BWBN model-based bridge pier nonlinear earthquake random response analysis method

Abstract

The invention provides a bridge pier nonlinear earthquake random response analysis method based on a BWBN model, and belongs to the field of civil engineering structure random vibration analys.The method comprises the steps that discretization processing is conducted on a concrete bridge pier structure, and a finite element nonlinear analysis numerical model is constructed; constructing a BWBN model by combining linear and nonlinear differential equations, setting a hysteresis parameter value of the BWBN model, simulating by adopting an Euler-Bernoulli beam unit, and establishing a degraded beam finite element model; obtaining a unit hysteresis vector, constructing a system motion balance equation, solving an equivalent linearization coefficient, constructing a covariance matrix of random response of the concrete pier structure, performing random response analysis on the concrete pier structure, and exploring typical working conditions of random excitation. According to the method, the operation time is shortened, the calculation precision is improved, and meanwhile, the response characteristics of the reinforced concrete pier structure under nonlinear earthquake random excitation can be truly represented.

Description

Technical Field

[0001] The present invention belongs to the field of random vibration analysis of civil engineering structures, and in particular relates to a nonlinear seismic random response analysis method of bridge piers based on a BWBN model. Background Art

[0002] As key hubs in regional highway transportation networks, ensuring their seismic performance and structural safety is of paramount engineering importance. Reinforced concrete (RC) piers, crucial load-bearing components supporting bridge superstructures, exhibit complex nonlinear seismic response characteristics and failure mechanisms under strong earthquakes, making them susceptible to damage and failure, jeopardizing the overall service safety of the bridge structure. Therefore, research on the seismic performance of RC piers has become a key and challenging issue in modern bridge seismic theory.

[0003] Currently, the nonlinear time-history analysis (NTHA) method based on the fiber beam model is commonly used to address the nonlinear dynamic response of RC bridge pier structures under strong earthquakes. This method simulates the failure process of RC bridge pier structures under deterministic excitation. However, as a nonlinear analysis method based on discrete elements, the fiber beam model's core lies in the layered numerical integration of discretized cross-sections, which significantly increases the computational complexity. In addition, the stress-strain relationship of the material under different states affects the analysis results, requiring the precise definition of material parameters and the establishment of a complex nonlinear constitutive model, resulting in a significant contradiction between computational efficiency and model accuracy. Therefore, under strong earthquakes, accurately simulating the mechanical behavior of RC bridge pier structures after they enter the nonlinear state using a refined numerical model is crucial for evaluating the seismic safety of RC bridge pier structures. The Bouc-Wen-Baber-Noori (BWBN) hysteretic model, derived from the Bouc-Wen hysteretic model, has been widely used in the engineering field because it can comprehensively consider hysteretic characteristics such as strength degradation, stiffness degradation, and pinching effect. By introducing a highly nonlinear and parameterized constitutive relationship, the BWBN hysteretic model effectively avoids stress assessment at the cross-sectional level and can simulate the nonlinear mechanical behavior of different cross-sections using a unified differential equation through calibrated hysteresis parameters, significantly reducing computational complexity.

[0004] At the same time, to ensure effective deterministic results for nonlinear time-history analysis of RC bridge pier structures, a large sample of seismic time-history excitations should be selected, and the nonlinear dynamic equations should be solved through stepwise integration. This makes the nonlinear time-history analysis process very time-consuming and limits the efficiency of the analysis. More importantly, the output of nonlinear time-history analysis is a time-history curve (such as displacement and acceleration), which requires post-processing to extract statistical information (such as peak value and envelope), but it is difficult to directly obtain the probability distribution of the response or reliability indicators. It is important to note that seismic excitation is essentially a nonstationary random process. The nonlinear random response analysis (NRSRVA) of RC bridge pier structures under nonstationary excitation can reflect the actual response characteristics. Because it is based on a frequency-domain method, the statistical characteristics of the seismic motion are characterized by the power spectral density (PSD), and the root mean square value and peak distribution of the response can be directly obtained, eliminating the need for stepwise integration in the time domain. This improves computational efficiency while ensuring analysis accuracy.

[0005] However, the non-stationary random response analysis of RC bridge pier structures has traditionally been simplified to a single degree of freedom (SDOF) system, while the analysis of the non-stationary random response of multi-degree-of-freedom (MDOF) systems remains insufficient. In fact, the non-stationary random response analysis of multi-degree-of-freedom (MDOF) systems overcomes the limitations of SDOF systems in terms of missing vibration modes, excitation modeling, and reliability quantification by constructing a multidimensional random process framework. It can also capture the dynamic interactions between different degrees of freedom and consider the non-uniform damage of various parts of the RC bridge pier structure (such as displacement angle distribution and plastic hinge distribution), providing accurate solutions for the non-stationary random response analysis and seismic performance assessment of RC bridge pier structures.

[0006] When analyzing the nonstationary random response of RC bridge pier structures, analytical solutions to the nonlinear system's differential equations of motion are difficult to obtain directly due to their mathematical complexity, requiring the use of specialized approximate solutions. Compared to traditional methods such as Markov analysis (MCA) and standard Monte Carlo simulation (MCS), the high-order stochastic equivalent linearization (SEL) method utilizes statistical linearization and stochastic averaging to determine the power spectral density (PSD) of the system response, avoiding the generation of a large number of random samples. This method offers significant advantages in computational efficiency and parameter sensitivity, making it suitable for analyzing the random response of MDOF hysteretic systems of RC bridge pier structures under nonstationary excitation. Existing research has primarily focused on solving the nonlinear hysteretic behavior of the BWBN model when the smoothness parameter n = 1 is used. To truly reflect the hysteretic characteristics of RC bridge pier structures, it is urgent to consider the nonlinear hysteretic behavior of the BWBN hysteretic model when it exhibits higher-order smoothness (n = 2). Furthermore, the high-order SEL method is employed to iteratively solve the equivalent stiffness and damping matrices to improve analytical accuracy. Summary of the Invention

[0007] In response to the above-mentioned deficiencies in the prior art, the present invention provides a nonlinear seismic random response analysis method for bridge piers based on the BWBN model, which solves the problems of the existing nonlinear seismic random response analysis method for bridge piers, such as long running time, low calculation accuracy, and inability to truly characterize the response characteristics of reinforced concrete bridge pier structures under nonlinear seismic random excitation.

[0008] In order to achieve the above objectives, the present invention adopts a technical solution: a nonlinear seismic random response analysis method of bridge piers based on the BWBN model, comprising the following steps: S1. A concrete bridge pier structure with a preset pier height is selected as the object of seismic random response analysis. The concrete bridge pier structure is discretized and a finite element nonlinear analysis numerical model considering the hysteresis effect is constructed. S2. Combining linear and nonlinear differential equations, a BWBN model is constructed to obtain the influence law of the hysteresis curve, and the hysteresis parameter value of the BWBN model is set to obtain a characterized BWBN model. The Euler-Bernoulli beam element is used for simulation to establish a degenerate beam finite element model; S3. Obtain the unit hysteresis vector, and construct the system motion equilibrium equation based on the degenerate beam finite element model, solve the equivalent linearization coefficient, and construct a high-order random equivalent linearization form of the hysteretic motion equation based on the BWBN model by defining the first state vector; S4. Based on the random process theory and the high-order random equivalent linearization form of the hysteretic motion equation, a random process model of earthquake motion is constructed by defining the second state vector. Regression analysis is performed to obtain the site condition parameters. The covariance matrix of the random response of the concrete pier structure is obtained by setting and updating the augmented state vector. S5. Based on the finite element nonlinear analysis numerical model, the covariance matrix of the random response of the concrete pier structure, and the site condition parameters, the random response analysis of the concrete pier structure is carried out, and the typical working conditions of random excitation are explored to complete the nonlinear seismic random response analysis of the pier.

