A Method for Analyzing Nonlinear Random Vibration and Seismic Vulnerability of Bridge Structures

Abstract

A method for analyzing the nonlinear random vibration and seismic vulnerability of bridge structures was developed. A near-fault random seismic motion model containing velocity pulses was established, and the model parameters and their probability distribution were identified, generating near-fault random seismic excitations. Based on the principle of probability conservation, the probability density integral equation for the bridge system under near-fault random seismic excitations was derived. A nonlinear finite element model of the bridge structure was established using the open-source software OpenSees. Using the direct probability integral method, the dynamic equations and the probability density integral equation for the bridge system under near-fault random seismic excitations were solved simultaneously, efficiently and accurately obtaining information such as the probability density function, mean, and standard deviation of the random response of the bridge structure. Finally, based on the definition of seismic vulnerability analysis, the equivalent extreme value mapping was combined with the direct probability integral method to conveniently calculate the dynamic reliability values of the bridge structure under different earthquake intensities, thereby obtaining the seismic vulnerability curve of the bridge structure.

Description

Technical Field

[0001] The present invention relates to a method for analyzing nonlinear random vibration and seismic vulnerability of a bridge structure under the action of near-fault random earthquake motion, and is applicable to the technical field of bridge engineering seismic resistance. Background Art

[0002] With the increasing development of transportation in my country, the number of bridges being built has also increased rapidly. However, numerous post-earthquake disaster investigations have shown that near-fault ground motion can cause significant damage to bridge structures located near faults. Given the significant threat of near-fault ground motion to bridges, it is crucial to analyze the seismic response of bridge structures to near-fault ground motion.

[0003] Considering the strong variability of near-fault seismic motions, which falls within the realm of random uncertainty, deterministic seismic response analysis methods are difficult to address. Therefore, it is particularly important to employ random vibration analysis methods to reveal the changing patterns of the seismic response of bridge structures under near-fault earthquakes. Similarly, due to the strong randomness of near-fault seismic motions, it is difficult for researchers to accurately predict the seismic response magnitudes of bridge components to achieve different performance targets. Therefore, seismic vulnerability analysis methods are employed to calculate the probability of achieving or exceeding a given performance target and to express the damage consequences of near-fault earthquake disasters in a probabilistic manner. This approach is more easily accepted by stakeholders in bridge engineering projects and provides a new approach for analyzing the random seismic response and seismic reliability of bridge structures. Summary of the Invention

[0004] This method analyzes measured seismic motion records to establish a near-fault random seismic motion model containing velocity pulses, identify the model's parameters and their probability distribution, and generate near-fault random seismic excitations. Based on the principle of probability conservation, the probability density integral equation for the bridge system under near-fault random seismic excitation is derived. A finite element model of the bridge is established using the open source software OpenSees, and the nonlinear dynamic equations of the bridge structure are derived. Subsequently, the direct probability integral method is used to simultaneously solve the nonlinear dynamic equations and the probability density integral equation of the bridge model. The probability density function, mean, and standard deviation of the bridge structure's response to near-fault random seismic excitation are efficiently and accurately obtained. Furthermore, based on the definition of seismic fragility analysis, an equivalent extreme value mapping is introduced. The dynamic reliability of the bridge structure under different earthquake intensities is calculated using the direct probability integral method, thereby obtaining the seismic fragility curve of the bridge structure.

[0005] The present invention provides a method for analyzing nonlinear random vibration and seismic vulnerability of a bridge structure under near-fault random ground motion excitation, comprising the following steps:

[0006] (1) A near-fault random seismic motion model containing velocity pulses was established, and random representative points were substituted into the model to generate near-fault random seismic excitations; including:

[0007] By analyzing near-fault seismic records at Jiji, Taiwan, we obtained a linear combination of the continuous wavelet transform coefficients of the two horizontal components, and determined the strongest pulse direction of the near-fault seismic motion. The velocity history in this direction was then divided into low-frequency and high-frequency components for fitting. Statistical analysis of the low-frequency and high-frequency components was then performed to obtain the probability distribution of the low-frequency pulse parameters and the high-frequency statistical power spectrum parameters. Furthermore, a two-step point selection technique based on GF deviation was employed to generate representative points according to the statistical probability distribution of each parameter. The generated random representative point data were then substituted into the near-fault random seismic motion model, thereby generating near-fault seismic excitations with corresponding assigned probabilities.

