Method for quickly determining seismic response of bridge cantilever pier structure

Abstract

This invention relates to the field of bridge design technology, specifically to a method for rapidly determining the seismic response of a bridge cantilever pier structure. The method includes the following steps: simplifying the basic seismic-resistant structure of a continuous beam into a seismic-resistant force model with concentrated upper mass and homogeneous lower mass; based on this seismic-resistant force model, establishing the structural mode shape function of the pier structure in the direction of seismic force; determining the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure; determining the shear force function and bending moment function for shear force and bending moment at any position of the pier structure based on the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force; and obtaining the shear force and bending moment at the bottom of the uniform cross-section cantilever pier based on the shear force function and bending moment function. This method can solve the problems of large errors in the solution results of simplified formulas in existing technologies and the low efficiency of finite element analysis methods.

Description

Technical Field

[0001] This invention relates to the field of bridge design technology, and specifically to a method for rapidly determining the seismic response of a bridge cantilever pier structure. Background Technology

[0002] Bridge seismic response refers to the response of a bridge structure under seismic loading, including parameters such as displacement, velocity, and acceleration.

[0003] The principles of bridge seismic isolation and mitigation technology can be summarized in three points based on the fundamental laws of structural system seismic response: First, extending the natural period of the structure. This is achieved by designing flexible support structures to extend the structural period, thereby reducing the structural acceleration response and weakening the seismic ground motion response. Second, reducing structural displacement. Extending the natural period inevitably leads to an increase in structural displacement. To reduce this increased displacement, damper-type energy dissipation elements can be scientifically and rationally applied in the structural design. Third, designing reasonable stiffness values. While designing flexible support structures, it is essential to ensure the rationality of the bridge structure's stiffness under normal service loads. This can be achieved by adding specialized spacers to the bridge structure to support the entire structure.

[0004] Based on extensive experience with earthquake damage and theoretical research findings, current bridge seismic design codes provide simplified calculation formulas for the seismic response of regular bridges or employ finite element analysis.

[0005] However, the simplified formula uses as many as eight analysis parameters, the physical meaning of the formula is unclear, and no specific solution algorithm is given for the parameters, making it inconvenient for engineering applications. In addition, the simplified formula is an approximate solution, not a theoretical solution, and has a large error. Using finite element analysis requires the establishment of a finite element model and the performance of finite element calculations, which is very inefficient. Summary of the Invention

[0006] To address the shortcomings of existing technologies, the present invention aims to provide a method for rapidly determining the seismic response of bridge cantilever pier structures, which can solve the problems of large errors in the solution results of simplified formulas in existing technologies and low efficiency of finite element analysis methods.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0008] This invention provides a method for rapidly determining the seismic response of a bridge cantilever pier structure, comprising the following steps:

[0009] The basic seismic structure of continuous beam is simplified into a seismic stress model of a cantilever pier with uniform cross-section, consisting of a concentrated upper mass and a homogeneous lower mass. Based on this seismic stress model, the structural vibration mode function of the pier structure in the direction of seismic force is established.

[0010] Based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure, the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force are determined.

[0011] Based on the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force, the shear force function and bending moment function for the shear force and bending moment at any position of the pier structure are determined.

[0012] Based on the shear force function and bending moment function, the shear force and bending moment at the bottom of the cantilever pier with uniform cross-section are obtained.

[0013] In some alternative schemes, based on this seismic stress model, the structural vibration mode function of the bridge pier structure in the direction of seismic force is established as follows: in, Let x be the vibration displacement of the cantilever beam at position x, where x is the distance from the bottom of the pier and L is the pier height.

[0014] In some alternative solutions, determining the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure includes:

[0015] Based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure, the projection of the pier mass onto the mode shape, the generalized mass and generalized stiffness of the pier structure are determined.

[0016] The fundamental frequency of the bridge pier structure is determined based on its generalized mass and generalized stiffness.