[0009] The beneficial effects of the present invention are as follows: based on high-order smooth characteristics and a parameterized BWBN hysteresis model, the present invention constructs a refined numerical model that can effectively simulate the bending moment-curvature of a multi-degree-of-freedom concrete pier structure, significantly improving the simulation accuracy of the nonlinear complex hysteretic behavior of the concrete pier structure, while effectively improving the efficiency of numerical calculations, reducing the running time of the seismic random response analysis method, improving the calculation accuracy, and realizing the true characterization of the response characteristics of the reinforced concrete pier structure under nonlinear seismic random excitation.

[0010] Furthermore, the S1 includes the following steps: S101. Select a concrete pier structure with a preset pier height, discretize the concrete pier structure using a cantilever column model, and simulate it using a nonlinear beam element to obtain a simulated concrete pier model. S102. Based on the simulated concrete pier model, the sum of the half-span masses of the adjacent span main beams is used as an equivalent mass, and the equivalent mass is applied to the top of the concrete pier in a discrete distribution manner to obtain a processed concrete pier model. S103. Apply a constant vertical force to the treated concrete pier model, establish a bending moment and curvature relationship curve, use an idealized bifold line model to perform equivalent simplification on the bending moment and curvature relationship curve to obtain a simplified bending moment and curvature relationship curve, and use the energy equivalence principle to calculate the equivalent yield curvature and the ultimate curvature; S104. Use the equivalent yield curvature to normalize the hysteretic curvature to obtain the normalized hysteretic curvature. Based on the processed concrete pier model, the equivalent yield curvature, the limit curvature and the normalized hysteretic curvature, a finite element nonlinear analysis numerical model considering the hysteretic effect is constructed.

[0011] Furthermore, the S2 includes the following steps: S201. Combining linear and nonlinear differential equations, a BWBN model is constructed. By setting the hysteresis parameter value, the hysteresis constitutive relation of the bending moment and curvature is characterized, and the influence law of the hysteresis curve is obtained; S202. According to the physical meaning of each hysteresis parameter and the influence of the hysteresis curve, the hysteresis parameter values ​​of the BWBN model are set, and the hysteresis constitutive relationship of the bending moment and curvature is characterized using each hysteresis parameter to obtain a characterized BWBN model. S203. Based on the discretized concrete pier structure, a consistent double-node Euler-Bernoulli beam element is used for simulation. Combined with the characterized BWBN model, a degenerate beam finite element model with the hysteretic evolution equation of the BWBN model is established.

[0012] Furthermore, the S203 includes the following steps: S2031. Based on the discretized concrete pier structure and Euler-Bernoulli, calculate the cross-sectional curvature of the beam end and the hysteretic relationship between the inelastic bending moment and curvature, and construct the plastic constitutive relationship for axial deformation. S2032. Introduce additional hysteresis degrees of freedom to obtain node displacements and hysteresis degrees of freedom vectors, and construct a displacement field within the unit. Obtain the total curvature and the deformation form of the central axis through the displacement field. S2033. Construct a coordinated interpolation function based on the plastic effect distributed along the length of the element; S2034. Using the principle of virtual work, combining virtual generalized strain with virtual node displacement vector and external node load vector, a virtual relationship is derived; S2035. Substitute the plastic constitutive relation related to the inelastic bending moment and curvature hysteresis relationship and the axial deformation into the virtual relation, and combine the coordinated interpolation function and the variation form of the total curvature and the central axis to obtain the constitutive matrix relationship; S2036. Using the transformation matrix, transform the constitutive matrix relationship to the global coordinate system to obtain the constitutive matrix relationship of the global coordinates; S2037. Based on the constitutive matrix relationship of the global coordinates and combined with the characterized BWBN model, a degenerate beam finite element model with the hysteretic evolution equation of the BWBN model is obtained.

[0013] The beneficial effects of this further approach are: constructing a multi-degree-of-freedom random process framework, breaking through the limitations of single-degree-of-freedom systems in terms of missing vibration modes, excitation modeling, and reliability quantification. At the same time, it can capture the dynamic interactions between different degrees of freedom and consider the non-uniform damage of various parts of the concrete pier structure, providing an accurate solution for the non-stationary random response analysis and seismic performance evaluation of concrete pier structures.

[0014] Furthermore, the S3 includes the following steps: S301. Ignore the axial nonlinear behavior of the concrete pier structure and the hysteretic strain corresponding to the axial deformation to obtain the unit hysteresis vector. Based on the degenerate beam finite element model, combined with the unit connection topology, mass distribution, and boundary constraints, construct the system motion equilibrium equation to obtain the hysteretic curvature vector. S302, performing high-order equivalent linearization on the hysteresis curvature vector to obtain an equivalent linearization coefficient; S303. Ignore the partial derivatives of the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter in the BWBN model, and replace them with the mean approximation to obtain the expected values ​​of the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter. S304: In response to the expected values ​​of the calculated strength degradation parameter, the stiffness degradation parameter, the degree pinching parameter, and the width pinching parameter, a partial derivative operation is performed on the equivalent linearization coefficient, and a formula is derived using the curvature variation and the hysteresis curvature to obtain an expression for the partial derivative coefficient; S305, defining a first state vector, and converting the system motion equation into a state space equation; S306. Combining the expected values, partial derivative coefficient expressions and state-space equations, a high-order stochastic equivalent linearized form of the hysteretic motion equation based on the BWBN model is constructed.

[0015] The beneficial effects of the above-mentioned further scheme are as follows: the present invention utilizes statistical linearization and random averaging through the experimental high-order random equivalent linearization method to avoid the generation of a large number of random samples, and shows significant advantages in computational efficiency and parameter sensitivity. It is suitable for the random response analysis of concrete pier structures under non-stationary excitation in multi-degree-of-freedom hysteresis systems of concrete pier structures.

[0016] Furthermore, the S4 includes the following steps: S401. Based on random process theory and the high-order random equivalent linearization form of the hysteretic motion equation, the seismic acceleration function and time modulation function are constructed. The stationary power spectral density function under earthquake excitation is constructed as a Kanei Kiyoshi model of a series second-order filter. A shaping filter excited by white noise is introduced, and the second state vector is defined to construct a random process model of seismic motion. S402. Based on representative strong earthquake records and the conditions of each site, the mean, standard deviation, and quantile of time domain parameters and frequency domain parameters are calculated respectively. A prediction equation is established for regression analysis. Using a statistical distribution estimation method, characteristic ground motion parameters corresponding to the site conditions are obtained. Filter parameters related to soil stiffness are selected to obtain site condition parameters. S403, setting an augmented state vector, obtaining a covariance matrix of the augmented state vector, and dynamically iteratively updating the equivalent matrix by solving the Lyapunov function to obtain a covariance matrix of the random response of the concrete pier structure.