[0008] (2) Based on the principle of probability conservation, the probability density integral equation of the bridge system under near-fault random ground motion excitation is derived; including:

[0009] Based on the probability conservation principle and the Dirac function, the probability density integral equation of the bridge system under near-fault random seismic motion is derived. The probability density integral equation of the multi-degree-of-freedom bridge system is subjected to dimensionality reduction, thereby obtaining the probability density integral equation for calculating any single response of the bridge system.

[0010] (3) Based on the open source software OpenSees, a deterministic finite element model of the bridge structure was established. The nonlinear dynamic equations and probability density integral equations of the bridge structure were solved simultaneously to obtain the probability density function information, mean, and standard deviation of the random vibration response of the bridge structure; including:

[0011] A nonlinear finite element model of the bridge structure was established using the open-source software OpenSees and known structural parameters. Combining the generalized variational principle with the OpenSees finite element model, the nonlinear dynamic equations of the bridge model were derived. Using the direct probability integral method, the nonlinear dynamic equations and the probability density integral equation for the bridge model were solved simultaneously, yielding information such as the probability density function of the bridge structure's response to near-fault random ground motion excitation.

[0012] (4) Based on the definition of seismic vulnerability analysis, the direct probability integral method is combined with the equivalent extreme value mapping to solve the dynamic reliability of the bridge structure under near-fault random ground motion excitation under different earthquake intensities, and finally the seismic vulnerability curve of the bridge structure is obtained; including:

[0013] Based on the equivalent extreme value mapping and probability density integral equation, a Heaviside function-based formula was constructed to calculate the dynamic reliability of bridge structures subjected to near-fault random seismic motion. Based on this formula, the dynamic reliability values of bridge structures under different earthquake intensities were calculated. Based on the definition of bridge seismic vulnerability analysis, a seismic vulnerability curve for bridge structures subjected to near-fault random seismic motion was then constructed.

[0014] The specific steps are as follows:

[0015] Step (1): Establish a random model of near-fault ground motion. First, use wavelet transform to obtain the velocity history in the direction of the strongest velocity pulse. After obtaining the velocity history with the strongest velocity pulse, the Gabor wavelet analytical function is used to fit the low-frequency components in the velocity history. The fitting formula is:

[0016]

[0017] In formula (1), V p ,T p ,N c ,T pk and Represent pulse peak value, pulse period, pulse cycle number, pulse time and pulse phase angle respectively; where T p ,N c ,T pk and It can be regarded as an independent random variable to calculate its probability distribution characteristics; and V p It is expressed as the peak velocity with the strongest velocity pulse, which has a certain correlation with the seismological parameters. First, it is assumed that the model regression residual σ lnPGV If it satisfies the normal distribution, then V p The relationship between seismological parameters can be expressed as:

[0018]

[0019] In formula (2), M w Expressed as matrix level, R is the fault distance, (M w , R) can be represented as a set of earthquake scenarios. c1, c2, c3, and c4 are regression parameters, and σ is the regression residual. By using different earthquake scenarios, a regression equation for the peak velocity in the direction of the strongest velocity pulse is obtained.

[0020] In the near-fault ground motion random model, the component model simulating the long-period velocity pulse contains a total of five random variables. The nonlinear least squares method can be used to fit the low-frequency velocity pulse function in formula (2) to obtain the pulse parameter T. p ,N c ,Tpk and Minimize the sum of squared errors in the entire time domain:

[0021]

[0022] After separating the long-period pulse component from the strongest pulse velocity history, the power spectrum of the residual velocity component of the high-frequency component is obtained. After differential processing, the non-stationary power spectrum of the residual acceleration history of the high-frequency component can be obtained. Using a function-based spectrum expression method, the residual acceleration history of the high-frequency component is obtained:

