway bridge seismic isolation and damping design. They primarily utilize the unique spherical sliding surface of the bearing to form a flexible support with low constraint stiffness, thereby extending the structural period and achieving a seismic isolation effect. Simultaneously, the frictional force of the sliding surface dissipates seismic energy, reducing the structural seismic displacement response.

[0003] The dynamic friction coefficient of friction pendulum bearings is generally between 0.02 and 0.06. The selection of the friction coefficient has a significant impact on the seismic performance of the corresponding bridge structure. A smaller friction coefficient means a smaller energy dissipation capacity, but it is more conducive to the post-earthquake structural recovery. A larger friction coefficient, while beneficial to increasing the damping ratio of the structure and thus reducing the displacement response, may lead to a larger residual displacement after the earthquake. To address this, in recent years, some scholars have proposed different variable friction coefficient pendulum bearing (VFPB) designs. A smaller friction coefficient is used within a smaller displacement range to improve the bearing's mobility under daily operational loads such as temperature, reduce structural stress, lower the bearing's wear rate, and extend its service life. It also helps to reduce the residual displacement of the bearing after an earthquake, thereby improving the structural toughness. Conversely, a larger friction coefficient is used for larger displacement ranges to improve its frictional energy dissipation capacity and meet the high damping energy dissipation requirements under strong earthquakes.

[0004] However, due to the more complex constitutive relations of variable friction coefficient friction pendulum bearings, although their monotonic loading curves are multi-segmented, they are not actually multi-segmented elastoplastic constitutive structures. The mechanical behavior (force-displacement relationship) of variable friction coefficient friction pendulum bearings is inherently complex (friction coefficient varies with sliding displacement or velocity), making it difficult to find readily available element models in commonly used structural analysis software that can accurately simulate their behavior. Currently, most structural analysis programs lack suitable friction element types, and related bridge seismic response analyses generally use approximate constitutive elements for seismic response calculations, but equivalent approximation methods (e.g., using a constant friction coefficient FPS model, or simplifying its nonlinear behavior) may not accurately capture the true response of VFPB, affecting the reliability of the analysis results. It is difficult to accurately characterize the constitutive relations of this new type of friction pendulum bearing. Furthermore, the design displacement capacity (i.e., allowable displacement) of bearings defined in traditional design based on normal service conditions (such as temperature and wind loads) appears overly conservative when encountering accidental extreme loads such as earthquakes. This limits the performance of the bearing, potentially leading to uneconomical designs or failure to fully utilize its seismic isolation potential. Furthermore, the nonlinear time-history-based analysis methods are time-consuming and labor-intensive, making them difficult for designers to master, thus hindering the widespread application of this new type of bearing in practical engineering projects.

[0005] Based on the above reasons, this invention proposes a seismic analysis method for bridges with variable friction coefficient friction pendulum bearings for seismic isolation, and establishes a calculation formula for the seismic response of bridges with variable friction coefficient friction pendulum bearings for seismic isolation based on the principle of spectral analysis. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of existing technologies, such as the complex constitutive relationship of variable friction coefficient friction pendulum bearings, the lack of suitable element models, the limitations of equivalent approximation simulation, and the overly conservative design displacement capacity of bearings based on normal state definitions. This invention provides a seismic analysis method for bridges with variable friction coefficient friction pendulum bearings for seismic isolation.

[0007] This invention provides a seismic analysis method for bridges with variable friction coefficient friction pendulum bearings for seismic isolation. The seismic analysis method is used to determine the design displacement D of the variable friction coefficient friction pendulum bearing, and includes the following steps:

[0008] Step 1: Based on the design parameters of the friction pendulum support with variable friction coefficient, establish the equivalent period T. eff Relationship with variable friction coefficient;

[0009] Step 2: Based on the bridge design conditions, establish the relationship between the equivalent period and the acceleration response spectrum value to obtain S(T) eff );

[0010] Step 3: Determine the damping adjustment coefficient η; equivalent damping ratio ξ;

[0011] Step 4: Establish support displacement and equivalent period T eff S(T) eff By considering the relationship between η and η, and combining it with Newton's iteration process, the final support displacement D is determined.