[0017] Furthermore, the step S401 includes the following steps: S4011. Based on the theory of random processes and the high-order random equivalent linearization form of the hysteretic motion equation, the ground acceleration function and time modulation function are constructed by ignoring the time-varying characteristics of the frequency. S4012. Based on the CP model, a bilateral evolutionary power spectral density function of earthquake excitation is constructed. Using the Jinjingqing model with a series second-order filter, a stationary power spectral density function of earthquake excitation in the CP model is constructed. The power spectrum intensity of the input white noise random process in the Jinjingqing model is related to the peak acceleration of the ground motion. S4013. Introducing a shaping filter, based on the power spectrum intensity of a constant input white noise random process, constructing an expression for the joint response process of a bilinear filter under white noise excitation; S4014. Define a second state vector, convert the expression of the joint response process of the bilinear filter under white noise excitation and the seismic acceleration function into a state space equation, and construct a seismic random process model.

[0018] The beneficial effect of this further solution is that, by treating seismic excitation as a nonstationary random process, the nonlinear random response analysis of concrete pier structures under nonstationary excitation can reflect the actual response characteristics. Furthermore, by characterizing the statistical characteristics of seismic motion using power spectral density, the root mean square value and peak distribution of the response can be directly obtained, eliminating the need for stepwise integration in the time domain. This results in higher computational efficiency while maintaining analytical accuracy.

[0019] Furthermore, the S403 includes the following steps: S4031. Set the augmented state vector, establish a joint expression of the system and the excitation, and obtain the covariance matrix of the augmented state vector by solving; S4032. Setting the initial state of the covariance matrix of the augmented state vector, and obtaining the Lyapunov function of the dynamic response by recursion based on the backward difference method; S4033. By solving the Lyapunov function and dynamically iteratively updating the equivalent matrix, in response to the covariance matrix difference between the current and subsequent iterative augmented state vectors meeting the preset error accuracy requirement, the covariance matrix of the random response of the concrete pier structure is output.

[0020] The beneficial effect of the above further scheme is: the present invention uses a high-order stochastic equivalent linearization method and a steady-state solution of the Lyapunov equation to achieve iterative updating of the equivalent stiffness and damping matrices, thereby reducing the linearization cumulative error and improving the analysis accuracy.

[0021] Furthermore, the S5 includes the following steps: S501. Based on the finite element nonlinear analysis numerical model, the covariance matrix of the random response of the concrete pier structure, and site condition parameters, and using the Kanei Kiyoshi model of a series second-order filter, the random response characteristics of the structure under specific earthquake excitation intensities are explored. The random response analysis of the concrete pier structure is performed under strong earthquake excitation with peak acceleration corresponding to the power spectrum intensity, and the root mean square value of the pier bottom curvature and the average hysteresis time history of the pier bottom are obtained. S502. Using incremental dynamic analysis, the dual parameters of pier bottom curvature and pier top displacement are used as indicators for evaluating the seismic performance of concrete pier structures. S503. Using typical working conditions of random excitation, conduct comparative analysis to obtain differentiated contribution characteristics of basic vibration modes and higher-order vibration modes to the RMS peak value of the pier body curvature and the RMS peak value of the pier body displacement; S504. Select the random response under hard soil conditions, and based on the site condition parameters and the typical working conditions of random excitation, explore the nonlinear behaviors of the curvature response and displacement response of the concrete pier structure in different steps; S505. Based on strong earthquake excitation, sensitivity analysis of the hysteresis parameters of the bridge pier is performed to obtain the effect of the RMS peak value of the pier bottom curvature and the average hysteresis energy dissipation under specific excitation intensity, and complete the nonlinear seismic random response analysis of the bridge pier.

[0022] The beneficial effect of the above further scheme is that the present invention reveals the differentiated influences of different hysteresis parameters and site conditions on the random response of concrete pier structures, and provides a nonlinear random response analysis framework and targeted parameter optimization basis for concrete pier structures under strong earthquakes. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 Flow chart of the method of the present invention.

[0024] Figure 2 This is a numerical model diagram of the concrete pier structure in this embodiment.

[0025] Figure 3 This is the bending moment-curvature curve of the pier bottom section in this embodiment.

[0026] Figure 4 This is the finite element diagram of the Euler-Bernoulli beam in this embodiment.

[0027] Figure 5 Graph showing the power spectrum density function under different soil conditions in this embodiment.

[0028] Figure 6 Graphs showing the root mean square time history of the pier bottom curvature and the average hysteretic energy dissipation time history considering different hysteretic parameters under hard soil conditions in this embodiment.

[0029] Figure 7 Graphs showing the root mean square time history of pier bottom curvature and average hysteretic energy dissipation time history under hard soil, medium soil and soft soil conditions in this embodiment.

[0030] Figure 8 Graphs showing the incremental dynamic analysis curve of the pier bottom curvature and the locally enlarged incremental dynamic analysis curve of the curvature in this embodiment.

[0031] Figure 9 Graphs showing the incremental dynamic analysis curve of the pier top displacement and the incremental dynamic analysis curve of the local amplified displacement in this embodiment.

[0032] Figure 10 : is the curvature RMS peak distribution diagram along the height direction of the typical working condition in this embodiment.

[0033] Figure 11 : is the peak value distribution diagram of the displacement RMS along the height direction under the typical working condition in this embodiment.

[0034] Figure 12 This is the distribution diagram of the curvature and displacement root mean square response along the pier height in this embodiment. DETAILED DESCRIPTION

[0035] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

[0036] Before describing this embodiment, the following terms are explained: Euler–Bernoulli theory: Euler-Bernoulli theory; BWBN model: Bouc-Wen-Baber-Noori model, used to describe the hysteretic behavior of structural components under cyclic loading and ground motion excitation; SEL: equivalent linearization; Lyapunov function: Lyapunov function; RMS Peak: Root mean square peak value.

[0037] Example like Figure 1 As shown, the present invention provides a nonlinear seismic random response analysis method for bridge piers based on the BWBN model, and its implementation method is as follows: S1. A concrete bridge pier structure with a preset pier height is selected as the object of seismic random response analysis. The concrete bridge pier structure is discretized, and a finite element nonlinear analysis numerical model considering the hysteresis effect is constructed.

[0038] In this embodiment, a continuous beam bridge is used as the prototype, and the pier height is selected as the preset height. H The RC bridge pier structure is taken as the object of seismic random response analysis, and the following Figure 2 The finite element nonlinear analysis numerical model shown in the figure considers the hysteresis effect, ignores the pile-soil interaction effect, and regards the pier bottom constraint as a consolidated state. The specific steps for constructing the finite element numerical model of the RC bridge pier structure are as follows: The RC pier structure is discretized using the cantilever column model. The discretization divides the RC pier structure into The unit is equal in length and the height of each unit is cc. The nonlinear beam unit based on the BWBN hysteresis model is used for detailed simulation. Considering the influence of the mass of the main beam of the adjacent span on the pier column, the sum of the mass of the half spans of the adjacent spans, M1, is taken and concentrated on the pier top. Since the mass of the RC pier body is greater than the mass of the superstructure, the equivalent mass M1 is discretely distributed to the pier top node position; like Figure 3 As shown, a constant vertical axial force is applied and the elastic-plastic bending moment of the pier bottom section is established based on the existing section software. M -Curvature Relationship curve, and the idealized bifold line model is used to calculate the bending moment M -Curvature The relationship curve is simplified and the equivalent yield curvature of the pier bottom section is determined based on the energy equivalence principle. and limiting curvature ; The equivalent yield curvature obtained by calculation The plastic capacity of the pier bottom section is characterized. To facilitate analysis, this parameter is used to normalize the hysteretic curvature, thereby eliminating the influence of dimension and order of magnitude. At the same time, the ductility demand index can be directly obtained, and a finite element numerical model of the RC pier structure can be constructed.