[0023]

[0024] In formula (4), S(t,ω k ) is the non-stationary power spectrum of the residual acceleration component. Δω=(ω u -ω l ) / N ω ,ω k =ω l +kΔω.X k and Y k are two mutually orthogonal random variables, which are represented by a random variable γ that is uniformly distributed on [-π,π]:

[0025]

[0026] For the non-stationary power spectrum S(t,ω) shown in formula (4) k ) can be uniformly modulated in the following form:

[0027] S(t,w)=|f(t)| 2 |H h (ω)| 2 S K-T (ω) (6)

[0028] In formula (6), |H h (ω)| 2 Indicates Butterworth filter, S K-T (ω) is expressed as KT spectrum, and the specific expressions are:

[0029]

[0030] The parameters contained in the KT spectrum in formula (7) can be obtained by fitting the normalized power spectrum of the residual acceleration time history.

[0031] In the near-fault ground motion stochastic model, in order to characterize the non-stationary characteristics of the high-frequency component, an envelope function containing three random parameters is used to represent its variability:

[0032]

[0033] In formula (8), t0 represents the initial time of the near-fault ground motion, t pk represents the moment corresponding to the acceleration peak in the direction of the strongest velocity pulse, while α and β are random parameters that control the rising and falling segments of the envelope function, respectively.

[0034] In the near-fault ground motion random model, the component model simulating high-frequency acceleration contains three random variables. First, the high-frequency acceleration record should be Hilbert transformed to obtain the instantaneous amplitude of the ground motion record. Then the multi-peak point method is used to obtain the time domain p-level envelope of the instantaneous amplitude After smoothing and filtering, the envelope parameters in formula (8) are also fitted using the nonlinear least squares method to minimize the sum of square errors in the time domain. The random envelope parameters t are obtained. pk , α and β.

[0035] Finally, the long-period velocity pulse generated by the random model of near-fault seismic motion is analytically differentiated to obtain the low-frequency component acceleration time history record, which is superimposed with the high-frequency component velocity pulse and analytically differentiated to obtain the acceleration time history record of near-fault random seismic motion with typical velocity pulses.

[0036] Step (2): According to the principle of probability conservation, the probability carried by random events remains unchanged throughout the entire physical evolution process. Let Θ and Y represent the input random vector and the output random vector respectively. Then the principle of probability conservation in the dynamic system can be expressed as:

[0037]

[0038] where p Y (y,t),p Θ (θ, t) are the probability density functions of the output random vector Y and the input random vector Θ respectively. Ω Y and Ω Θ Denote the sample space corresponding to the input and output random vectors respectively. The dynamic system can be expressed as follows:

[0039]

[0040] Introducing the Dirac function into equation (10), the relationship between the output response PDF and the input random variable PDF can be established through further derivation:

[0041]

[0042] From this, we can get the probability density function of the random output response of the dynamic system at any time t:

[0043]

[0044] Equation (12) describes the probability evolution relationship from the random input variable Θ to the random output response Y during the evolution of the dynamic system in the form of an integral, which is also called the probability density integral equation. In the study of bridge seismic resistance, researchers often only focus on one or several random vibration response components. Therefore, by applying the properties of the Dirac function and performing marginal probability density integration on Equation (12), the dimensionality reduction of the probability density integral equation is achieved, and finally the dynamic system with respect to a certain random response component can be obtained. The probability density function of :

[0045]

[0046] The output response probability density function shown in Equation (13) contains the Dirac delta function and needs to be integrated the same number of times as the input random variable dimension. For practical engineering problems, it is often necessary to solve it numerically.