[0017] The peak value of the generalized inertial force of the pier structure is determined based on the structural mode shape function, the parameters of the cantilever pier structure with uniform cross section, the generalized mass and fundamental frequency of the pier structure, and the projection of the pier mass onto the mode shape.

[0018] In some alternative solutions, according to the formula Determine the projection of the pier mass onto the mode shape. Where m(x) is the mass per unit length of the bridge pier. Let x be the vibration displacement of the cantilever beam at position x.

[0019] In some alternative solutions, according to the formula Determine the generalized mass of the bridge pier structure

[0020] According to the formula Determining the generalized stiffness of bridge pier structures

[0021] Where EI(x) is the flexural stiffness of the pier at position x. Let x be the vibration acceleration of the cantilever beam at position x.

[0022] In some alternative solutions, according to the formula Determine the fundamental frequency ω of the bridge pier structure. For the generalized mass of the bridge pier structure, This refers to the generalized stiffness of the bridge pier structure.

[0023] In some alternative solutions, according to the formula Determine the peak value of the generalized inertial force f(x) of the bridge pier structure, where the correlation coefficient is... A = λω, where A is the reaction spectrum and the coefficient is... T g For the characteristic period, K h denoted as the peak ground acceleration coefficient for the horizontal fundamental ground motion, and g is the gravitational acceleration.

[0024] In some alternative schemes, the shear force function at any location of the pier structure is: ξ is the integration variable.

[0025] In some alternative schemes, the bending moment function of the shear force at any location on the pier structure is:

[0026] In some alternative schemes, based on the shear force function and bending moment function, the bottom shear force of the cantilever pier with uniform cross-section is obtained as follows: The bending moment at the pier base is: The fundamental frequency, or natural frequency, of the bridge pier structure is: m b Let M be the total mass of the cantilever pier, and M be the concentrated mass of the superstructure.

[0027] Compared with existing technologies, the advantages of this invention are as follows: This scheme simplifies the basic seismic structure of a continuous beam into a seismic force model with a concentrated upper load and a homogeneous lower mass. It establishes the structural mode shape function of the pier structure in the direction of seismic force, and then determines the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force based on the mode shape function and the parameters of the uniform cross-section cantilever pier structure. Finally, it determines the shear force and bending moment at any position of the pier structure based on the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force. This scheme does not require the establishment of a finite element model or finite element calculation analysis; it uses an analytical method to obtain the mathematical expressions for the shear force and bending moment at the bottom of the pier. The calculation results are similar to those of finite element analysis, and can replace the finite element analysis method, greatly improving analysis efficiency and thus increasing the design efficiency of this type of pier. Attached Figure Description

[0028] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0029] Figure 1 This is a flowchart of a method for rapidly determining the seismic response of a bridge cantilever pier structure in an embodiment of the present invention;

[0030] Figure 2 This is a schematic diagram of the basic seismic-resistant structure of a two-span continuous beam in an embodiment of the present invention;

[0031] Figure 3 This is a simplified schematic diagram of the intermediate bridge pier in an embodiment of the present invention;

[0032] Figure 4 This is a schematic diagram of the structural mode shape function in an embodiment of the present invention.

[0033] In the diagram: 1. Bridge pier; 2. Continuous beam; 3. Friction pendulum support. Detailed Implementation

[0034] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0035] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0036] like Figure 1 As shown, this invention provides a method for rapidly determining the seismic response of a bridge cantilever pier structure, comprising the following steps:

[0037] S1: The basic seismic structure of the continuous beam is simplified into a seismic stress model of a cantilever pier with uniform cross-section, consisting of a concentrated upper mass M and a homogeneous lower mass m(x). Based on this seismic stress model, the structural vibration mode function of the pier structure in the direction of seismic force is established.