[0039] S2. Combine linear and nonlinear differential equations to construct a BWBN model, obtain the influence law of the hysteresis curve, and set the hysteresis parameter value of the BWBN model to obtain a characterized BWBN model. Use the Euler-Bernoulli beam element for simulation to establish a degenerate beam finite element model. The specific steps are as follows: S201. Combining linear and nonlinear differential equations, a BWBN model is constructed. By setting the hysteresis parameter value, the hysteresis constitutive relationship between the bending moment and curvature is characterized, and the influence law of the hysteresis curve is obtained.

[0040] In this embodiment, a BWBN hysteresis model with strength degradation, stiffness degradation and pinching effect is constructed based on the MATLAB platform to accurately simulate the bending moment of the RC pier structure. M -Curvature Nonlinear hysteretic behavior; The BWBN hysteretic model combines the linear and nonlinear parts, the bending moment M -Curvature The specific expression of the relationship is as follows: ; in, represents the internal bending moment, represents the time step of the random response analysis, represents the yield curvature, express, represents the curvature, represents the equivalent yield curvature, represents the hysteresis component of curvature, represents the elastic-plastic stiffness ratio, represents the initial bending stiffness; The hysteretic curvature is governed by a nonlinear differential equation expressed as follows: ; in, represents the hysteresis component of the curvature after time differentiation, represents the curvature after derivative with respect to time, represents the pinching effect parameter under bending, represents the stiffness degradation parameter under bending, represents the strength degradation parameter under bending, It represents the parameter that controls the smoothness of the hysteresis curve. represents the parameter that controls the peak value of the hysteresis curve, represents the parameter that controls the shape of the hysteresis curve, Represents the sign function and obtains the influence law of the hysteresis curve.

[0041] In this embodiment, based on the theoretical assumption that the degradation rate is linearly related to the hysteresis energy dissipation rate, the strength degradation parameter under bending is and the stiffness degradation parameter under bending , expressed as: ; ; in, represents the strength degradation rate, Indicates the stiffness degradation rate. When the strength degradation rate and stiffness degradation rate are set to zero, it means there is no strength and stiffness degradation. It represents the accumulated hysteresis energy dissipation, and its expression is as follows: ; in, represents the integration of the hysteresis loop; The pinching effect is mainly caused by the damage and interaction of structural components under large deformation, as well as the closed or unclosed yielding of the compressive reinforcement before the cracks in the RC components are closed. Using the BWBN model, the pinching effect is simulated as follows: ; ; ; ; in, Indicates the degree of pinching effect, Indicates the width of the pinch diffusion area. represents the final value of the hysteresis component of curvature, which is obtained by Solved, 、 、 、 、 、 All represent constant parameters, Used to control the total slip, Used to control the pinch slope, Used to control the starting position of the pinch effect, Used to control the pinching range. Used to control the pinching rate, Used to control the magnitude relationship between the total pinch amount and the pinch rate.

[0042] S202. According to the physical meaning of each hysteresis parameter and the influence rule of the hysteresis curve, the hysteresis parameter value of the BWBN model is set, and the hysteresis constitutive relationship of the bending moment and curvature is characterized by using each hysteresis parameter to obtain a characterized BWBN model.

[0043] In this embodiment, the hysteresis parameter value of the BWBN model is reasonably set to accurately characterize the bending moment M-curvature of the RC pier structure. The hysteretic constitutive relationship between them can realize high-precision simulation of complex hysteretic behavior; The values ​​of each hysteresis parameter are determined based on the range of values ​​given by a large number of existing studies, according to the physical meaning of each hysteresis parameter and the influence of the plastic characteristic parameters on the hysteresis curve, including: , , , , , , , , , , , .

[0044] S203. Based on the discretized concrete pier structure, a consistent double-node Euler-Bernoulli beam element is used for simulation. Combined with the characterized BWBN model, a degenerate beam finite element model with the hysteretic evolution equation of the BWBN model is established.

[0045] In this embodiment, Figure 4As shown, for the equal-length elements divided above, a consistent two-node Euler-Bernoulli beam element is used for simulation, and a degenerate beam finite element numerical model with hysteretic evolution equations is established on the MATLAB platform. The purpose is to accurately simulate the nonlinear mechanical behavior of RC bridge pier structures under complex load states. This beam element contains two nodes 1 and 2. As a planar structural configuration, a single node has three degrees of freedom (longitudinal, transverse translation, and rotation), resulting in a total of six degrees of freedom for the element as a whole. The specific steps for constructing the degenerate beam finite element numerical model with hysteretic evolution equations are as follows: According to the basic assumptions of Euler–Bernoulli theory, the distance from the beam end is x The curvature of the cross section at can be expressed as: ,in w represents the lateral deflection of the beam; For distance from beam end x The inelastic bending moment of a specific cross section at -Curvature The hysteresis relationship can be expressed as: ; ; in, Indicates the distance from the beam end x The lateral deflection of the beam at ; For the plastic constitutive relation related to axial deformation, a similar method can be used for theoretical construction. The mathematical expression of the plastic constitutive relation of axial deformation is as follows: ; ; in, represents the axial force, represents the elastic-plastic axial stiffness ratio, represents the elastic modulus of concrete, represents the cross-sectional area, represents the axial centerline strain, represents the axial yield deformation, represents the axial pinching effect parameter, represents the axial stiffness degradation parameter, It represents the axial strength degradation parameter and realizes the implicit characterization of the coupling effect of axial force and bending moment; Constitutive models are used to determine inelastic behavior by establishing a relationship between stress and generalized displacement. This approach can accurately and efficiently simulate hysteretic cyclic behavior without having to return to the cross-section level for stress evaluation, significantly reducing run times.

[0046] In this embodiment, four additional hysteretic degrees of freedom are introduced to consider the axial and bending hysteretic deformations of each element. The node displacement and hysteretic degrees of freedom vector are expressed as follows: ; ; in, Represents the node displacement vector in the local coordinate system, including the longitudinal displacement , lateral displacement and corners , Represents the four additional hysteretic degrees of freedom vectors, which are composed of the axial hysteretic deformation and bending hysteresis deformation Composition, subscripts 1 and 2 represent the starting node and ending node of the corresponding unit respectively; Construct the displacement field within the unit cell, and the expression is as follows: ; in, , , , , , is a cubic polynomial shape function, represents the length of the structural unit. The displacement field expression describes a polynomial interpolation method. The total curvature and central axial deformation obtained through the displacement field are shown as follows: ; in, Indicates the distance from the beam end x The axial strain on the cross section, Indicates the distance from the beam end x The curvature of the cross section at x and xx Represents the spatial variables x The first and second derivatives of .