[0047] Step (3): Based on the data information of the bridge engineering structure, nonlinear finite element modeling of the bridge is performed based on the OpenSees finite element platform. In order to solve the probability density integral equation, the direct probability integral method is further proposed. Its key technologies are (1) probability space segmentation; (2) smoothing the Dirac function, and then obtaining the numerical calculation formula of formula (14)

[0048]

[0049] Where N represents the total number of representative points, θ q It refers to the representative value of the qth representative point in the input probability space, σ refers to the smoothing parameter of the approximate Gaussian function, Refers to the representative area Ω Θ,q The probability of obtaining the q ,t) refers to the deterministic mapping relationship in the random dynamic system. After completing the nonlinear modeling of the OpenSees bridge, the near-fault earthquake excitation generated in step (1) is input into the OpenSees finite element model to calculate the deterministic seismic response of the nonlinear bridge model, that is, the deterministic mapping relationship g(θ) in formula (14) is obtained. q ,t). With the help of the direct probability integration method, the probability density integral equation and the physical mapping can be decoupled and solved, and the probability density function of the random dynamic response of the bridge can be obtained efficiently and accurately.

[0050] The specific numerical calculation steps of the direct probability integration method are as follows:

[0051] 1) Use the two-step point selection method based on GF deviation to perform probability space segmentation and obtain representative points and the corresponding probability of attachment

[0052] 2) Based on numerical integration methods, such as using nonlinear dynamic time history analysis methods, solve the dynamic response of representative points.

[0053] 3) Substitute the probability assigned to the representative point and the dynamic response into equation (14) to obtain the probability density function of the response y at time t.

[0054] Step (4): Seismic vulnerability analysis is defined as the probability that the structural response exceeds the specified limit state under given earthquake intensity conditions. Therefore, the dynamic reliability values of the bridge structure under different earthquake intensities are calculated based on the direct probability integration method, and then the seismic vulnerability curve of the bridge structure can be obtained. For the bridge structure subjected to random earthquake motion near the fault that exceeds the dynamic reliability for the first time, the equivalent extreme value mapping can be introduced to obtain the functional function Z(t) for solution. Let: Θ→Y ext , Then the performance function Z(t) can be expressed as

[0055]

[0056] Where B is the threshold of dynamic reliability, l(θ,t) is the dynamic response function of the structure. The corresponding first-time exceedance dynamic reliability calculation formula is

[0057]

[0058] Define the probability density function of Z(t) as p z (z, t), then formula (16) can be expressed as

[0059]

[0060] Where p z (z, t) can be obtained by the probability density integral equation, that is,

[0061]

[0062] Substituting formula (18) into formula (17) we can obtain the solution formula of dynamic reliability based on probability density integral equation:

[0063]

[0064] Since the Dirac function can be integrated into the Heaviside function Then formula (17) becomes

[0065]

[0066] Since the Dirac function has been replaced by the Heaviside function in Equation (20), no smoothing is required. Therefore, by introducing the probability space partitioning technology, the system dynamic reliability solution formula based on the direct probability integration method can be obtained:

[0067]

[0068] According to the actual engineering requirements, Equation (21) is selected as the direct probability integration method for calculating the dynamic reliability. By calculating the dynamic reliability of the bridge structure under different earthquake intensities, the seismic vulnerability curve of the bridge structure is finally obtained.

[0069] The steps of bridge seismic vulnerability analysis based on direct probability integration method are as follows:

[0070] 1) Based on the two-step point selection method, the initial probability space is divided to obtain representative points and corresponding assigned probabilities;

[0071] 2) Solve the dynamic response of the structure and obtain the functional function of the structure based on the equivalent extreme value mapping and threshold B.

[0072] 3) Substituting the probability of the representative point and the corresponding performance function value into formula (21), the dynamic reliability of the system can be obtained.

[0073] 4) Using the near-fault ground motion stochastic model proposed in step (1), the peak seismic acceleration (PGA) of the model is continuously adjusted so that the PGA follows a certain distribution pattern. Steps 1-3 are performed under different PGAs to obtain the bridge dynamic reliability values under different PGAs. Based on the concept of seismic vulnerability analysis, the seismic vulnerability curve of the bridge structure is obtained.

[0074] This paper presents a method for analyzing the nonlinear random vibration and seismic vulnerability of bridge structures. This method can efficiently and accurately solve the probability density function of the dynamic response and conveniently obtain the system's seismic vulnerability curve. Furthermore, this method can be easily embedded in mainstream commercial finite element software, facilitating engineering applications. It provides an efficient computational tool for safety, reliability assessment, and risk analysis of long-span bridge structures in complex dynamic environments. By decoupling the system's governing equations (such as equilibrium equations) from the probability density integral equation, this method is non-invasive and easily embedded in mainstream commercial finite element software, facilitating engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 A schematic flow diagram of the present invention is shown.

