Method for evaluating overall system performance degradation probability of box girder bridge based on DBN and gamma distribution

Abstract

This invention discloses a method for assessing the overall performance degradation probability of box girder bridges based on DBN and Gamma distributions, belonging to the technical field of box girder bridge performance degradation probability assessment. The method includes: determining the composition of the box girder bridge; determining the dwell time of each component; calculating relevant parameters of the Gamma distribution probability density function; calculating the cumulative probability density function and cumulative survival probability function of each component; modeling the performance degradation of the box girder bridge to obtain the transition probability; calculating the state probability distribution of each component; forming the probability degradation curves of each component and the overall structure through a dynamic Bayesian network; and assessing the impact of each component on the probability degradation curves of its component and the overall structure. This invention, employing the aforementioned method for assessing the overall performance degradation probability of box girder bridges based on DBN and Gamma distributions, can scientifically predict the performance degradation of box girder bridges from component to component and from component to the whole, thereby predicting the service life of box girder bridges from a probabilistic perspective.

Description

Technical Field

[0001] This invention relates to the field of performance degradation probability assessment technology for box girder bridges, and in particular to a method for performance degradation probability assessment of the entire box girder bridge system based on DBN and Gamma distribution. Background Technology

[0002] As a core hub in modern transportation networks, the structural stability and safety of box girder bridges are fundamental to ensuring smooth traffic flow. Over the past four decades, approximately 80% of box girder bridges have been constructed, witnessing the rapid development of the nation's transportation infrastructure. However, with the passage of time, over 430,000 box girder bridges have been in service for more than 20 years, gradually exhibiting aging and deterioration problems, with some even facing severe conditions such as insufficient load-bearing capacity. Simultaneously, the booming development of the transportation industry has led to continuously increasing traffic loads, and the rapid growth in transportation demand has forced many box girder bridges to exceed their design lifespan. This not only poses a significant challenge to the structural safety of the box girder bridges themselves but also presents a potential threat to the stable operation of the entire transportation network. Therefore, paying close attention to the aging problem of box girder bridges and strengthening their daily inspection and maintenance are crucial to ensuring the safe and smooth operation of the transportation network.

[0003] According to the "Specifications for Maintenance of Highway Bridges and Culverts" (JTGH11-2004), "the regular inspection cycle shall be determined based on the technical condition, and shall not exceed three years." The newly revised "Specifications for Maintenance of Highway Bridges and Culverts" (JTG5120-2021) further clarifies that "for box girder bridges with a maintenance inspection status of Class I, the regular inspection cycle shall not exceed one year; for box girder bridges with a maintenance inspection status of Class II or III, the regular inspection cycle shall not exceed three years." Therefore, a massive amount of box girder bridge inspection data is generated nationwide every year. How to accurately estimate the degree of performance degradation of box girder bridges based on this inspection information and effectively predict the degradation trend after maintenance intervention has become a critical problem that urgently needs to be solved in the engineering field.

[0004] Currently, there are numerous performance evaluation and prediction models for box girder bridges. Methods such as reliability assessment based on physical models and Gamma curve degradation are typically only applicable to the analysis of individual components, lacking reliable theoretical support when evaluating components and the overall structure. Data-driven neural network methods suffer from poor model interpretability and limited generalization ability; the analytic hierarchy process (AHP) is heavily influenced by subjective factors, making index quantification difficult; the semi-Markov method, when analyzing bridge performance degradation, often sets the state dwell time as a Weibull distribution, performing distribution degradation analysis on individual components, parts, or the overall structure. However, this approach has significant drawbacks; relying solely on the Weibull distribution degradation of components makes it difficult to reasonably infer the degradation distribution of parts and the overall structure, failing to accurately grasp the overall degradation trend of bridge performance. Summary of the Invention

[0005] The purpose of this invention is to provide a method for assessing the probability of performance degradation of the entire box girder bridge system based on DBN and Gamma distribution. Starting from the analysis of components, it systematically solves the problems of assessing and predicting the performance degradation of components, parts and the overall structure, as well as the impact of component replacement and maintenance on the performance degradation of the parts and the overall structure.

[0006] To achieve the above objectives, this invention provides a method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distributions, comprising the following steps:

[0007] S1. Determine the composition of the box girder bridge;

[0008] S2. Determine the dwell time of each component of the box girder bridge;

[0009] S3. Using the dwell time of each component of the box girder bridge obtained from S2, calculate the relevant parameters of the probability density function of the Gamma distribution;

[0010] S4. Calculate the cumulative probability density function and cumulative survival probability function of each component of the box girder bridge using Monte Carlo simulation;

[0011] S5. Model the performance degradation of box girder bridges to obtain the transition probability;

[0012] S6. Calculate the state probability distribution of each component using the data obtained in S4;

[0013] S7. Based on the state probability distribution of each component obtained in S6, the probability degradation curves of each component and the overall structure are formed by a dynamic Bayesian network.

[0014] S8. Evaluate the impact of the component on the degradation curve of its constituent parts and the overall structure.

[0015] Preferably, the specific operation of S1 is as follows:

[0016] S11. A box girder bridge consists of a superstructure, a substructure, and a bridge deck system. The superstructure includes upper load-bearing components, upper general components, and supports. The substructure includes slope protection, abutments, and pier foundations. The bridge deck system includes bridge deck pavement, expansion joint devices, guardrails, and a drainage system.