[0047] In this embodiment, in order to meet the equilibrium condition and based on the plastic effect distributed along the length of the element, a coordinated interpolation function is constructed. The expression is as follows: ; ; Using the principle of virtual work, combined with virtual generalized strain, virtual node displacement vector and external node load vector, the virtual relationship is derived. The specific expression is as follows: ; ; in, represents the virtual node displacement vector, represents the external nodal load vector, and Indicates the distance from the beam end x The relevant internal virtual generalized strain of the cross section at represents the axial force at node 1 of the beam element, represents the axial force at node 2 of the beam element, represents the torque at node 1 of the beam element, represents the torque at node 2 of the beam element, represents the bending moment at node 1 of the beam element, represents the bending moment at node 2 of the beam element.

[0048] In this embodiment, the plastic constitutive relation related to the inelastic bending moment and curvature hysteresis and axial deformation is brought into the virtual relation, and combined with the coordinated interpolation function and the variation of the total curvature and the central axis, the constitutive matrix relationship is obtained, as shown below: ; ; ; in, represents the element stiffness matrix associated with the elastic part, represents the displacement vectors of the nodes at both ends, represents the unit matrix associated with the hysteresis part, represents the hysteresis vector associated with axial strain and curvature; The constitutive matrix relationship is characterized by decoupling the axial force, bending moment, and shear force of the elastic part from the hysteretic part. This relationship is transformed into the global coordinate system through the transformation matrix, and its expression is as follows: ; ; ; ; ; in, represents the standard coordinate transformation matrix, represents the element node load vector in the global coordinate system, represents the unit node displacement vector in the global coordinate system, represents the element stiffness matrix in the global coordinate system, represents the hysteresis matrix in the global coordinate system, represents elastic behavior based on stiffness reduction, represents the hysteresis part; According to the constitutive matrix relationship of the global coordinates and combined with the characterized BWBN model, a degenerate beam finite element model with the hysteretic evolution equation of the BWBN model is obtained.

[0049] S3. Obtain the unit hysteresis vector and construct the system motion equilibrium equation based on the degenerate beam finite element model. Solve the equivalent linearization coefficient and construct a high-order random equivalent linearization form of the hysteretic motion equation based on the BWBN model by defining the first state vector. The specific steps are as follows: S301. Ignore the axial nonlinear behavior and hysteresis strain corresponding to the axial deformation of the concrete pier structure to obtain the unit hysteresis vector. Based on the degenerate beam finite element model, combined with the unit connection topology, mass distribution and boundary constraints, construct the system motion equilibrium equation and obtain the hysteresis curvature vector.

[0050] In this embodiment, to improve computational efficiency, the axial nonlinear behavior of the concrete pier structure and the hysteretic strain corresponding to the axial deformation are ignored. Based on the degenerate beam finite element model, the system motion equilibrium equation is established. The equivalent linearization coefficient under zero-mean Gaussian random excitation is solved using the high-order SEL method on the MATLAB platform. The high-order random equivalent linearization form of the hysteretic motion equation based on the BWBN model is constructed by defining the first state vector. The specific steps are as follows: The seismic response analysis of RC pier structures does not consider its nonlinear behavior along the axial direction and ignores the hysteresis deformation corresponding to the axial deformation, that is, setting , based on this, i Units ( ) is only related to the hysteresis curvature and is expressed as follows: ; in, Indicates the i The unit hysteresis vector of the unit has the dimension , and The subscripts 1 and 2 correspond to the starting node and ending node of the unit respectively; For those with n A specific planar structural system with degrees of freedom (DOFs) is given After connecting the topology, mass distribution and boundary constraints of each unit, the motion equilibrium equation is constructed, and its expression is as follows: ; ; in, represents the structural acceleration response vector, represents the velocity response vector, represents the displacement response vector, represents the consistent mass matrix in the global coordinate system of the structure, represents the Rayleigh damping matrix in the global coordinate system of the structure, represents the stiffness matrix in the global coordinate system of the structure, represents the hysteresis matrix in the global coordinate system, Indicates earthquake motion The influence vector of The hysteresis curvature vector represents the hysteresis characteristic. The specific expression is as follows: ; in, represents the hysteresis curvature vector considering the hysteresis characteristics, represents the first derivative of curvature with respect to time.

[0051] S302. Perform high-order equivalent linearization on the hysteresis curvature vector to obtain an equivalent linearization coefficient.

[0052] In this embodiment, in order to adapt to the state space equation and calculate the nonlinear random response, it is necessary to characterize the hysteresis curvature vector The basic principle of SEL is to replace the nonlinear equations with a set of equivalent linear equations to minimize the difference between the two at the statistical level. The coupled evolution equations are the high-order stochastic equivalent linearized form of the hysteretic motion equation based on the BWBN model, and their expressions are as follows: ; in, represents the equivalent damping matrix corresponding to the hysteretic degrees of freedom of the structure, represents the equivalent stiffness matrix corresponding to the hysteretic degree of freedom of the structure; in order to obtain the equivalent linearization coefficient, it is assumed that the response and Approximately obeying the Gaussian distribution, the equivalent linearization coefficient obtained by the high-order SEL method is as follows: ; ; in, Indicates the expected value, i , j Respectively represent the unit i Duanhe j end.

[0053] S303. Ignore the partial derivatives of the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter in the BWBN model, and use mean approximation to replace the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter to calculate the expected values ​​of the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter.

[0054] In this embodiment, considering the BWBN model, the strength degradation parameter , stiffness degradation parameter , degree pinching parameters and the width pinch parameter , both and function of strength degradation parameters, but these parameters change slowly. , stiffness degradation parameter , degree pinching parameters and the width pinch parameter The partial derivatives of are ignored, and the parameters are replaced by 、 、 、 ,in, ; ; ; ; in, represents the mean hysteresis energy consumption, and its expression is: .

[0055] S304. In response to the expected values ​​of the calculated strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter, perform partial derivative operations on the equivalent linearization coefficients, and use the curvature variation and hysteresis curvature to derive formulas to obtain partial derivative coefficient expressions.

[0056] In this embodiment, when calculating the aforementioned expected values, it is assumed that the response process obeys a zero-mean Gaussian distribution, and the non-Gaussian response characteristics that may be caused by nonlinear terms are ignored; partial derivative operations are performed on the equivalent linearization coefficients, and formulas are derived using the curvature change and the hysteresis curvature to obtain the partial derivative coefficient expressions; The expression for performing partial derivative operation on the equivalent linearization coefficient is as follows: ; ; Among them, combined with the curvature change and hysteretic curvature The zero-mean value of the Gaussian probability density function is combined to obtain the linearization coefficient that characterizes the plastic characteristics of the pier. and The specific analytical expression of ; The specific expression is as follows: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; in, Indicates the rate of change of curvature The root mean square value of represents the hysteresis curvature The root mean square value of for and The correlation coefficient value of represents the integrand of n The second power arrive The integral difference within the range is given by the expression ,in, represents the integration domain parameter, defined as , by the correlation coefficient The algebraic relationship is uniquely determined. 、 、 、 、 、 as well as All represent constants; represents the Gauss-Laguerre quadrature formula, which is expressed as follows: ; Represents the standard summation formula, the expression is as follows: ; ; in, and Both represent the independent variables to be quadratured. The value of is usually 1 or -1. represents the coefficient of the quadratic term, represents the gamma function, represents the symbolic function, represents the complementary error function, represents the incomplete gamma function, represents the argument of the general gamma function, represents the upper limit of the integral of the gamma function; The constant in the above formula is 、 、 、 、 、 as well as The expression is as follows: , , ; , ; , ; in, represents the parameter that controls the starting position of the pinching effect in the BWBN hysteresis model, represents the final value of the curvature hysteresis component in the BWBN hysteresis model, represents the pinching parameter after the first-order approximation.