[0076] Figure 2 A geometric elevation drawing of a continuous beam bridge is shown.

[0077] Figure 3(a) shows the time history of the mean value of the near-fault ground motion excitation.

[0078] Figure 3(b) shows the time history of the standard deviation of the near-fault ground motion excitation.

[0079] Figure 4 Probability density function curves of the displacement of the mid-span node along the bridge at different times.

[0080] Figure 5(a) Mean displacement curve of the bridge mid-span node along the bridge direction.

[0081] Figure 5(b) shows the standard deviation curve of the displacement of the bridge mid-span node along the bridge direction.

[0082] Figure 6 The first exceedance dynamic reliability curves of continuous beam bridge supports at different thresholds (B) are shown.

[0083] Figure 7 The seismic fragility curves of the continuous beam bridge bearings are shown. DETAILED DESCRIPTION

[0084] The specific embodiments of the present invention are described in detail below in conjunction with the technical solutions and drawings.

[0085] Example: Nonlinear random vibration analysis and seismic vulnerability analysis of a continuous beam bridge under near-fault random ground motion.

[0086] The direct probability integration method can be used to perform nonlinear random vibration analysis of bridge structures under random seismic motion, accurately and efficiently obtaining the probability density function of the structural response and the seismic vulnerability curve. Taking the seismic response study of a continuous beam bridge under near-fault random seismic motion as an example, the specific calculation steps are as follows:

[0087] 1. The bridge is a four-span continuous beam bridge (geometric model see Figure 2 ), a nonlinear finite element model of the continuous beam bridge was established based on the open source software OpenSees, and the finite element model motion equation of the bridge was obtained by combining the generalized variational principle.

[0088]

[0089] Where [M], [C] and [K] represent the overall mass matrix, damping matrix and stiffness matrix of the continuous beam bridge structure respectively. {E} is the inertia force index vector of the structure, represents the near-fault earthquake acceleration containing random variables, and and {y} represent the acceleration, velocity and displacement responses of the continuous beam bridge structure, respectively.

[0090] 2. Based on the near-fault random ground motion model proposed above, near-fault earthquake excitations with assigned probabilities are generated. The time history changes of the mean and standard deviation of the near-fault ground motion excitations are shown in Figure 3.

[0091] 3. Solve the dynamic response of a nonlinear finite element model of a four-span continuous beam bridge subjected to near-fault random seismic excitation. In this example, a nonlinear dynamic time-history analysis method is used to obtain the deterministic seismic response of the bridge structure.

[0092] 4. Solve the probability density function of the bridge structure response. For example, taking the displacement response of the main girder span of a four-span continuous beam bridge as an example, after obtaining its deterministic dynamic response along the bridge direction, substitute it into the direct probability integration method to calculate Equation (14) to obtain the probability density function of the response at different times. Figure 4 The probability density function (PDF) distribution curves of the displacement response of the mid-span nodes of continuous beam bridges along the longitudinal direction of the bridge are given at t = 5s, t = 25s, and t = 45s. The mean and standard deviation of the random vibration response of the mid-span nodes along the longitudinal direction of the bridge are calculated and compared with the results obtained by Monte Carlo simulation. The calculated results are shown in Figure 5.

[0093] 5. Solve the dynamic reliability of the bridge structure. For example, after solving the dynamic response of the finite element model of the continuous beam bridge, introduce the equivalent extreme value mapping to obtain the system's performance function under different thresholds B, substitute it into the direct probability integration method dynamic reliability calculation formula (21), and obtain the first exceedance dynamic reliability curve of the system under different thresholds B, as shown in the figure below: Figure 6 shown.