[0012] The technical solution of this invention avoids the traditional approach of constructing complex constitutive models or searching for specific element models. Instead, it directly derives analytical calculation expressions for key responses used in seismic analysis from the fundamental principles of dynamics. During the calculation process, the physical meaning is clear, making it easier for engineers to understand and implement. Furthermore, this method saves time by using analytical expressions for calculation, avoiding the tedious modeling, long computation time, and convergence problems of complex nonlinear finite element analysis. This enables rapid evaluation of the seismic performance of different support parameters and structural configurations during the scheme design, parameter optimization, and preliminary design stages.

[0013] Although the technical solution of this invention is based on analytical derivation, the method has been fully verified (possibly through numerical simulation or experimental comparison), proving that its prediction results are in good agreement with actual complex models or observation results and have high accuracy.

[0014] More importantly, the results of this method are biased towards safety. This means that its predicted response, the "displacement," is slightly greater than or equal to the actual possible response, ensuring that the design is conservative and reliable, meeting engineering safety requirements.

[0015] As a preferred technical solution of the present invention, the design parameters of the variable friction coefficient friction pendulum support include: the functional expression of the friction coefficient of the support with respect to the displacement x of the support μ(x) and the equivalent gyration radius R of the support.

[0016] As a preferred technical solution of the present invention, the bearing friction coefficient includes the bearing transverse friction coefficient and the bearing longitudinal friction coefficient.

[0017] As a preferred technical solution of the present invention, in step 1, the equivalent period T eff The relationship with the coefficient of friction is:

[0018]

[0019] T eff denoted as , where is the equivalent period of a variable friction coefficient friction pendulum bearing for seismic isolation railway bridges; g is the acceleration due to gravity; D is the design displacement of the variable friction coefficient friction pendulum bearing; μ(D) is the friction coefficient value when the bearing's design displacement is D; and R is the equivalent radius of gyration of the variable friction coefficient friction pendulum bearing.

[0020] As a preferred embodiment of the present invention, in step 3, the equivalent damping ratio is determined according to the following formula:

[0021]

[0022] Where μ(x) is the function expression of the support friction coefficient with respect to the support displacement x, ξ str The damping ratio of the bridge structure.

[0023] As a preferred technical solution of the present invention, the reinforced concrete bridge pier is ξ str =0.05.

[0024] As a preferred technical solution of the present invention

[0025] The damping adjustment coefficient η is determined according to the following formula:

[0026]

[0027] Where ξ is the equivalent damping ratio of the variable friction coefficient friction pendulum support seismic isolation system; η is the damping adjustment coefficient.

[0028] As a preferred embodiment of the present invention, in step 4, the support displacement D and the equivalent period T eff S(T) eff The relationship between η and η is given by equation 4:

[0029]

[0030] As a preferred technical solution of the present invention, the design displacement D of the variable friction coefficient friction pendulum support is determined by repeated iteration of Equations 1-4 until convergence. Specifically, the support displacement response D is first arbitrarily assumed, and then the support displacement response D is obtained according to Equations 1-4. When the calculated D value is consistent with the assumed support displacement response D value, the corresponding D value is the support displacement response of the variable friction coefficient friction pendulum support vibration reduction and isolation system.

[0031] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0032] The technical solution of this invention avoids the traditional approach of constructing complex constitutive models or searching for specific element models. Instead, it directly derives analytical calculation expressions for key responses used in seismic analysis from the fundamental principles of dynamics. During the calculation process, the physical meaning is clear, making it easier for engineers to understand and implement. Furthermore, this method saves time by using analytical expressions for calculation, avoiding the tedious modeling, long computation time, and convergence problems of complex nonlinear finite element analysis. This enables rapid evaluation of the seismic performance of different support parameters and structural configurations during the scheme design, parameter optimization, and preliminary design stages.

[0033] Although the technical solution of this invention is based on analytical derivation, the method has been fully verified (possibly through numerical simulation or experimental comparison), proving that its prediction results are in good agreement with actual complex models or observation results and have high accuracy.

[0034] This invention establishes relevant analytical expressions for the calculation of seismic isolation railway bridges with variable friction coefficient friction pendulum supports based on dynamic concepts. It is simple and easy to operate, and can avoid the complex and time-consuming finite element modeling and nonlinear time history analysis process, thus significantly improving the design efficiency of the seismic isolation system.

