Railway concrete bridge seismic reliability analysis method based on probability density evolution

Abstract

The application discloses a railway concrete bridge seismic reliability analysis method based on probability density evolution, which comprises the following steps: obtaining the random variables of external load of a railway concrete bridge and concrete material, and obtaining basic random variables of a system by integration; dividing the overall probability space formed by the probability space of all basic random variables to obtain a plurality of probability subspaces and the probability assigned thereto, and then performing fatigue response analysis and seismic response analysis; constructing a failure criterion function of the random bridge system based on the fatigue response analysis result and the seismic response analysis result, obtaining a probability dissipation factor of the random bridge system based on the failure criterion function; and obtaining the overall reliability of the railway concrete bridge structure based on the probability dissipation factor. The application evaluates the overall reliability of an existing damaged railway concrete bridge under the action of an earthquake, and provides a theoretical and technical basis for the seismic safety evaluation and maintenance and reinforcement of an existing bridge structure.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of structural engineering, and particularly relates to a railway concrete bridge seismic reliability analysis method based on probability density evolution. BACKGROUND

[0002] The railway concrete bridge long-term bears high-density and heavy-load train load, the internal fatigue damage accumulates continuously, which can cause the structure to present nonlinear behavior characteristics, and cause the bridge bearing capacity to degrade day by day. In addition, the external load borne by the structure and the mechanical properties of the structure material both have significant randomness, and the two influence each other, which can cause the diversity and complexity of the nonlinear behavior of the railway concrete bridge stress. In addition, the earthquake activity is frequent in China, and the existing damaged railway concrete bridge faces the threat of the earthquake action in its service period. How to scientifically evaluate the seismic safety of the existing damaged railway concrete bridge structure is crucial in establishing a quantitative evaluation method of the overall reliability of the structure, objectively reflecting the material fatigue, the structure damage condition, and the mutual influence and action between each component, and comprehensively understanding the overall reliability level of the structure.

[0003] The existing overall reliability evaluation method includes two aspects of work of identification of the main failure mode and calculation of the system failure probability. The existing method usually starts from the damage consequences of the structure, searches for the failure section, failure path and failure mode by determining the failure probability maximum, and finds all the main failure modes. With the increasing complexity of the engineering structure, this method will face two bottlenecks of exponential growth of the failure mechanism number and the failure path probability correlation, and cannot solve the overall reliability of the complex nonlinear structure.

[0004] Therefore, it is urgent to propose a railway concrete bridge seismic reliability analysis method based on probability density evolution. SUMMARY

[0005] To solve the above technical problems, the present application proposes a railway concrete bridge seismic reliability analysis method based on probability density evolution, applies the probability density evolution theory, establishes a structure failure criterion which can comprehensively describe the high-cycle and low-cycle fatigue damage modes based on the random damage evolution law and the random failure mechanism of the railway concrete bridge, develops an overall reliability analysis method of the railway concrete bridge structure under the train load and the earthquake action according to this, and thus accurately evaluates the remaining life and the overall reliability of the existing damaged railway concrete bridge structure under the earthquake action, so as to solve the problems existing in the prior art.

[0006] To achieve the above purpose, the present application provides a railway concrete bridge seismic reliability analysis method based on probability density evolution, which includes the following steps:

[0007] Obtaining external loads of a railway concrete bridge and random variables of a concrete material, and obtaining basic random variables of a random bridge system by integration;

[0008] Carrying out partition on a total probability space formed by probability spaces of all basic random variables, and obtaining a plurality of probability subspaces and probability assigned thereto;

[0009] Based on the plurality of probability subspaces and the probability assigned thereto, fatigue response analysis and earthquake response analysis are carried out;

[0010] Based on fatigue response analysis results and earthquake response analysis results, a failure criterion function of the random bridge system is constructed, and based on the failure criterion function, a probability dissipation factor of the random bridge system is obtained;

[0011] Based on the probability dissipation factor, a train fatigue load and an earthquake are jointly used to obtain an overall reliability of a railway concrete bridge structure under the action of the train fatigue load and the earthquake by using a probability density evolution theory.

[0012] Optionally, the random variables of the external loads include random variables of a train fatigue load and random variables of an earthquake action, wherein the random variables of the train fatigue load are circular frequencies and phase angles in a random harmonic function, and the random variables of the earthquake action are a time interval between adjacent earthquakes and an acceleration peak value of each earthquake;

[0013] The random variables of the concrete material are a homogenization surface energy parameter in a concrete micro-meso random fatigue damage constitutive model, a coefficient in a decay term reflecting micro-crack interaction, and a fracture strain field.