[0094] 6. Calculate the seismic vulnerability curve of the bridge structure. For example, adjust the seismic peak acceleration (PGA) of the near-fault random ground motion model, calculate the bridge dynamic reliability value under different seismic peak accelerations, and then obtain the failure probability of the bridge components beyond different damage states. Based on the concept of seismic vulnerability analysis, obtain the seismic vulnerability curve of the bridge structure, such as Figure 7 shown.

Claims

1. A bridge structure nonlinear random vibration and seismic vulnerability analysis method, characterized by the following steps: (1) A near-fault random seismic motion model containing velocity pulses was established, and random representative points were substituted into the model to generate near-fault random seismic excitations; including: By analyzing the near-fault seismic records, the linear combination of the continuous wavelet transform coefficients of the two horizontal components is obtained, and the strongest pulse direction of the near-fault seismic motion is obtained. The velocity time history in this direction is divided into low-frequency and high-frequency parts for fitting. The parameters of the low-frequency and high-frequency parts are statistically analyzed to obtain the probability distribution of the low-frequency pulse parameters and the high-frequency statistical power spectrum parameters. On this basis, a two-step point selection technique based on GF deviation is adopted to generate representative points according to the probability distribution of each parameter statistics. The generated random representative point data are substituted into the near-fault random seismic motion model, thereby generating near-fault seismic excitations with corresponding assigned probabilities. (2) Based on the principle of probability conservation, the probability density integral equation of the bridge system under near-fault random ground motion excitation is derived; including: Based on the probability conservation principle and the Dirac function, the probability density integral equation of the bridge system under near-fault random seismic motion is derived. The probability density integral equation of the multi-degree-of-freedom bridge system is subjected to dimensionality reduction, thereby obtaining the probability density integral equation for calculating any single response of the bridge system. (3) Based on the open source software OpenSees, a deterministic finite element model of the bridge structure was established. The nonlinear dynamic equations and probability density integral equations of the bridge structure were solved simultaneously to obtain the probability density function information, mean, and standard deviation of the random vibration response of the bridge structure; including: A nonlinear finite element model of the bridge structure was established based on the open source software OpenSees and known architectural parameters of the bridge structure. The nonlinear dynamic equations of the bridge model were derived by combining the generalized variational principle with the OpenSees finite element model. Using the direct probability integral method, the nonlinear dynamic equations and the probability density integral equation of the bridge model were solved simultaneously to obtain the probability density function and mean information of the bridge structure's response to near-fault random seismic excitation. (4) Based on the definition of seismic vulnerability analysis, the direct probability integral method is combined with the equivalent extreme value mapping to solve the dynamic reliability of the bridge structure under near-fault random ground motion excitation under different earthquake intensities, and finally the seismic vulnerability curve of the bridge structure is obtained; including: Based on the equivalent extreme value mapping and probability density integral equation, a calculation formula for the dynamic reliability of bridge structures under near-fault random seismic motions was constructed based on the Heaviside function. According to the calculation formula for the dynamic reliability of bridge structures, the dynamic reliability values of bridge structures under different earthquake intensities were calculated. According to the definition of bridge seismic vulnerability analysis, the seismic vulnerability curve of bridge structures under near-fault random seismic motions was constructed.