[0038] like Figure 2As shown, taking the basic seismic structure of a two-span continuous beam as an example, the basic seismic structure of a two-span continuous beam includes three rows of piers 1, a continuous beam 2, and friction pendulum bearings 3 between piers 1 and continuous beam 2. The continuous beam is a regular bridge. Among the three piers 1, there are two side piers N1 and N3, and one middle pier N2. The friction pendulum bearings 3 between the side piers N1 and N3 and the continuous beam 2 are longitudinal movable bearings, and the friction pendulum bearings 3 between the middle pier N2 and the continuous beam 2 are longitudinal fixed bearings.

[0039] In this example, as Figure 3 As shown, the seismic force model of a continuous beam with a simplified seismic basic structure consisting of a concentrated upper mass and a homogeneous lower mass includes: simplifying the superstructure load of the pier to a concentrated load M; simplifying the pier body to a pier with a homogeneous mass m(x) = m and a homogeneous stiffness EI(x) = EI; the dimension of the concentrated load M is kg; the dimension of the homogeneous mass m(x) is kg / m; when the pier cross-section is constant, m(x) is a constant, i.e., m(x) = m, and the pier cross-section of the object studied in this invention is constant; when the pier cross-section is variable, m(x) is a function of x, and the x-coordinate axis is shown in the figure below, with the origin located at the bottom of the pier, the pier height L, and the range of x values ​​[0, L].

[0040] like Figure 4 As shown, without considering the influence of higher-order vibration modes on the seismic response of the structure, based on the uniform cross-section cantilever pier structure, the structural vibration mode function of the pier structure in the seismic force direction is established as follows: in, Let x be the vibration displacement of the cantilever beam at position x, where x is the distance from the bottom of the pier and L is the length of the pier.

[0041] S2: Based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure, determine the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force.

[0042] In some optional embodiments, step S2 includes:

[0043] S21: Based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure, determine the projection of the pier mass onto the mode shape, the generalized mass of the pier structure, and the generalized stiffness.

[0044] According to the formula Projection of pier mass onto mode shape Where m(x) is the mass per unit length of the bridge pier (kg / m), Let x be the vibration displacement of the cantilever beam at position x.

[0045] Considering the bottom effect of cantilever piers, for Figure 2 The seismic calculation model for the uniform cross-section cantilever pier shown is obtained by substituting the structural material parameters of the uniform cross-section cantilever pier. The projection of the pier mass onto the mode shape is:

[0046]

[0047] Where M is the concentrated load derived from the simplified superstructure load of the bridge pier.

[0048] According to the formula Determine the generalized mass of the bridge pier structure

[0049] According to the formula Determining the generalized stiffness of bridge pier structures

[0050] Where EI(x) is the flexural stiffness of the pier at position x. Let x be the vibration acceleration of the cantilever beam at position x.

[0051] Considering the bottom effect of cantilever piers, for Figure 2 The seismic calculation model of the cantilever pier with uniform cross-section shown is obtained by substituting the structural material parameters of the cantilever pier with uniform cross-section into it.

[0052] Generalized mass of bridge pier structure:

[0053] Generalized stiffness of bridge pier structures:

[0054] S22: Determine the fundamental frequency of the bridge pier structure based on its generalized mass and generalized stiffness.

[0055] According to the formula Determine the fundamental frequency ω of the bridge pier structure. For the generalized mass of the bridge pier structure, This refers to the generalized stiffness of the bridge pier structure.

[0056] Generalized mass of bridge pier structure and generalized stiffness: available

[0057] S23: Determine the peak value of the generalized inertial force of the pier structure based on the structural mode shape function, the parameters of the uniform cross-section cantilever pier structure, the generalized mass and fundamental frequency of the pier structure, and the projection of the pier mass onto the mode shape.

[0058] According to the formula Determine the peak value of the generalized inertial force f(x) of the bridge pier structure, where the correlation coefficient is... A = λω, where A is the response spectrum, which can be found according to seismic design codes. T can be obtained from the "Code for Seismic Design of Highway Engineering".g For the characteristic period, K h denoted as the peak ground acceleration coefficient for the horizontal fundamental ground motion, and g is the gravitational acceleration.