[0017] S12. Establish a topology containing circular nodes, line segments, and connecting arrows to form a dynamic Bayesian network. The circular nodes represent the probabilities of each state of a component, part, or overall structure at a certain time point, the connecting arrows represent the transition probabilities between nodes, and the nodes at both ends of a line segment have the same meaning. The topology consists of a component layer, a part layer, and an overall structure layer from bottom to top. The component layer is the observation layer, the part layer is the first hidden layer, and the overall structure layer is the second hidden layer.

[0018] Preferably, the specific operation of S2 is as follows:

[0019] S21. Determine the composition weights of the component layer and the part layer in the topology structure in S12, and use them as the transition probabilities from the component layer to the part layer and from the part layer to the overall structure layer in the dynamic Bayesian network, respectively. Divide the detection score of the component layer in the topology structure into 5 score intervals, which correspond to the 5 states of the component layer.

[0020] S22. In S21, since the time value corresponding to the 5th scoring interval is infinite, only the first 4 scoring intervals are linearly fitted according to the detection score. The endpoint scores of the first 4 scoring intervals are substituted into the linear scoring curve fitted by the first 4 scoring intervals respectively, and the time points corresponding to the endpoint scores of the scoring intervals are extracted from the scoring curves. The dwell time of each component in each state is calculated based on the time points, as shown in formula (1):

[0021] (1);

[0022] in, Representing state The length of stay Representing state The right endpoint of the time interval of stay Representing state The right endpoint of the time interval of stay.

[0023] Preferably, the specific operation of S3 is as follows:

[0024] S31. Let the dwell time of each component of the box girder bridge in each state be a random variable. t , t If it follows a Gamma distribution, then the state Depending on the length of stay t The probability density function of the change is given by formula (2), and the state... The survival probability function is given by formula (4), and the probability density function of the Gamma distribution is given by formula (3). Formulas (2), (3), and (4) are expressed as follows:

[0025] (2);

[0026] (3);

[0027] (4);

[0028] Represents random variables t The probability density function, The proportional parameter represents the probability density function of the Gamma distribution. This represents the shape parameter of the probability density function of the Gamma distribution. Represents an exponential function. Represents the survival probability function;

[0029] S32. By combining formulas (5) and (6), the state is calculated using a genetic algorithm. The proportional parameter of the probability density function of the Gamma distribution and shape parameters Formulas (5) and (6) are expressed as follows:

[0030] (5);

[0031] (6);

[0032] Where min represents minimization. Indicates time, Representing state The average survival probability, Representing state The length of stay.

[0033] Preferably, the specific operation of S4 is as follows:

[0034] S41. Denote the first four states as State 1, State 2, State 3, and State 4. Use Monte Carlo simulation to calculate the cumulative probability density function of each component in the superstructure in State 1, from State 1 to State 2, from State 1 to State 3, and from State 1 to State 4. Similarly, the cumulative probability density functions of each component of the substructure and bridge deck system in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4 are calculated. ;

[0035] S42. Calculate the cumulative survival probability function of each component of the superstructure in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4 using the integral formula. Similarly, calculate the cumulative survival probability function of each component of the substructure and bridge deck system in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4. The integral formula is shown in formula (7).

[0036] (7);

[0037] in, Represents the transition from state 1 to state 2. The cumulative survival probability function; Represents the probability density function. Represents state 1 to state 2. The cumulative probability density function.

[0038] Preferably, the specific operation of S5 is as follows:

[0039] S51. Calculating the conditional probability of a box girder bridge: The performance of a box girder bridge decreases over time, and its state transitions from low to high. The transition between adjacent states exhibits the Markov property, meaning the next state depends only on the current state and is independent of past states. Therefore, the conditional probability expression for a box girder bridge is:

[0040] (8);

[0041] in, Represents conditional probability; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; Indicates that in the known In all previous states The conditional probability at that time; Indicates that in the known In the state of time The conditional probability at that time;

[0042] S52. Based on the Markov property of S51, a Markov chain is used to pair states. The performance degradation model of the box girder bridge is performed. Under this model, it is considered that in a time series with fixed time intervals, the box girder bridge will change from its current state at the next moment, that is, one time unit apart from the current moment. Transition to another state transition probability The conditional probability is expressed as shown in formula (9); when multiple states transition simultaneously, a transition probability matrix is ​​formed, as shown in formula (10).

[0043] (9);

[0044] (10);

[0045] in, Indicates from state Transition to state The transition probability, , Represents conditional probability. express OK The transition probability matrix of the column. Indicates the total number of states. , .

[0046] Preferably, the specific operation of S6 is: to change the state Corresponding stay time As a random variable, then in At that time, the box girder bridge is in the next time step. From state Transition to the next adjacent state The probability is given by formula (11). Formula (12) is used to calculate the state probability distribution of the upper load-bearing component, the upper general component, and the support, forming a probability degradation curve. Formulas (11) and (12) are expressed as follows:

[0047] (11);

[0048] (12);

[0049] in, express Time state To state The transition probability, Indicates the time step. Represents state 1 to state 2. The cumulative probability density function, Representing components No. The state probability distribution of the year Representing components No. The state probability distribution of the year The transition probability matrix is ​​obtained by substituting formula (11) into formula (10). Represents a time variable. Let be a vector, when hour, , indicating that the component is in The probability of being in each state, in The probability of being in state 1 is 1, and the probability of being in any of the other four states is 0.