2. The novel method for analyzing nonlinear random vibration and seismic vulnerability of bridge structures according to claim 1 is characterized by the following steps: Step (1): To establish a random model of near-fault ground motion, wavelet transform is first used to obtain the velocity history in the direction of the strongest velocity pulse. After obtaining the velocity history with the strongest velocity pulse, the Gabor wavelet analytical function is used to fit the low-frequency components in the velocity history. The fitting formula is: In formula (1), V p ,T p ,N c ,T pk and Represent pulse peak value, pulse period, pulse cycle number, pulse time and pulse phase angle respectively; where T p ,N c ,T pk and It can be regarded as an independent random variable to calculate its probability distribution characteristics; and V p It is expressed as the peak velocity with the strongest velocity pulse, which has a certain correlation with the seismological parameters. First, it is assumed that the model regression residual σ lnPGV If it satisfies the normal distribution, then V p The relationship between seismological parameters is expressed as: In formula (2), M w Expressed as matrix level, R is the fault distance, (M w ,R) can be expressed as a set of earthquake scenarios; c1, c2, c3 and c4 are regression parameters, and σ is the regression residual; by using different earthquake scenarios, the regression relationship of the peak velocity in the direction of the strongest velocity pulse is obtained; In the near-fault ground motion random model, the component model simulating the long-period velocity pulse contains a total of five random variables. The nonlinear least squares method is used to fit the low-frequency velocity pulse function in formula (2) to obtain the pulse parameter T. p ,N c ,T pk and Minimize the sum of squared errors in the entire time domain: After separating the long-period pulse component from the strongest pulse velocity history, the power spectrum of the residual velocity component of the high-frequency component is obtained. After differential processing, the non-stationary power spectrum of the residual acceleration history of the high-frequency component can be obtained. Using the function-based spectral expression method, the residual acceleration history of the high-frequency component is obtained: In formula (4), S(t,ω k ) is the non-stationary power spectrum of the residual acceleration component; Δω=(ω u -ω l ) / N ω ,ω k =ω l +kΔω;X k and Y k are two mutually orthogonal random variables, which are represented by a random variable γ that is uniformly distributed on [-π,π]: For the non-stationary power spectrum S(t,ω) shown in formula (5) k ) is uniformly modulated in the following form: S(t,w)=|f(t)| 2 |H h (ω)| 2 S K-T (ω) (6) In formula (6), |H h (ω)| 2 Indicates Butterworth filter, S K-T (ω) is expressed as KT spectrum, and the specific expressions are: The parameters contained in the KT spectrum in formula (7) are obtained by fitting the normalized power spectrum of the residual acceleration time history; In the near-fault ground motion stochastic model, in order to characterize the non-stationary characteristics of the high-frequency component, an envelope function containing three random parameters is used to represent its variability: In formula (8), t0 represents the initial time of the near-fault ground motion, t pk represents the moment corresponding to the acceleration peak in the direction of the strongest velocity pulse, while α and β are random parameters that control the rising and falling segments of the envelope function, respectively; In the near-fault ground motion random model, the component model simulating high-frequency acceleration contains three random variables. First, the high-frequency acceleration record should be Hilbert transformed to obtain the instantaneous amplitude of the ground motion record. Then the multi-peak point method is used to obtain the time domain p-level envelope of the instantaneous amplitude After smoothing and filtering, the envelope parameters in formula (8) are also fitted using the nonlinear least squares method to minimize the sum of square errors in the time domain; The random envelope parameter t is obtained from this pk , α and β; Finally, the long-period velocity pulse generated by the near-fault random seismic model is analytically differentiated to obtain the low-frequency component acceleration time history record, which is then superimposed with the high-frequency component velocity pulse and analytically differentiated to obtain the near-fault random seismic acceleration time history record with typical velocity pulses. Step (2): According to the principle of probability conservation, the probability carried by random events remains unchanged throughout the entire physical evolution process; let Θ and Y represent the input random vector and output random vector respectively, then the principle of probability conservation in the dynamic system can be expressed as: where p Y (y,t),p Θ (θ, t) are the probability density functions of the output random vector Y and the input random vector Θ respectively; Ω Y and Ω Θ They represent the sample spaces corresponding to the input and output random vectors respectively; for dynamic systems, they can be expressed by formula (10): G:Y(t)=g(Θ,t) (10) Introducing the Dirac function into equation (10), the relationship between the output response PDF and the input response PDF can be established through further derivation as follows: From this, we can get the probability density function of the random output response of the dynamic system at any time t: Equation (12) describes the probability evolution relationship from random input variable Θ to random output response Y in the evolution process of the dynamic system in the form of an integral, which is also called the probability density integral equation. In the study of bridge seismic resistance, researchers often only focus on one or