[0059] Calculate the internal force effects at the bottom of the cantilever pier. Figure 2 The seismic calculation model established for the cantilever pier with uniform cross-section shown, when incorporating the structural material parameters of the cantilever pier with uniform cross-section, yields the following expression for the correlation coefficient:

[0060]

[0061] S3: Based on the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force, determine the shear force function and bending moment function for the shear force and bending moment at any position of the pier structure.

[0062] The shear force function for shear force at any location on the bridge pier structure is: ξ is the integration variable.

[0063] The shear force function for shear force at any location on the bridge pier structure is: ξ is the integration variable.

[0064] S4: Determine the shear force and bending moment at the bottom of the bridge pier structure based on the shear force function and bending moment function.

[0065] Calculate the internal force effect at the bottom of the cantilever pier. Figure 2 The seismic stress model established for the cantilever pier with uniform cross-section shown, when fitted with the structural material parameters of the cantilever pier with uniform cross-section, yields:

[0066] Pier base shear force:

[0067]

[0068] After simplification, the shear force at the pier base is:

[0069] Bending moment at the pier base:

[0070] After simplification, the bending moment at the pier base is:

[0071] The fundamental frequency, or natural frequency, of the bridge pier structure is:

[0072] Where, m b Let m be the total mass of the cantilever pier. b =m(x)L.

[0073] The following is a specific example:

[0074] A concrete cantilever pier has an elastic modulus E = 3.45 x 10⁻⁶. 7 kPa, the pier has a rectangular cross-section, 2m long in the longitudinal direction and 3m wide in the transverse direction, with an area A = 2 x 3 = 6m² 2 Moment of inertia of cross section I = 3x2 3 / 12=2m 4 The pier height L = 10m, and the concrete unit weight γ = 2.5t / m³. 3 The simplified mass of the upper structure is M = 6 tons.

[0075] The homogeneous mass of the concrete cantilever pier is m(x) = γA = 2.5x6 = 15 ton / m.

[0076] Pier body mass of concrete cantilever pier

[0077]

[0078] The fundamental frequency of the uniform cross-section cantilever pier can be obtained using the above formula for calculating ω.

[0079]

[0080] Coefficients related to the reaction spectrum:

[0081]

[0082] The shear force at the bottom of the cantilever pier with uniform cross-section can be obtained using the above formula for solving Q(0):

[0083]

[0084] The bending moment at the bottom of the cantilever pier with uniform cross-section can be obtained using the above formula for solving M(0):

[0085]

[0086] If the traditional finite element method is used, a calculation model needs to be established with the help of calculation software. The internal force calculation results of the cantilever pier bottom are Q(0)=2087kN; M(0)=19800kN·m. However, this method takes a long time to calculate and the calculation model is complicated.

[0087] The internal force calculation results obtained by using the method of this patent are Q(0)=1994.4kN (error 4.6%); M(0)=19646.9kN·m (error 0.8%); the calculation is simple and convenient, and the analysis accuracy is high.

[0088] This scheme simplifies the basic seismic-resistant structure of a continuous beam into a seismic-resistant force model with a concentrated upper load and a homogeneous lower mass. It establishes the structural mode shape function of the pier structure in the direction of seismic force. Then, based on the mode shape function and the parameters of the uniform cross-section cantilever pier structure, it determines the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force. Finally, based on the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force, it determines the shear force and bending moment at any location on the pier structure. This scheme does not require establishing a finite element model or performing finite element calculations; it uses analytical calculations to obtain the shear force and bending moment at the bottom of the pier. The calculation results are similar to those of finite element analysis and can replace the finite element analysis method, greatly improving analysis efficiency. Therefore, it can improve the design efficiency of this type of pier.