[0050] Preferably, the specific operation of S7 is as follows:

[0051] S71. Based on the state probability distribution of each component obtained in S6, the state probability distribution of the superstructure, substructure, and bridge deck system is calculated using formula (13) through a dynamic Bayesian network, forming the probability degradation curve of each component. The expression of formula (13) is as follows:

[0052] (13);

[0053] in, Indicates components The state probability distribution curve, Representing components Transfer to component The probability, q Indicates the total number of components contained in the component;

[0054] S72. Based on the state probability distribution of each component obtained in S71, the state probability distribution of the overall structure is calculated using formula (14) through a dynamic Bayesian network, forming the overall structure probability degradation curve. The expression of formula (14) is:

[0055] (14);

[0056] in, The state probability distribution curve representing the overall structure. Indicates components The state probability distribution curve, Indicates components The probability of transferring to the overall structure, r This indicates the total number of components contained in the overall structure.

[0057] Preferably, the specific operation of S8 is as follows: at a specific service time of the box girder bridge, one of the components is replaced or repaired, and the influence of the component on the probability degradation curve of its component and the overall structure is calculated, thereby assessing its sensitivity to the overall structural performance.

[0058] This invention scientifically determines the compositional weights of each component and part of a box girder bridge and uses them as the transition probabilities from component to part and from part to the overall structure. This breaks through the limitations of the traditional single distribution form and can reveal the underlying logic of bridge performance degradation more deeply and comprehensively. In addition, compared with the Weibull distribution, applying the Gamma distribution to describe the state dwell time of components has unique advantages.

[0059] From a theoretical perspective, a box girder bridge is a complex structural system with close interrelationships among its components and parts. The performance degradation of a component does not occur in isolation but rather affects the components through certain transmission mechanisms, thereby impacting the overall structural performance. Using component weights as the transfer probability from component to part, and from part to the overall structure, aligns with the fundamental principles of structural mechanics and systems engineering. Furthermore, compared to the Weibull distribution, the Gamma distribution is more flexible. In bridge structures, when components are subjected to multiple factors with cumulative effects, the Gamma distribution better describes their state dwell time. Finally, the parameter estimation of the Gamma distribution is less affected by noise or missing bridge inspection data.

[0060] Therefore, the present invention employs the above-mentioned method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distributions, which has the following beneficial effects:

[0061] (1) The present invention uses a box girder bridge performance degradation probability assessment method, adopts box girder bridge state scoring, and is based on a dynamic Bayesian Network (DBN) and a semi-Markov process with Gamma distribution performance degradation. The method is simple and easy to operate, and can scientifically predict the performance degradation of box girder bridges from components to parts and from parts to the whole, thereby predicting the service life of box girder bridges from a probabilistic perspective.

[0062] (2) It can predict the impact of component replacement or repair on the degradation trend of the performance of the component and the overall structure, providing a technical basis for the scientific maintenance and upkeep of box girder bridges;

[0063] (3) Performing a separate Gamma distribution degradation analysis on the degradation of the underlying components can provide a more detailed understanding of the performance degradation of each component, thereby more accurately assessing the overall performance of the box girder bridge.

[0064] (4) The prediction result is no longer a fixed value, but a probability of being in different states, which is more scientific and reasonable.

[0065] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0066] Figure 1 This is the dynamic Bayesian network diagram of the present invention;

[0067] Figure 2 This is a graph of the cumulative probability density function of the upper load-bearing component of the present invention;

[0068] Figure 3 This is a graph showing the cumulative survival probability function of the upper load-bearing component of this invention;

[0069] Figure 4 This is a probability degradation curve of the upper load-bearing component of the present invention;

[0070] Figure 5 This is a cumulative probability density function diagram of the upper general components of this invention;

[0071] Figure 6 This is a graph showing the cumulative survival probability function of the upper general components of this invention;

[0072] Figure 7 This is a probability degradation curve diagram of the upper general components of this invention;

[0073] Figure 8 This is the cumulative probability density function diagram of the support of the present invention;

[0074] Figure 9 This is a graph of the cumulative survival probability function of the support of the present invention;

[0075] Figure 10 This is a graph showing the probability degradation of the support in this invention.

[0076] Figure 11 This is a probability degradation curve of the upper structure of the present invention;

[0077] Figure 12 This is a probability degradation curve of the lower structure of the present invention;

[0078] Figure 13 This is a probability degradation curve of the bridge deck system of the present invention;

[0079] Figure 14 This is a probability degradation curve of the overall structure of the box girder bridge of the present invention;

[0080] Figure 15 This is a graph showing the probability degradation of the support (after support replacement) according to the present invention.