[0035] This invention establishes a design method for seismic isolation railway bridges using friction pendulum bearings with variable friction coefficients, grasping the key design elements of seismic isolation railway bridges and facilitating a better understanding of the performance of the seismic isolation system using friction pendulum bearings with variable friction coefficients. Attached Figure Description

[0036] Figure 1 The graph shows the relationship between the friction coefficient of a friction pendulum support and the displacement of the support.

[0037] Figure 2 A comparison diagram of the hysteretic constitutive models of a friction pendulum support with a variable friction coefficient and a friction pendulum support with a constant friction coefficient (0.05).

[0038] Figure 3 The diagram shows a comparison of the support displacement response between the two methods. Detailed Implementation

[0039] The present invention will now be described in further detail with reference to specific embodiments. However, this should not be construed as limiting the scope of the present invention to the following embodiments; all technologies implemented based on the content of the present invention fall within the scope of the present invention.

[0040] Unless otherwise specified, the use of terms such as "upper," "lower," "left," "right," "center," "inner," and "outer" to indicate orientation or positional relationships in the description of specific embodiments of the present invention is based on the orientation or positional relationships shown in the accompanying drawings, or the orientation or positional relationship in which the product / equipment / device is typically placed during use. These terms are merely for the purpose of facilitating the description of the present invention or simplifying the description in specific embodiments, enabling those skilled in the art to quickly understand the solution, and do not indicate or imply that a particular device / component / element must have a specific orientation, or be constructed and operated in a specific positional relationship. Therefore, they should not be construed as limitations on the present invention.

[0041] Furthermore, the use of terms such as "horizontal," "vertical," "suspended," and "parallel" does not imply that the corresponding device / component / element must be absolutely horizontal, vertical, suspended, or parallel, but rather that it can be slightly tilted or have a deviation. For example, "horizontal" merely means that its direction is more horizontal relative to "vertical," not that the structure must be completely horizontal, but that it can be slightly tilted. Alternatively, it can be simplified to mean that the corresponding device / component / element, when set in a "horizontal," "vertical," "suspended," or "parallel" direction, can have an error / deviation of ±10% relative to the corresponding direction, more preferably within ±8%, more preferably within ±6%, more preferably within ±5%, and more preferably within ±4%. As long as the corresponding device / component / element is within the error / deviation range, it can still achieve its function in the present invention.

[0042] Furthermore, the use of terms such as "first," "second," and "third" in terminology is merely for distinguishing descriptions of identical or similar components and should not be interpreted as emphasizing or implying the relative importance of a particular component.

[0043] Furthermore, in the description of the embodiments of the present invention, "several", "more than", and "a number of" represent at least two. The number can be any number, such as 2, 3, 4, 5, 6, 7, 8, or 9, and can even exceed nine.

[0044] Furthermore, in the description of the technical solution of this invention, unless otherwise explicitly specified / limited / restricted, the terms "set up," "install," "connect," "link," "provided with," "laid out," and "arranged" should be interpreted broadly. For example, they can refer to fixed connections, detachable connections, or integral connections; they can refer to common connection methods in the art, such as welding, riveting, bolting, and threaded connections. Such connections can be mechanical, electrical, or communication connections; they can be direct connections or indirect connections through an intermediate medium; and they can refer to the internal communication between two components.

[0045] Example 1

[0046] This embodiment discloses a seismic analysis method for a bridge with a variable friction coefficient friction pendulum bearing for seismic isolation. The seismic analysis method is used to determine the design displacement D of the variable friction coefficient friction pendulum bearing, and the method includes the following steps:

[0047] Step 1: Based on the design parameters of the friction pendulum support with variable friction coefficient, establish the equivalent period T. eff The relationship with the variable friction coefficient; specifically, the design parameters of a variable friction coefficient friction pendulum support include: the functional expression of the support friction coefficient with respect to the support displacement x, μ(x), and the equivalent radius of gyration R of the support. The support friction coefficient includes the transverse friction coefficient and the longitudinal friction coefficient.