[0089] In the description of this application, it should be noted that the terms "upper," "lower," etc., indicating the orientation or positional relationship are based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this application. Unless otherwise expressly specified and limited, the terms "installed," "connected," and "linked" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication between two elements. For those skilled in the art, the specific meaning of the above terms in this application can be understood according to the specific circumstances.

[0090] It should be noted that in this application, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0091] The above description is merely a specific embodiment of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. A method for rapidly determining the seismic response of a bridge cantilever pier structure, characterized by, Includes the following steps: The basic seismic structure of continuous beam is simplified into a seismic stress model of a cantilever pier with uniform cross-section, consisting of a concentrated upper mass and a homogeneous lower mass. Based on this seismic stress model, the structural vibration mode function of the pier structure in the direction of seismic force is established. Based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure, the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force are determined. According to the formula Determine the projection of the pier mass onto the mode shape. ,in, The mass per unit length of the bridge pier. For cantilever beams in x The vibration displacement of the position; According to the formula Determine the generalized mass of the bridge pier structure ; According to the formula Determine the generalized stiffness of the bridge pier structure ; in, For the bridge piers x The bending stiffness at the location, For cantilever beams in x The vibration acceleration of the position, , x The distance from the bottom of the pier is the spacing. For pier height; Based on the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force, the shear force function and bending moment function for the shear force and bending moment at any position of the pier structure are determined. Based on the shear force function and bending moment function, the bottom shear force and bending moment of the cantilever pier with uniform cross-section are obtained; Based on the shear force function and bending moment function, the shear force at the bottom of the cantilever pier with a uniform cross-section is obtained as follows: The bending moment at the bottom of the pier is: The fundamental frequency of the bridge pier structure, i.e., its natural frequency, is: m b The total mass of the cantilever pier. M For the concentrated mass at the top, the coefficient .

2. The method for rapidly determining the seismic response of a bridge cantilever pier structure as described in claim 1, characterized in that, Based on this seismic stress model, the structural vibration mode function of the bridge pier structure in the direction of seismic stress is established as follows: ,in, For cantilever beams in x Vibration displacement of position, x The distance from the bottom of the pier is the spacing. The height of the pier.

3. The method for rapidly determining the seismic response of a bridge cantilever pier structure as described in claim 1, characterized in that, The determination of the projection of the pier mass onto the mode shape, the fundamental frequency of the pier structure, and the peak value of the generalized inertial force based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure includes: Based on the structural mode shape function and the parameters of the uniform cross-section cantilever pier structure, the projection of the pier mass onto the mode shape, the generalized mass and generalized stiffness of the pier structure are determined. The fundamental frequency of the bridge pier structure is determined based on its generalized mass and generalized stiffness. The peak value of the generalized inertial force of the pier structure is determined based on the structural mode shape function, the parameters of the cantilever pier structure with uniform cross section, the generalized mass and fundamental frequency of the pier structure, and the projection of the pier mass onto the mode shape.

4. The method for rapidly determining the seismic response of a bridge cantilever pier structure as described in claim 1, characterized in that, According to the formula Determine the fundamental frequency of the bridge pier structure , For the generalized mass of the bridge pier structure, This refers to the generalized stiffness of the bridge pier structure.

5. The method for rapidly determining the seismic response of a bridge cantilever pier structure as described in claim 4, characterized in that, According to the formula Determine the peak value of the generalized inertial force of the bridge pier structure. Among them, the correlation coefficient , , For the reaction spectrum, the coefficients , For characteristic period, The horizontal peak ground acceleration coefficient is the basic ground motion coefficient. This is the acceleration due to gravity.

6. The method for rapidly determining the seismic response of a bridge cantilever pier structure as described in claim 5, characterized in that, The shear force function for shear force at any location on the bridge pier structure is: , It is the integral variable.

7. The method for rapidly determining the seismic response of a bridge cantilever pier structure as described in claim 6, characterized in that, The bending moment function of the shear force at any location of the bridge pier structure is: .

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