[0081] Figure 16 This is a probability degradation curve of the superstructure of the present invention (after support replacement);

[0082] Figure 17 This is a probability degradation curve of the overall structure of the box girder bridge (after bearing replacement) according to the present invention;

[0083] Figure 18 This is a probability degradation curve of the upper load-bearing component (after reinforcement) of the present invention;

[0084] Figure 19 This is a probability degradation curve of the superstructure of the present invention (after reinforcement of the upper load-bearing component);

[0085] Figure 20 This is the overall probability degradation curve of the box girder bridge (after reinforcement of the upper load-bearing components) according to the present invention. Detailed Implementation

[0086] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0087] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0088] Example

[0089] This invention provides a method for evaluating the probability of performance degradation of the entire box girder bridge system based on DBN and Gamma distributions, including the following steps:

[0090] S1. Determine the composition of the box girder bridge. The specific steps are as follows:

[0091] S11. A box girder bridge consists of a superstructure, a substructure, and a bridge deck system. The superstructure includes upper load-bearing components, upper general components, and supports. The substructure includes slope protection, abutments, and pier foundations. The bridge deck system includes bridge deck pavement, expansion joint devices, guardrails, and a drainage system.

[0092] S12. Establish a topology containing circular nodes, line segments, and connecting arrows to form a dynamic Bayesian network. The circular nodes represent the probabilities of each state of a component, part, or overall structure at a certain time point, the connecting arrows represent the transition probabilities between nodes, and the nodes at both ends of a line segment have the same meaning. The topology consists of a component layer, a part layer, and an overall structure layer from bottom to top. The component layer is the observation layer, the part layer is the first hidden layer, and the overall structure layer is the second hidden layer.

[0093] The dynamic Bayesian network formed therein is, for example... Figure 1 As shown in the figure; Indicates the time step. Indicates the overall structure at the 1st The state probability at time t. Indicates the overall structure at time... The state probability, Indicates the overall structure at time... The state probability, Indicates the time of the 0th component The state probability, Indicates the time of the 0th component The state probability, Indicates the time of the 0th component The state probability, Indicates the first Each component at time The state probability, Indicates the first Each component at time The state probability, Indicates the first Each component at time The state probability, Indicates the time of the 0th component The state probability, Indicates the time of the 0th component The state probability, Indicates the time of the 0th component The state probability, Indicates the first Component at time The state probability, Indicates the first Each component at time The state probability, Indicates the first Each component at time The state probability.

[0094] S2. Determine the dwell time of each component of the box girder bridge. The specific operation is as follows:

[0095] S21. Referencing the "Standard for Technical Condition Assessment of Highway Bridges (JTG / T H21—2011)," determine the composition weights of the component layer and part layer in the topology of S12. These weights will be used as the transition probabilities from the component layer to the part layer and from the part layer to the overall structure layer in the dynamic Bayesian network, respectively. The detection scores of the component layer within the topology obtained from the road network company will be divided into 5 scoring intervals, classified into 1-5 categories according to the standards of intact function, minor damage, moderate damage, severe damage to serious damage, corresponding to the 5 states of the component layer, as shown in Table 1.

[0096] Table 1. Classification Boundaries of Component Layer Conditions for Box Girder Bridges

[0097] ;

[0098] S22. In S21, since the time value corresponding to the 5th scoring interval can be infinitely large, only the first 4 scoring intervals are linearly fitted according to the detection score. The endpoint scores of the first 4 scoring intervals are substituted into the linear scoring curve fitted by the first 4 scoring intervals respectively, and the time points corresponding to the endpoint scores of the scoring intervals are extracted from the scoring curves. The dwell time of each component in each state is calculated based on the time points, as shown in formula (1):

[0099] (1);

[0100] in, Representing state The length of stay Representing state The right endpoint of the time interval of stay Representing state The right endpoint of the time interval of stay.

[0101] For the upper load-bearing component, the dwell times for the first four states are calculated according to formula (1). t 1 =9 years t 2 =14 years t 3 =20 years and t 4 =25 years, and the length of stay of the remaining components is detailed in Table 2.

[0102] Table 2. Dwell Time of Various Components in Different States of Box Girder Bridge

[0103] ;

[0104] S3. Calculate the relevant parameters of the Gamma distribution probability density function based on the dwell time of each component of the box girder bridge obtained in S2. The specific operation is as follows:

[0105] S31. Let the dwell time of each component of the box girder bridge in each state be a random variable. t , t If it follows a Gamma distribution, then the state Depending on the length of stay t The probability density function of the change is given by formula (2), and the state... The survival probability function is given by formula (4), and the probability density function of the Gamma distribution is given by formula (3). Formulas (2), (3), and (4) are expressed as follows:

[0106] (2);

[0107] (3);

[0108] (4);

[0109] Represents random variables t The probability density function, The proportional parameter represents the probability density function of the Gamma distribution. This represents the shape parameter of the probability density function of the Gamma distribution. Represents an exponential function. This represents the survival probability function.

[0110] S32. By combining formulas (5) and (6), the state is calculated using a genetic algorithm. The proportional parameter of the probability density function of the Gamma distribution and shape parameters Formulas (5) and (6) are expressed as follows:

[0111] (5);

[0112] (6);

[0113] Where min represents minimization. Indicates time, Representing state The average survival probability, Representing state The length of stay.