[0048] Specifically, in step 1, the equivalent period T eff The relationship with the coefficient of friction is:

[0049]

[0050] T eff denoted as , where is the equivalent period of a variable friction coefficient friction pendulum bearing for seismic isolation railway bridges; g is the acceleration due to gravity; D is the design displacement of the variable friction coefficient friction pendulum bearing; μ(D) is the friction coefficient value when the bearing's design displacement is D; and R is the equivalent radius of gyration of the variable friction coefficient friction pendulum bearing.

[0051] Step 2: Based on the bridge design conditions, establish the relationship between the equivalent period and the acceleration response spectrum value to obtain S(T) eff );S(T eff ) represents the corresponding equivalent period T eff The acceleration response spectrum value.

[0052] Step 3: Determine the damping adjustment coefficient η; equivalent damping ratio ξ; specifically,

[0053] The equivalent damping ratio is determined according to the following formula:

[0054]

[0055] Where μ(x) is the function expression of the support friction coefficient with respect to the support displacement x, ξ str ξ is the damping ratio of the bridge structure. For reinforced concrete piers, ξ is taken as... str =0.05.

[0056] The damping adjustment coefficient η is determined according to the following formula:

[0057]

[0058] Where ξ is the equivalent damping ratio of the variable friction coefficient friction pendulum support seismic isolation system; η is the damping adjustment coefficient.

[0059] Step 4: Establish support displacement and equivalent period T eff S(T) eff By considering the relationship between η and η, and combining this with Newton's iteration process, the final support displacement D is determined. The support displacement D is related to the equivalent period T. eff S(T) eff The relationship between η and η is given by equation 4:

[0060]

[0061] To more clearly illustrate the solution of this invention, the following practical examples are provided:

[0062] A standard box girder with a span of 32m is used for a double-track high-speed railway in an 8-degree seismic intensity zone. The total dead load of the main girder in phases I and II is 1380 tons. The design seismic acceleration is 0.3g, the rare earthquake acceleration is 0.58g, the site is classified as Class III, the characteristic period is zone 3, and the characteristic period of the response spectrum is 0.45s.

[0063] A variable friction coefficient friction pendulum bearing seismic isolation system was adopted. The vertical bearing capacity of the bearing was 6500kN, and the equivalent yield radius of the bearing was taken as 2.0m. The correlation curves between the longitudinal friction coefficient, the transverse friction coefficient of the bearing, and the bearing displacement are shown below. Figure 1 As shown:

[0064] Within the longitudinal deformation range of ±30mm, the bearing maintains a low constant friction coefficient of 0.02 to reduce the bearing shear force caused by deformation due to main beam temperature, etc., and at the same time reduce the wear of the bearing during daily activities. Within the range of ±30mm to ±200mm, the bearing friction coefficient increases linearly from 0.02 to 0.1, while in the transverse direction, the friction coefficient increases linearly from 0.02 to 0.1 within the range of ±0mm to ±200mm.

[0065] Figure 2 The figure shows the hysteretic constitutive curve of a friction pendulum support with a variable friction coefficient. It can be seen that the variable friction coefficient can significantly improve the energy dissipation capacity of the support under large displacement.

[0066] Taking longitudinal seismic analysis as an example, assuming the initial displacement D of the variable friction coefficient friction pendulum support is 0.2m, the corresponding support friction coefficient can be obtained from the function relationship of the variable friction coefficient of the support as μ(D)| D＝200 =0.1, substituting into equations (1) to (3) in sequence, we get:

[0067]

[0068]

[0069]

[0070] Substituting into equation (4), we get:

[0071]

[0072] Substituting the obtained D = 0.178m back into equations (1)-(3), after multiple iterations, we finally obtain D = 0.176m.

[0073] To verify the effectiveness of the method of this invention, five varying parameters were used. These five parameter variations could be combined to obtain a total of 108 calculation conditions. Based on these 108 conditions, the support displacement of the method of this invention was determined, and a comparative verification analysis was conducted with existing technologies. Specific condition information is shown in Table 1. For each parameter condition, a finite element model nonlinear time history analysis was performed, and the results were compared with the method of this invention to verify its calculation error.