[0057] S305, defining a first state vector, and converting the system motion equation into a state space equation; S306. Combining the expected values, partial derivative coefficient expressions and state-space equations, a high-order stochastic equivalent linearized form of the hysteretic motion equation based on the BWBN model is constructed.

[0058] In this embodiment, the first state vector is defined as , transforming the system motion equations into state space equations; Combining the expected values, partial derivative coefficient expressions and state-space equations, a high-order stochastic equivalent linearization form of the hysteretic motion equation based on the BWBN model is constructed.

[0059] S4. Based on the random process theory and the high-order random equivalent linearization form of the hysteretic motion equation, a random process model of earthquake motion is constructed by defining the second state vector. Regression analysis is performed to obtain the site condition parameters. The covariance matrix of the random response of the concrete pier structure is obtained by setting and updating the augmented state vector. The specific steps are as follows: S401. Based on the random process theory and the high-order random equivalent linearization form of the hysteretic motion equation, the seismic acceleration function and time modulation function are constructed. The stationary power spectral density function under seismic excitation is constructed as the Jin Jingqing model of a series second-order filter, and a shaping filter excited by white noise is introduced. By defining the second state vector, a random process model of seismic motion is constructed.

[0060] In this example, based on random process theory, ground motion acceleration and time modulation functions were constructed on the MATLAB / SIMULINK platform. The stationary PSD function under earthquake excitation was constructed as a Jinjingqing model with a series second-order filter. A shaping filter excited by white noise was introduced. By defining the second state vector, a ground motion random process model with clear physical meaning was constructed. The specific steps are as follows: Ignoring the time-varying characteristics of frequency, the earthquake excitation is regarded as a zero-mean uniformly modulated stationary Gaussian random process. The expression is as follows: ; in, represents the time modulation function, which is used to characterize the time-varying intensity characteristics of non-stationary random processes. represents a stationary random process; The expression is: , It represents the morphological constant related to magnitude and epicentral distance, and its value is determined based on the statistical analysis corresponding to the local site conditions.

[0061] In this embodiment, in seismic analysis, the frequency domain-based method is preferred to represent the random process. Therefore, the bilateral evolution PSD function of the seismic excitation adopts the CP model, which is expressed as follows: ; in, Expressed as The stationary PSD function of Constructed as the Jinjingqing model, the expression is as follows: ; in, represents the power spectrum intensity of the input white noise random process, Indicates the main frequency of the venue corresponding to the high frequency band, Indicates the damping coefficient corresponding to the high frequency band; The power spectrum intensity of the input white noise random process in the Jinjingqing model is compared with the peak acceleration of the earthquake motion. Establish a relationship, the expression is as follows: ; Introducing a shaping filter, It is considered as the joint response process of the bilinear filter under the action of white noise excitation, and its power spectrum amplitude remains constant. , construct the expression of the joint response process of the bilinear filter under white noise excitation; the specific expression is as follows: ; ; in, represents white noise excitation.

[0062] In this embodiment, the second state vector is defined as , the expression of the joint response process of the bilinear filter under white noise excitation and the seismic acceleration function are transformed into a state space equation, and the seismic random process model is constructed.

[0063] S402. Based on representative strong earthquake records and the conditions of each site, the mean, standard deviation, and quantile of the time domain parameters and frequency domain parameters are calculated respectively, and a prediction equation is established for regression analysis. The statistical distribution estimation method is used to obtain the earthquake motion characteristic parameters corresponding to the site conditions, and the filtering parameters related to the soil stiffness are selected.

[0064] In this embodiment, based on representative strong earthquake records, the mean, standard deviation, and quantile of time domain parameters and frequency domain parameters are calculated for each type of site condition (hard soil, medium soil, and soft soil), and a prediction equation is established for regression analysis. The characteristic parameters of the earthquake motion corresponding to the site conditions are determined by the statistical distribution estimation method, that is, the morphological constants related to the magnitude and epicenter distance. , and select the filter parameters related to soil stiffness ; Site conditions have a significant impact on the amplitude and spectral characteristics of the excitation. When a white noise process is used to simulate earthquake excitation, the Gaussian random process after filtering can effectively characterize the earthquake action on the structure, such as Figure 5 As shown in the figure, based on the filtering parameters under different site conditions, the peak ground acceleration (PGA) of three types of site conditions (a hard soil, b medium soil and c soft soil) is calculated. The PSD functions corresponding to the three types of site conditions are shown in Figure 2. The results show that the PSD of the hard soil site exhibits a broadband characteristic. Although the high-frequency energy accounts for a low proportion, it still contains significant high-frequency components. On the other hand, the PSD of the soft soil site exhibits a narrow-band process, with a relatively high proportion of high-frequency energy.

[0065] S403, setting an augmented state vector, obtaining a covariance matrix of the augmented state vector, and dynamically iteratively updating the equivalent matrix by solving the Lyapunov function to obtain a covariance matrix of the random response of the concrete pier structure.

[0066] In this embodiment, the augmented state vector is set , establish the joint expression of system and incentive as follows: ; ; in, , as well as ; The covariance matrix of the augmented state vector is used as a key indicator to evaluate the performance of the RC pier structure. The covariance matrix of the augmented state vector is obtained by solving the equation, and the expression is as follows: ; ; Under the assumption of deterministic initial conditions, the initial state of the covariance matrix is ​​set to zero, and based on the simple and unconditionally stable backward difference method, at each time step Within, recursion is obtained at a given time t The dynamic response of can be solved recursively to obtain the covariance of the non-stationary structural response; when When , the earthquake excitation turns into a stationary random process, and its steady-state solution is directly solved by the following expression: ; The equivalent linearized form is used in the analysis of nonlinear control systems, where the state matrix Depends on the equivalent damping matrix and the equivalent stiffness matrix , the above two equivalent matrices are covariance matrices function, so the given time t The dynamic response expression of is determined by an iterative method, applying the initial conditions of the nonlinear system to the corresponding linear system and determining the initial equivalent matrix and , an iterative update mechanism is established by setting the maximum number of iteration steps to 3000. During the iteration process, the equivalent matrix of the current iteration step is dynamically updated based on the covariance matrix generated by the previous iteration step. and When the difference between the covariance matrices of the two iteration steps meets the error accuracy requirement, that is, when the error accuracy is less than 1E-15, the covariance matrix of the random response (curvature, displacement) of the RC pier structure is output. , the expression is as follows: .