[0114] S4. Calculate the cumulative dwell time and cumulative survival probability function of each component of the box girder bridge using Monte Carlo simulation. The specific operation is as follows:

[0115] S4. Calculate the cumulative probability density function and cumulative survival probability function of each component of the box girder bridge using Monte Carlo simulation. The specific steps are as follows:

[0116] S41. Denote the first four states as State 1, State 2, State 3, and State 4. Use Monte Carlo simulation to calculate the cumulative probability density function of each component in the superstructure in State 1, from State 1 to State 2, from State 1 to State 3, and from State 1 to State 4. The cumulative probability density function curves of the upper load-bearing components, the upper general components, and the supports are as follows: Figure 2 , Figure 5 , Figure 8 As shown. From Figure 2 It can be seen that as the number of states increases, the variance of the cumulative probability density function of the upper load-bearing component increases, and the cumulative probability density function of states 1 to 4 reaches its maximum in 1969; from Figure 5 It can be seen that, compared to the upper load-bearing components, the time corresponding to the peak value of the cumulative probability density function of the upper general components is shorter, and the cumulative probability density function from state 1 to state 4 reaches its maximum at 36 years; from Figure 8 It can be seen that, compared to the superstructure load-bearing components and general superstructure components, the cumulative probability density function of the supports corresponds to the shortest time, and the cumulative probability density function from state 1 to state 4 reaches its maximum in 22 years. Similarly, the cumulative probability density functions of each component of the substructure and bridge deck system in state 1, from state 1 to state 2, from state 1 to state 3, and from state 1 to state 4 can be calculated. .

[0117] S42. Calculate the cumulative survival probability function of each component of the superstructure in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4 using the integral formula (7). The graphs of the cumulative survival probability functions of the superstructure load-bearing components, superstructure general components, and supports are shown below. Figure 3 , Figure 6 , Figure 9 As shown. From Figure 3 It can be seen that the downward trend of the four curves is initially slow, then rapid, and finally slows down again, with the downward trend gradually slowing down as the cumulative state increases. The average survival probability (survival probability function value of 50%) corresponding to the first four cumulative states takes 68 years, and the survival probability drops to 0 after 120 years. Figure 6 , Figure 9 The downward trend of the curve and Figure 3 Similarly. Among them, Figure 6 The cumulative survival probability of the upper general components drops to 0 in the first 4 states, which is 80 years, 40 years less than that of the upper load-bearing components. Figure 9 In the first four states, the cumulative survival probability of the bearings decreased to zero in a shorter time, only 47 years. Similarly, the cumulative survival probability functions of each component of the substructure and bridge deck system in states 1, 1 to 2, 1 to 3, and 1 to 4 were calculated, with the integral formula shown in formula (7):

[0118] (7);

[0119] in, Indicates the transition from state 1 to state 2. The cumulative survival probability function; Represents the probability density function. Represents state 1 to state 2. The cumulative probability density function.

[0120] S5. Model the performance degradation of the box girder bridge to obtain the transition probability. The specific operation is as follows:

[0121] S51. Calculating the conditional probability of a box girder bridge: The performance of a box girder bridge will decrease over time, meaning the bridge's state will transition from low to high. The transition between adjacent states exhibits the Markov property, meaning the box girder bridge's current state depends only on the current state and is independent of past states. Therefore, the conditional probability expression for a box girder bridge is:

[0122] (8);

[0123] in, Represents conditional probability; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; Indicates that in the known In all previous states The conditional probability at time t; Indicates that in the known In the state of time The conditional probability at that time.

[0124] S52. Based on the Markov property of S51, a Markov chain is used to pair states. The performance degradation model of the box girder bridge is performed. Under this model, it is considered that in a time series with fixed time intervals, the box girder bridge will change from its current state at the next moment (i.e., one fixed time unit away from the current moment). Transition to another state The transition probability can be represented by the conditional probability, as shown in formula (9); when multiple states transition simultaneously, a transition probability matrix is ​​formed, as shown in formula (10);

[0125] (9);

[0126] (10);

[0127] in, Indicates from state Transition to state The transition probability, , Represents conditional probability. Represents the transition probability matrix ( OK List), Indicates the total number of states. , .

[0128] S6. Calculate the state probability distribution of each component using the data obtained in S4. Specifically, this involves: [The text abruptly ends here, likely due to an incomplete sentence or missing information.] Corresponding stay time As a random variable, then in At that time, the box girder bridge passed the next time step. From state Transition to the next adjacent state The probabilities are given in formula (11). Formula (12) is used to calculate the state probability distribution of the upper load-bearing member, the upper general member, and the support, forming probability degradation curves. The probability degradation curves of the upper load-bearing member, the upper general member, and the support are shown in the figure. Figure 4 , Figure 7 , Figure 10 As shown.

[0129] Compare Figure 4 , Figure 7 and Figure 10 It can be observed that the three types of components, namely the upper load-bearing components, the upper general components, and the supports, gradually change from a low state to a high state over time, and the peak probability of states 1 to 4 decreases step by step. At any given time, the sum of the probabilities of all states is always 1. Figure 4 The upper and middle load-bearing components begin to enter state 5 in the 40th year and fully enter state 5 in the 120th year. Figure 7 The upper and middle general components begin to enter state 5 in the 18th year and fully enter state 5 in the 80th year; Figure 10 The middle support begins to enter state 5 in the 12th year and fully enters state 5 in the 47th year. Formulas (11) and (12) are expressed as follows:

[0130] (11);

[0131] (12);

[0132] in, express Time state To state The transition probability, Indicates the time step. Represents state 1 to state 2. The cumulative probability density function, Representation of components u No. The state probability distribution of the year Representing components No. The state probability distribution of the year The transition probability matrix can be obtained by substituting formula (11) into formula (10). Represents a time variable. Let be a vector, when hour, , indicating that the component is in The probability of being in each state, in The probability of being in state 1 is 1, and the probability of being in any of the other four states is 0.