[0014] Optionally, the basic random variables of the random bridge system include basic random variables of the train fatigue load, basic random variables of the earthquake action, and basic random variables of the concrete material;

[0015] The obtaining process of the basic random variables of the train fatigue load includes: a random harmonic function method is used to simulate a train fatigue load process, a plurality of circular frequency random variables and phase angle random variables are obtained, and the two are combined as the basic random variables of the train fatigue load;

[0016] The obtaining process of the basic random variables of the earthquake action includes: a Monte Carlo method is used to generate an earthquake sequence process, a plurality of time interval random variables and acceleration peak value random variables are obtained, and the two are combined as the basic random variables of the earthquake action.

[0017] Optionally, the process of obtaining a plurality of probability subspaces and their assigned probabilities comprises: dividing the total probability space based on a probability space division method of Voronoi region to obtain a plurality of probability subspaces; selecting a representative point for each probability subspace based on a point selection strategy of GF deviation, wherein the representative point comprises a train fatigue load random variable sample and an earthquake sequence random variable sample; obtaining a corresponding train fatigue load sample based on the train fatigue load random variable sample and obtaining a corresponding earthquake sequence sample based on the earthquake sequence random variable sample; and a combined probability of each train fatigue load sample and the corresponding earthquake sequence sample is the assigned probability of the probability subspace where the representative point is located.

[0018] Optionally, the process of fatigue response analysis comprises: establishing a basic physical equation of fatigue response analysis of the railway concrete bridge structure under long-term train fatigue load based on a concrete micro-meso random fatigue damage constitutive model; solving the basic physical equation of fatigue response analysis based on a finite element numerical method and a cyclic jump type fatigue acceleration algorithm with adaptive precision control to obtain a fatigue response analysis result; wherein the basic physical equation of fatigue response analysis is as follows:

[0019]

[0020] wherein, is a Nabla operator; σ is a stress tensor; b represents a body force component per unit mass; u is a displacement component; ρ is a material density; η is a viscous damping coefficient; I is a unit tensor; D is a damage tensor; C0 is an initial stiffness tensor of the material; ε and ε p are a total strain tensor and a plastic strain tensor, respectively; is a displacement on the boundary surface; n is a unit normal vector on the boundary surface; is an external force on the boundary surface;

[0021] The process of seismic response analysis comprises: taking the fatigue response analysis result as an initial condition, establishing a basic physical equation of seismic response analysis of the existing damage railway concrete bridge structure, and solving the basic physical equation of seismic response analysis by using a finite element numerical analysis method to obtain a seismic response analysis result; wherein the basic physical equation of seismic response analysis is as follows:

[0022]

[0023] wherein, is a Nabla operator; σ is a stress tensor; b represents a body force component per unit mass; ε and ε p are a total strain tensor and a plastic strain tensor, respectively; u is a unit displacement tensor; I is a unit tensor; D is a damage tensor; C0 is an initial stiffness tensor of the material; is the displacement on the boundary surface; n is the unit normal vector on the boundary surface; is the external force on the boundary surface.

[0024] Optionally, the failure criterion function of the random bridge system is as shown in the following formula:

[0025]

[0026] wherein H(·) is the Heaviside function; d ED represents the damage of the damage concentration zone; represents the damage of the plastic hinge zone; d th,f is the fatigue damage threshold; d th,e is the seismic damage threshold.

[0027] Optionally, the process of obtaining the overall reliability of the railway concrete bridge structure under the combined action of train fatigue load and earthquake includes: for the train fatigue load displacement of the random bridge system, an equation group of the probability density distribution after probability dissipation is constructed and solved to obtain the train fatigue displacement probability density distribution; for the seismic displacement of the random bridge system, an equation group of the probability density distribution after probability dissipation is constructed and solved to obtain the seismic displacement probability density distribution; the train fatigue displacement probability density distribution and the seismic displacement probability density distribution are superimposed to obtain the probability density distribution of the final displacement of the railway concrete bridge structure; the overall integral of the probability density distribution of the final displacement is obtained to obtain the overall reliability of the railway concrete bridge structure under the combined action of train fatigue load and earthquake.

[0028] Optionally, for the train fatigue load displacement of the random bridge system, the equation group of the probability density distribution after probability dissipation is:

[0029]

[0030] wherein p(-) represents probability; u m,f is the train fatigue load displacement of the random bridge system; t f is the evolution time of the train fatigue load displacement, and the unit is year; Θ is the basic random variable of the system; S mu (t) is the failure criterion function of the system, H mu (S mu (t)) is the probability dissipation factor of the system; u m,f (t0) is the displacement corresponding to the initial time t0 of the train fatigue load action; t coin is the time when the train fatigue load and earthquake meet; u m,e (t→(t coin ) + ) is the displacement generated by the earthquake when the earthquake action ends.