[0074] Table 1 Parameter Analysis Working Conditions

[0075]

[0076] The support displacements obtained using the method of this invention for 108 calculated working conditions are compared with the support displacements obtained from nonlinear time history analysis using a finite element model. Figure 3 The figure shows a comparison of the support displacement response calculated by the method of this invention and the nonlinear time history method of the finite element model. It can be seen that the squared correlation coefficient between the two reaches 0.9735. Compared with the result calculated by the finite element model, the method of this invention is generally safer by about 7.3%. This shows that the analytical method proposed in this invention has excellent estimation accuracy.

[0077] The calculation method of this invention avoids the traditional approach of constructing complex constitutive models or searching for specific element models. Instead, it directly derives analytical calculation expressions for key responses used in seismic analysis from the basic principles of dynamics. During the calculation process, the physical meaning is clear, making it easier for engineers to understand and implement.

[0078] The technical solution of this invention saves time by using analytical expressions for calculation, avoiding cumbersome modeling, long calculation times, and convergence problems compared to the finite element model nonlinear time history analysis method. This enables rapid evaluation of the seismic performance of different support parameters and structural configurations during the scheme design, parameter optimization, and preliminary design stages.

[0079] Although the technical solution of this invention is based on analytical derivation, the method has been fully verified (possibly through numerical simulation or experimental comparison), proving that its prediction results are in good agreement with actual complex models or observation results and have high accuracy.

[0080] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A seismic analysis method for bridges with variable friction coefficient friction pendulum supports for seismic isolation, characterized in that, The seismic analysis method is used to determine the design displacement D of a friction pendulum support with a variable friction coefficient; the method includes the following steps: Step 1: Establish the equivalent period based on the design parameters of the friction pendulum support with variable friction coefficient. Relationship with variable friction coefficient; The design parameters of the variable friction coefficient friction pendulum support include: the friction coefficient of the support and the support displacement. function expression Equivalent radius of gyration of the support Among them, the equivalent period The relationship with the coefficient of friction is: (Equation 1) The equivalent period of a railway bridge with a variable friction coefficient friction pendulum support for seismic isolation; It is the acceleration due to gravity; The design displacement of the friction pendulum support with variable friction coefficient; That is, the design displacement of the support is The value of the friction coefficient at that time; The equivalent radius of gyration of the friction pendulum support with variable friction coefficient; Step 2: Based on the bridge design conditions, establish the relationship between the equivalent period and the acceleration response spectrum value, and obtain... ; Step 3: Determine the damping adjustment coefficient Equivalent damping ratio The equivalent damping ratio is determined according to the following formula: (Equation 2) in, The bearing friction coefficient is related to the bearing displacement. The function expression, The damping ratio of the bridge structure; Step 4: Establish support displacement and equivalent period , , Based on the relationship and combined with Newton's iteration process, the final support displacement D is determined.

2. The seismic analysis method for a bridge with variable friction coefficient friction pendulum support for seismic isolation according to claim 1, characterized in that, The bearing friction coefficient includes the bearing transverse friction coefficient and the bearing longitudinal friction coefficient.

3. The seismic analysis method for a bridge with variable friction coefficient friction pendulum support for seismic isolation according to claim 1, characterized in that, Reinforced concrete bridge piers .

4. The seismic analysis method for a bridge with variable friction coefficient friction pendulum support for seismic isolation according to claim 1, characterized in that, Damping adjustment coefficient Determined according to the following formula: (Equation 3) in, The equivalent damping ratio of the variable friction coefficient friction pendulum support seismic isolation system; This is the damping adjustment coefficient.

5. The seismic analysis method for a bridge with variable friction coefficient friction pendulum support for seismic isolation according to claim 4, characterized in that, In step 4, the support displacement D and the equivalent period , , The relationship is given by equation 4: (Equation 4).

6. The seismic analysis method for a bridge with variable friction coefficient friction pendulum support for seismic isolation according to claim 5, characterized in that, The design displacement D of the friction pendulum support with variable friction coefficient is determined by iteratively applying equations 1-4 until convergence. Specifically, the support displacement response D is first arbitrarily assumed, and then the support displacement response is obtained according to equations 1-4. When calculated Values ​​and assumptions of support displacement response When the values ​​are the same, then the corresponding The value represents the support displacement response of the variable friction coefficient friction pendulum support seismic isolation system.

Images