[0133] S7. Based on the state probability distribution of each component obtained in S6, a probability degradation curve is generated using a dynamic Bayesian network. The specific operation is as follows:

[0134] S71. Based on the state probability distribution of each component obtained in S6, the state probability distribution of the superstructure, substructure, and bridge deck system is calculated using formula (13) through a dynamic Bayesian network, forming the probability degradation curves of each component. The probability degradation curves of the superstructure, substructure, and bridge deck system are as follows: Figure 11 , Figure 12 , Figure 13 As shown, comparison Figure 11 , Figure 12 and Figure 13 It can be seen that the curves in the three graphs have both similarities and differences. The similarities are: the peak probability of states 1 through 4 decreases progressively over time; at any given time point, the sum of the probabilities of all states is always 1; and the maximum probability of states 1 and 5 is also 1. The differences are: Figure 11 In the process, the curves of states 4 and 5 are slightly less smooth than those of states 1 and 2. The superstructure enters state 5 on average in the 70th year and fully enters state 5 in the 120th year. Figure 12 In the middle, the curves of states 3, 4 and 5 of the lower structure are compared to Figure 11 and Figure 13 The curve corresponding to the middle state is smoother. The speed of entering state 5 is relatively slow in the first 60 years, but the speed accelerates in the later period, and state 5 is fully entered by the 120th year. Figure 13 Among them, states 3, 4, and 5 have the largest peak descent gradients, with state 4 showing a more severe left skew. After entering state 5, the initial rise is faster, but slows down later, and the state is fully entered in the 50th year. The expression of formula (13) is:

[0135] (13);

[0136] in, Indicates components The state probability distribution curve, Representing components Transfer to component The probability, q This indicates the total number of components contained in the component; the transition probability is obtained from Table 3:

[0137] Table 3 Weight values ​​(transition probabilities) of various components in box girder bridges

[0138] ;

[0139] S72. Based on the state probability distribution of each component obtained in S71, the state probability distribution of the overall structure is calculated using formula (14) through a dynamic Bayesian network, forming the overall structure probability degradation curve, such as... Figure 14 As shown in the figure, after the overall structure enters state 3, the probability increase trend is relatively slow from the 9th to the 18th year; the probability increase in state 4 also slows down from the 10th to the 38th year; and state 5 shows a relatively slow upward trend from the 20th to the 61st year. This indicates that the overall degradation law of the bridge structure is: rapid degradation in the early stage, slow degradation in the middle stage, and rapid degradation again in the later stage over a relatively long period. In the 72nd year, the bridge has a 50% probability of entering a severely damaged state (state 5). Therefore, it can be inferred that the average lifespan of the bridge is about 72 years, and there is less than a 10% probability that it can survive to 100 years. The expression of formula (14) is:

[0140] (14);

[0141] in, The state probability distribution curve representing the overall structure. Indicates components The state probability distribution curve, Indicates components The probability of transferring to the overall structure, r The total number of components in the overall structure is shown in Table 4, and the transition probabilities are as follows:

[0142] Table 4 Weights (Transition Probabilities) of Each Component in a Box Girder Bridge

[0143] ;

[0144] S8. Evaluate the impact of each component on the probability degradation curve. Specifically, at a specific service time of the box girder bridge, replace or repair one of the components, calculate the degree of impact of that component on the probability degradation curve of its component and the overall structure, and then evaluate its sensitivity to the overall structural performance.

[0145] Assumption 1: Assume that the box girder bridge bearings are replaced with new bearings in the 15th year. Therefore, the bearings will degrade again after the 15th year. The probability degradation curve of the bearings after the re-degradation is as follows: Figure 15 As shown in the figure, the supports that had degraded again completely degraded in the 60th year.

[0146] Using a dynamic Bayesian network, the probability degradation curve of the superstructure after support replacement can be obtained as follows: Figure 16 As shown, comparison Figure 11 and Figure 16It can be seen that the probability of the superstructure being in state 1 has increased, but the increase in its duration is not significant. Meanwhile, the time when the probability of the superstructure being in state 2 decreases is delayed, but the duration of that state also does not show a significant increase. Overall, during the period from the 15th to the 60th year after the support replacement, the state of the superstructure has improved to some extent, but the effect is not very significant. Furthermore, since the support has completely degraded after the 60th year and can no longer affect the performance of the superstructure, from the 60th year onwards, the superstructure will continue to develop according to its original degradation curve.

[0147] Using a dynamic Bayesian network, the overall structural probability degradation curve after support replacement can be obtained as follows: Figure 17 As shown, comparison Figure 17 and Figure 14 It was found that, apart from slight fluctuations in states 1 and 2 in the 15th year, states 3 to 5 remained almost unchanged. From the perspective of structural performance transfer, since the probability of the support performance state being transferred to the superstructure is small, and the probability of it being transferred from the superstructure to the overall structure is even smaller, the support replacement measure does not have a significant effect on improving the overall structural performance.