[0031] Optionally, for the seismic displacement of the random bridge system, the equation group of the probability density distribution after the probability dissipation is:

[0032]

[0033] In the formula, p(-) represents probability; u m,e is the seismic displacement under the combined action of train load and earthquake; t e is the evolution time of the seismic displacement, and the unit is second; Θ is the basic random variable of the system; S mu (t) is the failure criterion function of the system, H mu (S mu (t)) is the probability dissipation factor of the system; u m,f (t0) is the displacement corresponding to the initial time t0 of the train fatigue load action; t coin is the time when the train fatigue load and the earthquake meet;

[0034] u m,e (t→(t coin ) - ) is the displacement generated by the train fatigue load when the train fatigue load and the earthquake meet.

[0035] Compared with the prior art, the present application has the following advantages and technical effects:

[0036] Based on the probability density evolution theory, the present application considers various damage modes of the bridge structure, proposes a failure criterion for the railway concrete bridge structure, establishes an overall reliability evaluation method of the railway concrete bridge under the action of the train fatigue load and the earthquake, and evaluates the overall reliability of the existing damaged railway concrete bridge under the action of the earthquake, thereby providing a theoretical and technical basis for the seismic safety evaluation and maintenance and reinforcement of the existing bridge structure. BRIEF DESCRIPTION OF DRAWINGS

[0037] The drawings constituting a part of the present application are used to provide further understanding of the present application, the illustrative embodiments of the present application and the description thereof are used to explain the present application, and do not constitute improper limitation on the present application. In the drawings:

[0038] Figure 1 is the overall reliability analysis flowchart of the railway concrete bridge system of the embodiment of the present application;

[0039] Figure 2 is the general layout schematic diagram of the Qingshuihe Bridge of the embodiment of the present application;

[0040] Figure 3 is the main beam support pier and mid-span section view of the embodiment of the present application;

[0041] Figure 4 is the pier body section view of the embodiment of the present application;

[0042] Figure 5 A schematic diagram of a random load combination process based on probability space subdivision according to an embodiment of the present application;

[0043] Figure 6 A flowchart of a bridge structure fatigue response analysis according to an embodiment of the present application;

[0044] Figure 7 A flowchart of a seismic response analysis of an existing damaged railway concrete bridge according to an embodiment of the present application;

[0045] Figure 8 A schematic diagram of total probability density evolution results of vertical displacement at the midspan of the Qingshuihe Bridge according to an embodiment of the present application, wherein (a) is a probability density evolution surface diagram, and (b) is a probability density contour diagram;

[0046] Figure 9 A schematic diagram of overall reliability results of the Qingshuihe Bridge under the combined action of fatigue load and earthquake according to an embodiment of the present application. DETAILED DESCRIPTION

[0047] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict. The present application will be described in detail below with reference to the accompanying drawings and in combination with embodiments.

[0048] It should be noted that the steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a group of computer executable instructions, and although a logical order is shown in the flowchart, in some cases, the steps shown or described herein can be executed in an order different from that shown herein.

[0049] Embodiment One

[0050] A railway concrete bridge seismic reliability analysis method based on probability density evolution is provided in the present embodiment, which is applicable to the case where fatigue load is a stationary random excitation, and includes the following steps:

[0051] Step 1, determining the basic random variables of external load and concrete material;

[0052] Step 2, obtaining the basic random variables of the system;

[0053] Step 3, selecting representative samples and calculating the assigned probability of the probability subspace;

[0054] Step 4, performing random load combination analysis to obtain train load and earthquake samples;

[0055] Step 5, performing bridge structure random response analysis on the representative samples to obtain the displacement evolution speed and damage evolution process of each sample;

[0056] Step 6, obtaining system failure criterion function and screening operator;

[0057] Step 7, obtaining probability density distribution of train fatigue load displacement and seismic displacement;

[0058] Step 8, superimposing to obtain probability density distribution of system displacement;

[0059] Step 9, calculating structural reliability.

[0060] The flow is shown as Figure 1 .