several random vibration response components. Therefore, by applying the properties of the Dirac function and performing marginal probability density integration on Equation (12), the dimensionality reduction processing of the probability density integral equation is achieved, and finally the dynamic system with respect to a certain random response component can be obtained. The probability density function of : The output response probability density function shown in Equation (13) contains the Dirac delta function and needs to be integrated the same number of times as the input random variable dimension. For practical engineering problems, it is often necessary to solve it numerically. Step (3): Based on the data information of the bridge engineering structure, nonlinear finite element modeling of the bridge is performed based on the OpenSees finite element platform; in order to solve the probability density integral equation, the direct probability integral method is further proposed; its key technologies are (1) probability space segmentation; (2) smoothing the Dirac function, and then obtaining the numerical calculation formula of formula (14) Where N represents the total number of representative points, θ q It refers to the representative value of the qth representative point in the input probability space, σ refers to the smoothing parameter of the approximate Gaussian function, Refers to the representative area Ω Θ,q The probability of obtaining the q ,t) refers to the deterministic mapping relationship in the random dynamic system; After completing the nonlinear modeling of the OpenSees bridge, the near-fault ground motion excitation generated in step (1) is input into the OpenSees finite element model to calculate the deterministic seismic response of the nonlinear bridge model, that is, the deterministic mapping relationship g(θ) in Equation (14) is obtained. q ,t); With the help of direct probability integration method, the probability density integral equation and physical mapping can be decoupled and solved, and the probability density function of the random dynamic response of the bridge can be obtained efficiently and accurately; The specific numerical calculation steps of the direct probability integration method are as follows: 1) Use the two-step point selection method based on GF deviation to perform probability space segmentation and obtain representative points and the corresponding probability of attachment 2) Based on numerical integration methods, such as nonlinear dynamic time history analysis methods, solve the dynamic response of representative points; 3) Substitute the probability assigned to the representative point and the dynamic response into equation (14) to obtain the probability density function of the response y at time t; Step (4): Seismic vulnerability analysis is defined as the probability that the structural response exceeds the specified limit state under given earthquake intensity conditions; therefore, the dynamic reliability values of the bridge structure under different earthquake intensities are calculated based on the direct probability integration method, and then the seismic vulnerability curve of the bridge structure can be obtained; for the bridge structure subjected to random earthquake action near the fault that exceeds the dynamic reliability for the first time, the equivalent extreme value mapping can be introduced to obtain the function Z(t) for solution; let: Θ→Y ext , Then the performance function Z(t) can be expressed as: Z(t) = m(θ, t) = Bl(θ, t) (15) where B is the threshold of dynamic reliability and l(θ, t) is the dynamic response function of the structure. The corresponding first-time exceeding dynamic reliability calculation formula is: Define the probability density function of Z(t) as p z (z, t), then formula (16) can be expressed as Where p z (z, t) can be obtained by the probability density integral equation, that is, Substituting formula (18) into formula (17) we can obtain the solution formula of dynamic reliability based on probability density integral equation: Since the Dirac function can be integrated into the Heaviside function Η(·), Equation (17) becomes Since the Dirac function has been replaced by the Heaviside function in Equation (20), no smoothing is required; therefore, by introducing the probability space partitioning technology, the system dynamic reliability solution formula based on the direct probability integration method can be obtained: According to the actual engineering requirements, formula (21) is selected as the direct probability integration method for calculating the dynamic reliability. By calculating the dynamic reliability of the bridge structure under different earthquake intensities, the seismic vulnerability curve of the bridge structure is finally obtained. The steps of bridge seismic vulnerability analysis based on direct probability integration method are as follows: 1) Based on the two-step point selection method, the initial probability space is divided to obtain representative points and corresponding assigned probabilities; 2) Solve the dynamic response of the structure and obtain the structural performance function based on the equivalent extreme value mapping and threshold B; 3) Substituting the probability of the representative point and the corresponding performance function value into formula (21), the dynamic reliability of the system can be obtained; 4) Using the near-fault seismic random model proposed in step (1), the seismic peak acceleration (PGA) of the model is continuously adjusted so that the seismic peak acceleration is distributed according to a certain distribution law. Steps 1-3 are performed under different seismic peak accelerations to obtain the bridge dynamic reliability values under different seismic peak accelerations; based on the seismic vulnerability analysis concept, the seismic vulnerability curve of the bridge structure is obtained.

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