[0148] Assumption 2: It is assumed that the superstructure's load-bearing components gradually age over time, leading to a decline in their technical performance. To improve their condition, reinforcement and maintenance are performed on the superstructure in its 60th year of service. It is assumed that the reinforced superstructure's load-bearing components recover to 70% of their condition after reinforcement and maintenance, similar to when the bridge was new. The probability degradation curve of the reinforced superstructure's load-bearing components is shown below. Figure 18 As shown. Figure 18 and Figure 4 Comparative analysis shows that the component only changed from state 2 to state 5 in the 60th year. In the 115th year, the component has a 50% probability of entering state 5, and its complete degradation point is the 157th year. Compared with before maintenance, its service life has been extended by 37 years.

[0149] Using a dynamic Bayesian network, the state probability degradation curve of the reinforced superstructure can be obtained as follows: Figure 19 As shown in the figure, significant abrupt changes occurred from state 2 to state 5 in year 60, with abrupt changes of 0.426, 0.138, -0.486, and -0.081, respectively. This result indicates that the reinforcement and repair of the superstructure's load-bearing components significantly improved the condition of the superstructure and effectively enhanced its performance.

[0150] The state of the overall structure after the reinforcement of the superstructure load-bearing components is further analyzed using a dynamic Bayesian network, and its probability degradation curve is shown below. Figure 20 As shown. Figure 20 and Figure 14Comparative analysis revealed that in year 60, the probability degradation curve for the overall structure being in state 2 increased significantly, the probability of being in state 3 initially decreased and then slightly increased, the probability of being in state 4 decreased significantly, and the probability of being in state 5 decreased slightly. Furthermore, the time point corresponding to a 50% probability value in the state 5 curve was delayed from year 71 to year 81, resulting in a 10-year increase in the average lifespan of the structure. These analytical results clearly demonstrate that implementing reinforcement and maintenance measures on the superstructure load-bearing components can significantly extend the overall service life of the bridge and improve its durability and reliability.

[0151] Similarly, based on Markov theory and with the help of dynamic Bayesian networks, it is possible to conduct in-depth analysis on the improvement of the condition of other components and the overall bridge structure after replacement or maintenance, thereby providing a scientific basis for the maintenance and management of bridge structures.

[0152] Therefore, the present invention adopts the above-mentioned method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distribution, which can more scientifically predict the performance degradation of box girder bridges from components to parts and from parts to the whole, thereby predicting the service life of box girder bridges from a probabilistic perspective.

[0153] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for assessing the probability of performance degradation of a box girder bridge system based on DBN and Gamma distributions, characterized by: Includes the following steps: S1. Determine the composition of the box girder bridge; S2. Determine the dwell time of each component of the box girder bridge; S3. Using the dwell time of each component of the box girder bridge obtained from S2, calculate the relevant parameters of the probability density function of the Gamma distribution; The specific operation of S3 is as follows: S31. Let the dwell time of each component of the box girder bridge in each state be a random variable. t , t If it follows a Gamma distribution, then the state i Depending on the length of stay t The probability density function of the change is given by formula (2), and the state... i The survival probability function is given by formula (4), and the probability density function of the Gamma distribution is given by formula (3). Formulas (2), (3), and (4) are expressed as follows: (2)； (3)； (4)； Represents random variables t The probability density function, The proportional parameter represents the probability density function of the Gamma distribution. This represents the shape parameter of the probability density function of the Gamma distribution. Represents an exponential function. Represents the survival probability function; S32. By combining formulas (5) and (6), the state is calculated using a genetic algorithm. i The proportional parameter of the probability density function of the Gamma distribution and shape parameters Formulas (5) and (6) are expressed as follows: (5)； (6)； Where min represents minimization. Indicates time, Representing state i The average survival probability, t i Representing state i The length of stay; S4. Calculate the cumulative probability density function and cumulative survival probability function of each component of the box girder bridge using Monte Carlo simulation; S5. Model the performance degradation of box girder bridges to obtain the transition probability; The specific operation of S5 is as follows: S51. Calculating the conditional probability of a box girder bridge: The performance of a box girder bridge decreases over time, and its state transitions from low to high. The transition between adjacent states exhibits the Markov property, meaning the next state depends only on the current state and is independent of past states. Therefore, the conditional probability expression for a box girder bridge is: (8)； in, Represents conditional probability; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; express Time-state random variables; express The specific value of the state random variable at that time; Indicates that in the known In all previous states The conditional probability at that time; Indicates that in the known In the state of time The conditional probability at that time; S52. Based on the Markov property of S51, a Markov chain is used to pair states. i The performance degradation model of the box girder bridge is performed. Under this model, it is considered that in a time series with fixed time intervals, the box girder bridge will change from its current state at the next moment, that is, one time unit apart from the current moment. i Transition to another state j transition probability The conditional probability is expressed as shown in formula (9); when multiple states transition simultaneously, a transition probability matrix is ​​formed, as shown in formula (10). (9)； (10)； in, Indicates from state Transition to state j The transition probability, j≥i , Represents conditional probability. express m OK m The transition probability matrix of the column. m Indicates the total number of states. , ; S6. Calculate the state probability distribution of each component using the data obtained in S4; S7. Based on the state probability distribution of each component obtained in S6, the probability degradation curves of each component and the overall structure are formed by a dynamic Bayesian network. S8. Evaluate the impact of the component on the degradation curve of its constituent parts and the overall structure.