[0061] The railway concrete bridge seismic reliability analysis method based on probability density evolution provided in the embodiment establishes a structure failure criterion according to the random damage evolution law and random damage excitation of the railway concrete bridge, and evaluates the overall reliability of the existing damage railway concrete bridge structure under the action of the earthquake. The overall reliability evaluation method of the railway concrete bridge is illustrated by taking the Qingshuihe Bridge in the southwest of Guizhou Province as an example. The overall layout of the Qingshuihe Bridge is shown in Figure 2 . The main bridge of the bridge is taken as the analysis object, the main girder of the bridge is made of C50 concrete, and is a single-box single-cell variable cross-section prestressed concrete box girder. The cross-section diagram of the main girder at the support pier and the midspan of the main girder is shown in Figure 3 . The main bridge is provided with two piers, and the pier heights are 86m and 100m respectively. For the sake of convenience, the 86m high pier is recorded as the 1# pier, and the 100m high pier is recorded as the 2# pier. The piers of the main bridge are made of C30 concrete, and are variable cross-section rectangular hollow piers. The cross-section diagram of the pier body of the 2# pier is shown in Figure 4 .

[0062] The external load of the railway concrete bridge and the randomness of the concrete material are obtained, and the basic random variables of the random bridge system are obtained by integration. Specifically,

[0063] The external load of the railway concrete bridge considers the train fatigue load and the seismic action. The random variable Θ f of the train fatigue load takes the circular frequency ω f and the phase angle of the random harmonic function. The random variable Θ e of the seismic action takes the time interval T e between the occurrence of adjacent two earthquakes and the acceleration peak value y f at the occurrence of each earthquake. The randomness of the concrete material is the homogenization surface energy parameter Γ, the coefficient κ in the attenuation term reflecting the interaction of micro-cracks and the fracture strain field Δ(x) in the micro-meso random fatigue damage constitutive model of the concrete.

[0064] The basic random variables of the random bridge system can be obtained by integrating the external load and the random variables of concrete material, which are denoted as:

[0065]

[0066] The train fatigue load process is simulated by the random harmonic function method, and 20 circular frequency random variables ω f f,1 f,2 f,20 and 20 phase angle random variables are combined as the basic random variables of the train fatigue load The earthquake sequence process is generated by the Monte Carlo method, and 2 time interval random variables T e e,1 e,2 and 2 acceleration peak random variables y e e,1 e,2 are combined as the basic random variables of the earthquake action Θ e e e .

[0067] The overall probability space formed by the probability space of all basic random variables is divided to obtain a number of probability subspaces and their assigned probabilities, specifically:

[0068] The overall probability space formed by the probability space of all basic random variables is divided by the probability space division method of Voronoi region, and 482 probability subspaces Ω q q=1,2,…,482 are obtained; the representative points θ q =(θ f,q ,θ e,q ), θ f =(θ f,q ,θ w,q ) are selected by the point selection strategy of GF deviation, where θ f,q represents the train fatigue load random variable sample, θ e,q represents the earthquake sequence random variable sample, q=1,2,…,482; and the assigned probability P q of each point in the probability subspace is obtained, q=1,2,…,482.

[0069] Based on variable identification, the corresponding train fatigue load sample L f,q is obtained by the train fatigue load random variable sample θ tr,q q=1,2,…,482.​​​​​​​​, q = 1, 2,..., 482; by the earthquake sequence random variable sample θ e,q , q = 1, 2,..., 482 get the corresponding earthquake sequence sample L er,q (t), q = 1, 2,..., 482; each train fatigue load sample L tr,q and the corresponding earthquake sequence sample L er,q (t) is the combination probability of the corresponding variable sample, that is, the probability of the probability subspace P q . Figure 5 The random load combination process based on probability space subdivision is described. Table 1 shows the specific information of 4 groups of sample combinations.

[0070] Table 1

[0071]

[0072] Based on the probability subspace and its assigned probability, fatigue response analysis and seismic response analysis are carried out, specifically:

[0073] Fatigue response analysis:

[0074] Based on the micro-meso random fatigue damage constitutive model of concrete, the basic physical equation for fatigue response analysis of railway concrete bridge structure under long-term train fatigue load is established, and the finite element numerical method is used for solving, and a kind of self-adaptive precision control cycle jump type fatigue acceleration algorithm is used to improve the calculation efficiency, and finally the whole process of fatigue development of railway bridge structure under train fatigue load is obtained. In which, the finite element numerical calculation is completed by ABAQUS software, and the solution of cycle jump acceleration algorithm is realized by Python program.

[0075] As Figure 6 shown, the fatigue response analysis process is as follows:

[0076] The random fatigue load sample of the last step is converted into the corresponding equivalent constant amplitude load sample by using the equivalent constant amplitude load method.