[0067] S5. Based on the finite element nonlinear analysis numerical model, the covariance matrix of the random response of the concrete pier structure, and the site condition parameters, the random response analysis of the concrete pier structure is carried out, and the typical working conditions of random excitation are explored to complete the nonlinear seismic random response analysis of the pier.

[0068] In this embodiment, based on the finite element nonlinear analysis numerical model and the seismic random excitation in S401, the random response characteristics of the structure under a specific seismic excitation intensity are explored, and based on the covariance matrix of the random response of the concrete pier structure, the random corresponding root mean square (RMS) value matrix is ​​obtained. The corresponding peak ground acceleration (PGA) Under the strong earthquake excitation, the random response analysis of RC bridge pier structure is carried out, and the root mean square (RMS) value of the pier bottom curvature and the average hysteretic energy dissipation of the pier bottom of the RC bridge pier structure are obtained. Time course, in-depth study of the influence mechanism of hysteresis parameters and site effects on random response; taking hard soil conditions as an example, based on site condition parameters, such as Figure 6 As shown in the figure, the influence mechanism of hysteresis parameters on random response is explored, and four working conditions are compared and analyzed: 1. Without considering degradation and pinching effects, that is, ; 2. Consider only strength degradation, that is ; 3. Consider both strength and stiffness degradation, i.e. ; 4. Comprehensively consider the degradation and pinching effects, that is, ; Consider the degradation and pinching effects, such as Figure 7 As shown in the figure, the mechanism of site effect on random response is explored.

[0069] In this embodiment, Figure 8 and Figure 9 As shown in the figure, the incremental dynamic analysis (IDA) method is used to take the dual parameters of pier bottom curvature and pier top displacement as indicators to evaluate the seismic performance of RC pier structures, thereby revealing the deformation evolution process of RC pier structures under different earthquake excitation intensities; Select Peak Ground Acceleration (PGA) Conduct full-process random response analysis within the interval, based on the site characteristic frequency and damping ratio , based on the relationship between the power spectrum intensity of the input white noise random process and the peak acceleration of the earthquake, the power spectrum intensity is calculated , achieving peak ground acceleration (PGA) Power spectrum magnitude (PSD) At the same time, based on three types of site conditions (hard soil, medium soil, soft soil) and morphological constants related to magnitude and epicenter distance, , and select the filter parameters related to soil stiffness ,Incremental dynamic analysis (IDA) random response analysis was carried out to explore the influence of site effects on the yield characteristics of RC pier structures.

[0070] In this embodiment, Figure 10 and Figure 11 As shown in the figure, a comparative analysis is conducted on two typical working conditions (Case-I and Case-II) under random excitation. In the figure, a is when the bottom of the pier yields in Case-I, and b is when the middle of the pier yields in Case-II. This further explores the influence mechanism of site conditions on random response and reveals the differentiated contribution characteristics of the fundamental and higher-order vibration modes to the RMS peak value of the pier curvature and the RMS peak value of the pier displacement. There are two typical working conditions under the random excitation. Case-I corresponds to the pier bottom yield condition: When the load reaches 0.70g (hard soil), 0.49g (medium soil) and 0.52g (soft soil), the pier bottom enters the yield state; Case-II represents the yield condition of the middle part of the pier. When the load reaches 0.96g (hard soil), 0.70g (medium soil) and 0.87g (soft soil) respectively, the middle part of the pier body enters the yield state.

[0071] In this embodiment, Figure 12 As shown in the figure, the random response under hard soil conditions is selected as the analysis object, focusing on two typical working conditions, Case-I (yielding at the bottom of the pier) and Case-II (yielding in the middle of the pier body), to explore the nonlinear different behaviors between the curvature response (a) and displacement response (b) of the RC pier structure; It should be noted that by comparing the distribution differences of the curvature and displacement RMS responses along the pier height when the RMS peak of the pier bottom curvature and the RMS peak of the pier top displacement occur, the different behaviors between the curvature response and the displacement response are revealed; Under strong earthquake excitation, the hysteresis parameters of the bridge piers The value of has a significant impact on its hysteresis characteristics. Four types of hysteresis parameters are analyzed through sensitivity analysis to explore their effects on the RMS peak value of the pier bottom curvature and the average hysteresis energy dissipation under specific excitation intensity. The law of action; The value range of each hysteresis parameter is limited to: , , as well as When conducting single parameter sensitivity analysis with reference parameter values, the aforementioned strong earthquake excitation As a sensitivity analysis condition, the nonlinear seismic random response analysis of the bridge pier was completed.

Claims

1. A nonlinear seismic random response analysis method for bridge piers based on the BWBN model, characterized in that: The following steps are involved: S1. A concrete bridge pier structure with a preset pier height is selected as the object of seismic random response analysis. The concrete bridge pier structure is discretized and a finite element nonlinear analysis numerical model considering the hysteresis effect is constructed. S2. Combining linear and nonlinear differential equations, a BWBN model is constructed to obtain the influence law of the hysteresis curve, and the hysteresis parameter value of the BWBN model is set to obtain a characterized BWBN model. The Euler-Bernoulli beam element is used for simulation to establish a degenerate beam finite element model; S3. Obtain the unit hysteresis vector, and construct the system motion equilibrium equation based on the degenerate beam finite element model, solve the equivalent linearization coefficient, and construct a high-order random equivalent linearization form of the hysteretic motion equation based on the BWBN model by defining the first state vector; S4. Based on the random process theory and the high-order random equivalent linearization form of the hysteretic motion equation, a random process model of earthquake motion is constructed by defining the second state vector. Regression analysis is performed to obtain the site condition parameters. The covariance matrix of the random response of the concrete pier structure is obtained by setting and updating the augmented state vector. S5. Based on the finite element nonlinear analysis numerical model, the covariance matrix of the random response of the concrete pier structure, and the site condition parameters, the random response analysis of the concrete pier structure is carried out, and the typical working conditions of random excitation are explored to complete the nonlinear seismic random response analysis of the pier.

2. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 1 is characterized in that: Said S1 comprises the following steps: S101. Select a concrete pier structure with a preset pier height, discretize the concrete pier structure using a cantilever column model, and simulate it using a nonlinear beam element to obtain a simulated concrete pier model. S102. Based on the simulated concrete pier model, the sum of the half-span masses of the adjacent span main beams is used as an equivalent mass, and the equivalent mass is applied to the top of the concrete pier in a discrete distribution manner to obtain a processed concrete pier model. S103. Apply a constant vertical force to the treated concrete pier model, establish a bending moment and curvature relationship curve, use an idealized bifold line model to perform equivalent simplification on the bending moment and curvature relationship curve to obtain a simplified bending moment and curvature relationship curve, and use the energy equivalence principle to calculate the equivalent yield curvature and the ultimate curvature; S104. Use the equivalent yield curvature to normalize the hysteretic curvature to obtain the normalized hysteretic curvature. Based on the processed concrete pier model, the equivalent yield curvature, the limit curvature and the normalized hysteretic curvature, a finite element nonlinear analysis numerical model considering the hysteretic effect is constructed.

3. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 1 is characterized in that: The S2 comprises the following steps: S201. Combining linear and nonlinear differential equations, a BWBN model is constructed. By setting the hysteresis parameter value, the hysteresis constitutive relation of the bending moment and curvature is characterized, and the influence law of the hysteresis curve is obtained; S202. According to the physical meaning of each hysteresis parameter and the influence of the hysteresis curve, the hysteresis parameter values ​​of the BWBN model are set, and the hysteresis constitutive relationship of the bending moment and curvature is characterized using each hysteresis parameter to obtain a characterized BWBN model. S203. Based on the discretized concrete pier structure, a consistent double-node Euler-Bernoulli beam element is used for simulation. Combined with the characterized BWBN model, a degenerate beam finite element model with the hysteretic evolution equation of the BWBN model is established.

4. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 3 is characterized in that: The S203 includes the following steps: S2031. Based on the discretized concrete pier structure and Euler-Bernoulli, calculate the cross-sectional curvature of the beam end and the hysteretic relationship between the inelastic bending moment and curvature, and construct the plastic constitutive relationship for axial deformation. S2032. Introduce additional hysteresis degrees of freedom to obtain node displacements and hysteresis degrees of freedom vectors, and construct a displacement field within the unit. Obtain the total curvature and the deformation form of the central axis through the displacement field. S2033. Construct a coordinated interpolation function based on the plastic effect distributed along the length of the element; S2034. Using the principle of virtual work, combining virtual generalized strain with virtual node displacement vector and external node load vector, a virtual relationship is derived; S2035. Substitute the plastic constitutive relation related to the inelastic bending moment and curvature hysteresis relationship and the axial deformation into the virtual relation, and combine the coordinated interpolation function and the variation form of the total curvature and the central axis to obtain the constitutive matrix relationship; S2036. Using the transformation matrix, transform the constitutive matrix relationship to the global coordinate system to obtain the constitutive matrix relationship of the global coordinates; S2037. Based on the constitutive matrix relationship of the global coordinates and combined with the characterized BWBN model, a degenerate beam finite element model with the hysteretic evolution equation of the BWBN model is obtained.

5. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 1 is characterized in that: The S3 includes the following steps: S301. Ignore the axial nonlinear behavior of the concrete pier structure and the hysteretic strain corresponding to the axial deformation to obtain the unit hysteresis vector. Based on the degenerate beam finite element model, combined with the unit connection topology, mass distribution, and boundary constraints, construct the system motion equilibrium equation to obtain the hysteretic curvature vector. S302, performing high-order equivalent linearization on the hysteresis curvature vector to obtain an equivalent linearization coefficient; S303. Ignore the partial derivatives of the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter in the BWBN model, and replace them with the mean approximation to obtain the expected values ​​of the strength degradation parameter, stiffness degradation parameter, degree pinching parameter, and width pinching parameter. S304: In response to the expected values ​​of the calculated strength degradation parameter, the stiffness degradation parameter, the degree pinching parameter, and the width pinching parameter, a partial derivative operation is performed on the equivalent linearization coefficient, and a formula is derived using the curvature variation and the hysteresis curvature to obtain an expression for the partial derivative coefficient; S305, defining a first state vector, and converting the system motion equation into a state space equation; S306. Combining the expected values, partial derivative coefficient expressions and state-space equations, a high-order stochastic equivalent linearized form of the hysteretic motion equation based on the BWBN model is constructed.

6. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 1 is characterized in that: The S4 comprises the following steps: S401. Based on random process theory and the high-order random equivalent linearization form of the hysteretic motion equation, the seismic acceleration function and time modulation function are constructed. The stationary power spectral density function under earthquake excitation is constructed as a Kanei Kiyoshi model of a series second-order filter. A shaping filter excited by white noise is introduced, and the second state vector is defined to construct a random process model of seismic motion. S402. Based on representative strong earthquake records and the conditions of each site, the mean, standard deviation, and quantile of time domain parameters and frequency domain parameters are calculated respectively. A prediction equation is established for regression analysis. Using a statistical distribution estimation method, characteristic ground motion parameters corresponding to the site conditions are obtained. Filter parameters related to soil stiffness are selected to obtain site condition parameters. S403, setting an augmented state vector, obtaining a covariance matrix of the augmented state vector, and dynamically iteratively updating the equivalent matrix by solving the Lyapunov function to obtain a covariance matrix of the random response of the concrete pier structure.

7. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 6 is characterized in that: The S401 includes the following steps: S4011. Based on the theory of random processes and the high-order random equivalent linearization form of the hysteretic motion equation, the ground acceleration function and time modulation function are constructed by ignoring the time-varying characteristics of the frequency. S4012. Based on the CP model, a bilateral evolutionary power spectral density function of earthquake excitation is constructed. Using the Jinjingqing model with a series second-order filter, a stationary power spectral density function of earthquake excitation in the CP model is constructed. The power spectrum intensity of the input white noise random process in the Jinjingqing model is related to the peak acceleration of the ground motion. S4013. Introducing a shaping filter, based on the power spectrum intensity of a constant input white noise random process, constructing an expression for the joint response process of a bilinear filter under white noise excitation; S4014. Define a second state vector, convert the expression of the joint response process of the bilinear filter under white noise excitation and the seismic acceleration function into a state space equation, and construct a seismic random process model.

8. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 6 is characterized in that: The S403 includes the following steps: S4031. Set the augmented state vector, establish a joint expression of the system and the excitation, and obtain the covariance matrix of the augmented state vector by solving; S4032. Setting the initial state of the covariance matrix of the augmented state vector, and obtaining the Lyapunov function of the dynamic response by recursion based on the backward difference method; S4033. By solving the Lyapunov function and dynamically iteratively updating the equivalent matrix, in response to the covariance matrix difference between the current and subsequent iterative augmented state vectors meeting the preset error accuracy requirement, the covariance matrix of the random response of the concrete pier structure is output.

9. The nonlinear seismic random response analysis method of bridge piers based on the BWBN model according to claim 1, characterized in that: The S5 comprises the following steps: S501. Based on the finite element nonlinear analysis numerical model, the covariance matrix of the random response of the concrete pier structure, and site condition parameters, and using the Kanei Kiyoshi model of a series second-order filter, the random response characteristics of the structure under specific earthquake excitation intensities are explored. The random response analysis of the concrete pier structure is performed under strong earthquake excitation with peak acceleration corresponding to the power spectrum intensity, and the root mean square value of the pier bottom curvature and the average hysteresis time history of the pier bottom are obtained. S502. Using incremental dynamic analysis, the dual parameters of pier bottom curvature and pier top displacement are used as indicators for evaluating the seismic performance of concrete pier structures. S503. Using typical working conditions of random excitation, conduct comparative analysis to obtain differentiated contribution characteristics of basic vibration modes and higher-order vibration modes to the RMS peak value of the pier body curvature and the RMS peak value of the pier body displacement; S504. Select the random response under hard soil conditions, and based on the site condition parameters and the typical working conditions of random excitation, explore the nonlinear behaviors of the curvature response and displacement response of the concrete pier structure in different steps; S505. Based on strong earthquake excitation, sensitivity analysis of the hysteresis parameters of the bridge pier is performed to obtain the effect of the RMS peak value of the pier bottom curvature and the average hysteresis energy dissipation under specific excitation intensity, and complete the nonlinear seismic random response analysis of the bridge pier.

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