2. The method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distribution according to claim 1, characterized in that: The specific operation of S1 is as follows: S11. A box girder bridge consists of a superstructure, a substructure, and a bridge deck system. The superstructure includes upper load-bearing components, upper general components, and supports. The substructure includes slope protection, abutments, and pier foundations. The bridge deck system includes bridge deck pavement, expansion joint devices, guardrails, and a drainage system. S12. Establish a topology containing circular nodes, line segments, and connecting arrows to form a dynamic Bayesian network. The circular nodes represent the probabilities of each state of a component, part, or overall structure at a certain time point, the connecting arrows represent the transition probabilities between nodes, and the nodes at both ends of a line segment have the same meaning. The topology consists of a component layer, a part layer, and an overall structure layer from bottom to top. The component layer is the observation layer, the part layer is the first hidden layer, and the overall structure layer is the second hidden layer.

3. The method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distribution according to claim 2, characterized in that: The specific operation of S2 is as follows: S21. Determine the composition weights of the component layer and the part layer in the topology structure in S12, and use them as the transition probabilities from the component layer to the part layer and from the part layer to the overall structure layer in the dynamic Bayesian network, respectively. Divide the detection score of the component layer in the topology structure into 5 score intervals, which correspond to the 5 states of the component layer. S22. In S21, since the time value corresponding to the 5th scoring interval is infinite, only the first 4 scoring intervals are linearly fitted according to the detection score. The endpoint scores of the first 4 scoring intervals are substituted into the linear scoring curve fitted by the first 4 scoring intervals respectively, and the time points corresponding to the endpoint scores of the scoring intervals are extracted from the scoring curves. The dwell time of each component in each state is calculated based on the time points, as shown in formula (1): (1)； in, Representing state The length of stay Representing state The right endpoint of the time interval of stay Representing state The right endpoint of the time interval of stay.

4. The method for evaluating the probability of performance degradation of a box girder bridge based on DBN and Gamma distribution according to claim 3, characterized in that: The specific operation of S4 is as follows: S41. Denote the first four states as State 1, State 2, State 3, and State 4. Use Monte Carlo simulation to calculate the cumulative probability density function of each component in the superstructure in State 1, from State 1 to State 2, from State 1 to State 3, and from State 1 to State 4. Similarly, the cumulative probability density functions of each component of the substructure and bridge deck system in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4 are calculated. ; S42. Calculate the cumulative survival probability function of each component of the superstructure in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4 using the integral formula. Similarly, calculate the cumulative survival probability function of each component of the substructure and bridge deck system in state 1, state 1 to state 2, state 1 to state 3, and state 1 to state 4. The integral formula is shown in formula (7). (7)； in, Represents the transition from state 1 to state 2. The cumulative survival probability function; Represents the probability density function. Represents state 1 to state 2. The cumulative probability density function.

5. The method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distribution according to claim 4, characterized in that: The specific operation of S6 is: to change the state Corresponding stay time As a random variable, then in At that time, the box girder bridge is in the next time step. From state Transition to the next adjacent state The probability is given by formula (11). Formula (12) is used to calculate the state probability distribution of the upper load-bearing member, the upper general member, and the support, forming a probability degradation curve. Formulas (11) and (12) are expressed as follows: (11)； (12)； in, express Time state To state The transition probability, Indicates the time step. Represents state 1 to state 2. The cumulative probability density function, Representation of components u No. The state probability distribution of the year Representation of components u No. The state probability distribution of the year The transition probability matrix is ​​obtained by substituting formula (11) into formula (10). Represents a time variable. Let be a vector, when hour, , indicating that the component is in The probability of being in each state, in The probability of being in state 1 is 1, and the probability of being in any of the other four states is 0.

6. The method for evaluating the overall performance degradation probability of box girder bridges based on DBN and Gamma distribution according to claim 5, characterized in that: The specific operation of S7 is as follows: S71. Based on the state probability distribution of each component obtained in S6, the state probability distribution of the superstructure, substructure, and bridge deck system is calculated using formula (13) through a dynamic Bayesian network, forming the probability degradation curve of each component. The expression of formula (13) is as follows: (13)； in, Indicates components l The state probability distribution curve, Representation of components Transfer to component l The probability, q Indicates the total number of components contained in the component; S72. Based on the state probability distribution of each component obtained in S71, the state probability distribution of the overall structure is calculated using formula (14) through a dynamic Bayesian network, forming the overall structure probability degradation curve. The expression of formula (14) is: (14)； in, The state probability distribution curve representing the overall structure. Indicates components The state probability distribution curve, Indicates components The probability of transferring to the overall structure, r This indicates the total number of components contained in the overall structure.

7. The method for evaluating the probability of performance degradation of a box girder bridge system based on DBN and Gamma distribution according to claim 6, characterized in that: The specific operation of S8 is as follows: at a specific service time of the box girder bridge, one of the components is replaced or repaired, and the influence of the component on the probability degradation curve of its component and the overall structure is calculated, thereby assessing its sensitivity to the overall structural performance.

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