[0077] According to the equivalent constant amplitude load sample of the last step, the structure finite element model is established by ABAQUS / CAE, and the model file (.inp file) is generated;

[0078] At least three complete cycles are calculated by Python program calling ABAQUS to generate result data file (.odb file); in the solving process, ABAQUS calls the concrete fatigue random damage constitutive relationship through user-defined subprogram UMAT; the ABAQUS finite element numerical solution of structure physical equation is as follows:

[0079]

[0080] where, is the Nabla operator; σ is the stress tensor; b represents the body force component per unit mass; u is the displacement component; ρ is the material density; η is the viscous damping coefficient; I is the identity tensor; D is the damage tensor; C0 is the initial stiffness tensor of the material; ε and ε p are the total strain tensor and the plastic strain tensor, respectively; is the displacement on the boundary surface; n is the unit normal vector on the boundary surface; is the external force on the boundary surface.

[0081] Read the values of the variables of interest in the data file (.odb file) of the previous step results, use the extrapolation jump method to interpolate the relevant variables, and generate a data file (.dat file) that stores the values of the variables after jumping.

[0082] By calling the SDVINI subroutine in the UMAT subroutine to set the initial condition reading, and reading the values after jumping in the data file of the previous step, assign them as the new initial conditions to the relevant variables to perform the next step of ABAQUS progressive fatigue cycle calculation.

[0083] Repeat the above steps until the structure or component is damaged; integrate the jump results of each time to output the complete structure fatigue development process under the equivalent constant amplitude load sample.

[0084] According to energy equivalence, the structure fatigue process under random fatigue load sample is obtained, as shown in Figure 6 .

[0085] Seismic response analysis:

[0086] As shown in Figure 7 , the fatigue damage obtained by fatigue response analysis is used as the initial condition to establish the basic physical equation for seismic response analysis of existing damage railway concrete bridge structure, and the finite element numerical analysis method is used to solve the following basic physical equation based on the ABAQUS platform:

[0087]

[0088] where, is the Nabla operator; σ is the stress tensor; b represents the body force component per unit mass; ε and ε p are the total strain tensor and the plastic strain tensor, respectively; u is the unit displacement tensor; I is the identity tensor; D is the damage tensor; C0 is the initial stiffness tensor of the material; is the displacement on the boundary surface; n is the unit normal vector on the boundary surface; is the external force on the boundary surface.

[0089] Analysis results:

[0090] The train load and seismic sequence information of sample 34 are analyzed, and details are shown in Table 1. For sample 34, the meeting time of train load and seismic action is the 68th year of the service period of the structure. At this time, the number of train fatigue load is 1839600 times, and the peak acceleration of ground motion is 0.4g. The fatigue damage generated under the train fatigue load is taken as the initial damage of the bridge structure, and the San Fernando earthquake with an acceleration peak of 0.4g is applied to the structure to perform seismic response analysis. The analysis results show that at the time t = 6.42s, the plastic hinge first appears at the top of the 1# pier; then, at t = 7.72s, the plastic hinge appears at the bottom of the 1# pier; then, at t = 9.53s, the longitudinal reinforcement at the top of the 2# pier yields; then, at t = 10.13s, the hinge appears at the bottom of the 2# pier; finally, at t = 14.95s, the top of the 1# pier is damaged. Based on the fatigue response analysis results and the seismic response analysis results, the failure criterion function of the random bridge system is constructed, and based on the failure criterion function, the probability dissipation factor of the random bridge system is obtained, specifically:

[0091] For train fatigue load and seismic action, the fatigue failure criterion and the seismic failure criterion of the structure are defined as:

[0092]

[0093] In the formula, e i is the i th unit, ED is the damage concentration zone; L p,c is the plastic hinge area; d i is the damage variable of the i th unit; d th,f is the fatigue damage threshold; d th,e is the seismic damage threshold.

[0094] The failure criterion function of the system can be expressed as:

[0095]

[0096] In the formula, H(·) is the Heaviside function; d ED represents the damage of the damage concentration zone; represents the damage of the plastic hinge area; H(d th,f -d ED (t)) represents: if the fatigue damage of the unit in the damage concentration zone is greater than the fatigue damage threshold, the structure is damaged by fatigue; represents: if the unit damage in the plastic hinge area is greater than the seismic damage threshold, the structure is damaged by earthquake.

[0097] According to the failure criterion function, the probability dissipation factor of the system is obtained:

[0098]

[0099] The above formula shows that if the system does not have any form of damage, the system probability does not dissipate; on the contrary, if the system has fatigue damage or earthquake damage, the system will dissipate the probability.

[0100] Based on the probability dissipation factor, the probability density evolution theory is used to obtain the overall reliability of the railway concrete bridge structure under the combined action of train fatigue load and earthquake, specifically:

[0101] The system probability dissipation factor H obtained in step 6 is combined mu , and the equation set for solving the overall reliability of the railway concrete bridge structure under the combined action of train fatigue load and earthquake is obtained.

[0102] Among them, for the train fatigue load displacement of the random bridge system, the results obtained by the fatigue response analysis in the foregoing are substituted into the following generalized probability density evolution equation set after probability dissipation and solved:

[0103]

[0104] In the formula, p(-) represents probability; u m,f is the train fatigue load displacement of the random bridge system; t f is the evolution time of the train fatigue load displacement, with the unit of year; Θ is the basic random variable of the system; S mu (t) is the failure criterion function of the system, H mu (S mu (t)) is the probability dissipation factor of the system; u m,f (t0) is the displacement corresponding to the initial time t0 of the train fatigue load action; t coin is the time when the train fatigue load and earthquake meet; u m,e (t→(t coin ) + ) is the displacement generated by the earthquake when the earthquake action ends.

[0105] For the earthquake displacement of the random bridge system, the results obtained by the earthquake response analysis in the foregoing are substituted into the following generalized probability density evolution equation set after probability dissipation and solved:

[0106]

[0107] In the formula, p(-) represents probability; u m,e is the earthquake displacement under the combined action of train load and earthquake; t e is the evolution time of the earthquake displacement, with the unit of second; Θ is the basic random variable of the system; S mu(t) is the failure criterion function of the system, H mu (S mu (t) is the probability dissipation factor of the system; u m,f (t0) represents the displacement corresponding to the initial time t0 of the train fatigue load application; t coin The moment when train fatigue load coincides with an earthquake;

[0108] u m,e (t→(t coin ) - ) represents the displacement caused by the fatigue load of the train when it coincides with an earthquake.

[0109] Solving the two sets of equations above will yield the probability density distribution of train fatigue displacement. With earthquake displacement probability density distribution

[0110] The probability density distribution of train fatigue displacement obtained in step 7 With earthquake displacement probability density distribution By superimposing these values, the probability density distribution of the final displacement of the railway concrete bridge structure can be obtained. Right now:

[0111]

[0112] The final total probability density evolution results of the vertical displacement at mid-span of the Qingshuihe Bridge are shown in the attached figure. Figure 8 As shown.

[0113] Overall reliability R of railway concrete bridge structures under combined train loads and earthquakes m (t) can be obtained from the probability density distribution of the final displacement. The result obtained by performing a global integral is:

[0114]

[0115] The final overall reliability results of the Qingshuihe Bridge under the combined action of fatigue load and earthquake are shown in the attached figure. Figure 9 As shown.

[0116] This invention employs a probability density evolution reliability assessment method with a solid theoretical foundation to evaluate the seismic reliability of existing damaged railway concrete bridges. Compared to classical reliability analysis methods, this method uses probability density evolution theory to uniformly handle various randomnesses in the system, scientifically reflecting the propagation law of randomness in physical systems, and is more suitable for analyzing the reliability of complex nonlinear structures.

[0117] Theoretical basis of this invention:

[0118] Using classical mechanics, engineering structures can be viewed as conservative systems that obey the law of conservation of energy. Furthermore, since real-world engineering structures are always subject to random disturbances from both the external environment and their internal structure, the concept of conservative stochastic systems is introduced to describe them. In a conservative stochastic system, the system not only obeys the law of conservation of energy but also exhibits randomness. For a conservative stochastic system, if no new random factors are added and no existing random factors disappear during its state evolution, then probability conservation is satisfied during the system's state evolution. This principle is known as the principle of probability conservation.

[0119] The principle of probability conservation can be examined from two different perspectives: state-space description and random event description. The random event description of the probability conservation principle states that the probability measure of the same random event remains constant during mathematical and physical changes. The generalized probability density evolution equation is derived using this statement.

[0120] A general stochastic dynamical system can be described in the following form:

[0121]

[0122] In the formula, Z(t) is the physical quantity of interest; x and t are spatial and temporal variables; Θ = (Θ1, Θ2, ..., Θt). s ) represents the basic random variable in the system; λ represents the deterministic physical parameter in the system; and L(·) represents the differential operator describing the system.

[0123] According to the principle of probability conservation, the probability measure of a certain state of the system does not change within any time increment, that is:

[0124]

[0125] In the formula, Pr{·} represents the probability of a random event occurring, and Ω θ Let Ω be any region in the distribution space Θ. t Let × represent the corresponding region in the Z distribution space at time t, where × denotes Ω. θ With Ω t A probability space spanned by both.

[0126] Based on the principle of probability conservation, and through a series of derivations, the following generalized probability density evolution equation can be obtained:

[0127]

[0128] In the formula, Let m represent the j-th component of Z(t), and m be the number of components of Z(t).

[0129] The initial conditions for the above equation are:

[0130]

[0131] where, is a deterministic initial value.

[0132] In particular, when only one physical quantity is of interest (m = 1), the generalized probability density evolution equation is reduced to a one-dimensional partial differential equation:

[0133]

[0134] The initial condition of the above equation is:

[0135]

[0136] Based on the initial condition, the joint probability density distribution of the target physical state variable Z and the basic random variable Θ can be obtained by solving the above generalized probability density evolution equation: The time-varying process of the probability density of the target state variable can be obtained by integrating with respect to θ, i.e.:

[0137]

[0138] From the perspective of the one-dimensional generalized probability density evolution equation, the one-dimensional partial differential equation can be expressed as:

[0139]

[0140] From the above equation, it can be found that the joint probability density distribution The rate of change with respect to time t is proportional to the rate of change with respect to the target physical state variable z at each time, and the proportional coefficient is And Exactly represents the comprehensive rate of change of the target physical state variable at time t. In fact, the solution of the above equation is a characteristic line, which leads to the continuous redistribution of the probability carried by each sample to different value ranges of the target state variable over time, and further leads to the change of the probability distribution density of each state variable (probability redistribution).

[0141] For a random system, it is possible that the probability of the system is dissipated at any time during the evolution process. For the problem of structural dynamic stability, once the stability criterion is met, the probability of the corresponding structural response will be dissipated. Therefore, the probability density evolution equation for a probability dissipation system is proposed:

[0142]

[0143] In the equation, H is called the probability dissipation factor, which can be defined as:

[0144]

[0145] where S(t) is the structural failure criterion function; Ω S is the structural safety region; Ω D is the structural failure region; has

[0146] The initial condition of the above formula is:

[0147]

[0148] The one-dimensional form of the above formula is:

[0149]

[0150] The initial condition is:

[0151]

[0152] The residual probability density function of the system response can be obtained by solving the probability density evolution equation of the dissipation system under the initial condition:

[0153]

[0154] The recently developed probability density evolution theory scientifically reflects the randomness propagation law of the structural system, realizes the decoupling of the physical state of the system, does not require solving high-dimensional probability density evolution equations, and provides a new way for the reliability research of complex structures. Based on this, the randomness of material properties and external loads is also investigated, and the related parameters describing the two are taken as basic random variables; by introducing the failure criterion of the railway concrete bridge structure, the probability density distribution equation set of the structural system after the probability dissipation is constructed and solved; finally, the overall reliability analysis method of the complex nonlinear structural system is established, and the overall reliability evolution law of the existing railway concrete bridge structure under the action of the earthquake is correctly grasped.

[0155] The above is only the preferred specific implementation manner of the present application, but the protection scope of the present application is not limited to this, and any person skilled in the art can easily think of changes or replacements within the technical range disclosed by the present application, which should be covered in the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for seismic reliability analysis of railway concrete bridges based on probability density evolution, characterized in that, Includes the following steps: The random variables of external loads and concrete materials of railway concrete bridges are obtained, and the basic random variables of the random bridge system are obtained by integration. The overall probability space formed by the probability space of all basic random variables is divided to obtain several probability subspaces and their assigned probabilities. Based on the aforementioned probability subspaces and their assigned probabilities, fatigue response analysis and seismic response analysis are performed. Based on the fatigue response analysis results and the seismic response analysis results, a failure criterion function for a stochastic bridge system is constructed, and based on the failure criterion function, the probability dissipation factor of the stochastic bridge system is obtained. Based on the aforementioned probability dissipation factor, the overall reliability of railway concrete bridge structures under the combined action of train fatigue load and earthquake is obtained using the probability density evolution theory. The failure criterion function for a stochastic bridge system is shown in the following equation: ； in, For the Heaviside function; Damage representing the area of ​​concentrated damage; Damage representing the plastic hinge region; The fatigue damage threshold; This represents the earthquake damage threshold. Based on the failure criterion function, the probability dissipation factor of the system can be obtained as follows: 。 2. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 1, characterized in that, The random variables of the external load include random variables of train fatigue load and random variables of seismic action. The random variables of train fatigue load are the circular frequency and phase angle in the random harmonic function, and the random variables of seismic action are the time interval between two adjacent earthquakes and the peak acceleration at the time of each earthquake. The random variables of the concrete material are the homogenized surface energy parameters, the coefficients in the attenuation term reflecting the interaction of microcracks, and the fracture strain field in the micro-micro stochastic fatigue damage constitutive model of concrete.

3. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 2, characterized in that, The basic random variables of the random bridge system include the basic random variables of train fatigue load, seismic action, and concrete material. The process of obtaining the basic random variables of train fatigue load includes: simulating the train fatigue load process using the random harmonic function method, obtaining several circular frequency random variables and phase angle random variables, and combining the two as the basic random variables of train fatigue load; The process of obtaining the basic random variables of seismic action includes: generating an earthquake sequence using the Monte Carlo method, obtaining several time interval random variables and peak acceleration random variables, and combining the two as the basic random variables of seismic action.

4. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 1, characterized in that, The process of obtaining several probability subspaces and their assigned probabilities includes: dividing the overall probability space based on the probability space partitioning method of the Voronoi region to obtain several probability subspaces; selecting representative points for each probability subspace based on the GF bias point selection strategy, wherein the representative points include train fatigue load random variable samples and earthquake sequence random variable samples; obtaining corresponding train fatigue load samples based on the train fatigue load random variable samples, and obtaining corresponding earthquake sequence samples based on the earthquake sequence random variable samples; the combination probability of each train fatigue load sample and the corresponding earthquake sequence sample is the assigned probability of the probability subspace in which each representative point is located.

5. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 1, characterized in that, The fatigue response analysis process includes: establishing the basic physical equations for fatigue response analysis of railway concrete bridge structures under long-term train fatigue load based on the micro-micro stochastic fatigue damage constitutive model of concrete; solving the basic physical equations for fatigue response analysis based on the finite element numerical method and the cyclic jump fatigue acceleration algorithm with adaptive precision control; and obtaining the fatigue response analysis results. The basic physical equation for fatigue response analysis is as follows: In the formula, σ is the Nable operator; b is the stress tensor; u is the displacement component; The density of the material; It is the viscous damping coefficient; Unit tensor; For damage tensor; This is the initial stiffness tensor of the material; and These are the total strain tensor and the plastic strain tensor, respectively. denoted as , where is the displacement on the boundary surface; and is the unit normal vector on the boundary surface. External forces at the boundary surface; The process of seismic response analysis includes: using the fatigue response analysis results as initial conditions, establishing the basic physical equations for the seismic response analysis of existing damaged railway concrete bridge structures, and solving them using the finite element numerical analysis method to obtain the seismic response analysis results. The basic physical equations for seismic response analysis are as follows: in, σ is the Nable operator; σ is the stress tensor; b represents the volume component per unit mass. and These are the total strain tensor and the plastic strain tensor, respectively; u is the unit displacement tensor; Unit tensor; For damage tensor; This is the initial stiffness tensor of the material; denoted as , where is the displacement on the boundary surface; and is the unit normal vector on the boundary surface. For external forces at the boundary surface.

6. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 1, characterized in that, The process of obtaining the overall reliability of a railway concrete bridge structure under the combined action of train fatigue load and earthquake includes: for the train fatigue load displacement of a random bridge system, constructing and solving a system of equations for the probability density distribution after probability dissipation to obtain the probability density distribution of train fatigue displacement; for the earthquake displacement of a random bridge system, constructing and solving a system of equations for the probability density distribution after probability dissipation to obtain the probability density distribution of earthquake displacement; superimposing the probability density distributions of train fatigue displacement and earthquake displacement to obtain the probability density distribution of the final displacement of the railway concrete bridge structure; and performing a global integral on the probability density distribution of the final displacement to obtain the overall reliability of the railway concrete bridge structure under the combined action of train fatigue load and earthquake.

7. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 6, characterized in that, For the train fatigue load displacement of a stochastic bridge system, the equations for the probability density distribution after probability dissipation are as follows: in, Represents probability; For train fatigue load displacement in a random bridge system; Θ represents the evolution time of the train's fatigue load displacement; Θ represents the basic random variable of the system. Let be the failure criterion function of the system. The probability dissipation factor of the system; The initial moment of train fatigue load application The corresponding displacement; The moment when train fatigue load coincides with an earthquake; This represents the displacement caused by the earthquake when the seismic action ends.

8. The seismic reliability analysis method for railway concrete bridges based on probability density evolution according to claim 6, characterized in that, For the seismic displacement of a random bridge system, the equations for the probability density distribution after probability dissipation are as follows: in, Represents probability; This refers to the seismic displacement under the combined action of train load and earthquake. Θ represents the evolution time of the seismic displacement; Θ represents the basic random variable of the system. Let be the failure criterion function of the system. The probability dissipation factor of the system; The initial moment of train fatigue load application The corresponding displacement; The moment when train fatigue load coincides with an earthquake; This refers to the displacement caused by the fatigue load on the train when it coincides with an earthquake.

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