A method and system for evaluating performance of a main beam based on vehicle-induced deflection over-limit probability

Abstract

The application discloses a kind of main beam performance evaluation method and system based on vehicle-induced deflection over-limit probability, related to structural performance evaluation field, by collecting deflection and vehicle monitoring data and preprocessing, separate vehicle-induced deflection component;Adopting moving block self-help method, block self-sampling is retained, data autocorrelation;Combined with POT method and generalized Pareto distribution, determine the optimal threshold by multiple testing, calculate single test point over-limit probability;Again through hierarchical sampling, Monte Carlo simulation and Gaussian kernel density estimation, quantize single test point and main beam overall time-varying over-limit probability and 95% confidence interval, combine with specification to evaluate stiffness performance and identify high-risk cross section.The application can accurately calculate the over-limit probability distribution of time-varying vehicle-induced deflection, quantify over-limit risk and uncertainty.Not only improve the utilization degree of monitoring data, but also provide theoretical support for the optimization of bridge maintenance strategy and the improvement of service safety, which has important technical significance for bridge life prediction and long-term structural safety evaluation.

Description

Technical Field

[0001] This invention relates to the field of structural performance evaluation technology, and more specifically to a method and system for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding the limit. Background Technology

[0002] Currently, in bridge health monitoring, deflection, influenced by both the loads on the main girder and its mechanical properties, is key data for bridge health assessment and safety analysis. The probability of time-varying vehicle-induced deflection exceeding a certain threshold, as a deflection-based indicator, reflects the risk of a bridge becoming unusable and can effectively reflect the performance of the main girder, providing important evidence for bridge safety assessment and service life prediction. However, research on the probability of vehicle-induced deflection exceeding the limit has not adequately considered the autocorrelation of monitoring data and the coupling effect between multiple vehicle loads, and the reliability of its calculation results is difficult to quantify, raising questions about the reliability of the main girder performance evaluation results.

[0003] Existing studies on the probability of vehicle-induced deflection exceeding limits have the following shortcomings: they fail to reasonably consider the autocorrelation of monitoring data and the coupling effect between multiple vehicle loads, resulting in low accuracy of extreme value distribution fitting; they can only calculate the probability of exceeding limits at a single point, and cannot quantify the reliability of the calculation results, so the reliability of the main beam performance evaluation results is questionable; the traditional interval maximum method is significantly affected by the size of the time window, is highly subjective, and is prone to underestimating low-probability exceeding events; the traditional exceedance extreme value method (POT) does not introduce uncertainty analysis, and is prone to seriously underestimating the risk of exceeding limits of the main beam when predicting long-term service life.

[0004] Therefore, there is an urgent need to develop a main beam performance evaluation method that can take into account data autocorrelation, quantify the uncertainty of evaluation results, and accurately predict the probability of long-term over-limit performance. Summary of the Invention

[0005] In view of this, the present invention provides a method and system for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding the limit, in order to solve the problems existing in the background art.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits includes: Collect measured deflection data of the main beam of the bridge and vehicle monitoring data, preprocess the data to obtain vehicle-induced deflection component data; The vehicle-induced deflection component data and vehicle monitoring data are divided into blocks and self-sampled with replacement by setting the single sampling time length. The self-sampled deflection dataset and self-sampled vehicle dataset are then spliced ​​together according to the sampling order. A set of candidate thresholds with equal intervals is constructed. The average excess value corresponding to each threshold is calculated by the mean excess number method. The reasonable lower limit of the threshold is determined by combining the goodness of linear fitting. Within the reasonable threshold range, the optimal threshold is determined by the goodness of fitting test of the QQ plot. The independent extreme values ​​of the excess threshold are extracted and fitted with a generalized Pareto distribution. After passing the Anderson-Darling test, the probability of exceeding the limit of a single measurement point within the self-sampling period is calculated. Repeatedly calculate the out-of-limit probability of a single measurement point within the self-sampling period to obtain the set of out-of-limit probabilities for a single measurement point; set the evaluation time length, simulate the out-of-limit probability of multiple periods through hierarchical sampling, and combine Monte Carlo simulation and Gaussian kernel density estimation to calculate the most likely out-of-limit probability and 95% confidence interval of a single measurement point within the time length. The probability of exceeding the limit is calculated for all m deflection measurement points to obtain the set of probability of exceeding the limit for each measurement point; samples are independently extracted from the set of probability of exceeding the limit for each measurement point through hierarchical sampling, and the probability of exceeding the limit for a single sampling of the main beam is calculated by combining the series failure criterion; the most likely probability of exceeding the limit and the 95% confidence interval of the main beam within the time length L are obtained by Monte Carlo simulation and Gaussian kernel density estimation. Based on the most likely probability of the main beam exceeding the limit and the 95% confidence interval, combined with the deflection threshold specified in the highway bridge technical evaluation standard, the stiffness performance and service safety of the main beam are evaluated, and high-risk sections are identified.

[0007] Optionally, the preprocessing specifically includes: Data is subjected to outlier removal, missing value imputation, and decoupling of deflection components; Specifically, the deflection component decoupling involves extracting the temperature-induced deflection component from the measured deflection data, subtracting the temperature-induced deflection component from the original deflection data, and obtaining vehicle-induced deflection component data that includes vehicle-induced dynamic deflection and vehicle-induced static deflection.

[0008] Optionally, the single sampling time length t s The values ​​of satisfy: Where X is the total length of the bridge and y is the average speed of vehicles on the bridge, to ensure that the overlap rate between vehicle data and vehicle-induced deflection data is ≥95%; the number of self-sampling times is... Where t is the total duration of the original data, This is for rounding up. Optionally, constructing the equally spaced candidate threshold set {u} specifically involves: using the self-sampled deflection dataset... D ttsi Sort the data in ascending order and take the last one. n min The number of values ​​is used as the maximum threshold. u max Take the last one The values ​​are used as the minimum threshold. u minWhere H is the sampling frequency; in u min and u max Multiple threshold values ​​are taken at equal intervals to form a set of candidate thresholds. Optionally, the formula for calculating the probability of exceeding the limit for a single measurement point within the self-sampling period is:

[0009] in, Optimal threshold The corresponding number of independent extrema exceeding the threshold, This represents the total number of axles within the self-sampling period. The vehicle's deflection exceeded the threshold. Let be the cumulative distribution function of the generalized Pareto distribution. For shape parameters, For position parameters, This is the scale parameter.

[0010] Optionally, the calculation of the probability of exceeding the limit for a single measurement point within a time length L... The formula is:

[0011] in, The number of hierarchical samplings. To extract from the set of probabilities beyond the limit P f The Middle j The sample values ​​drawn this time. for P f The Middle One element. Optionally, the probability of the main beam exceeding the limit in a single sampling under the series failure criterion. The formula is:

[0012] Where m is the total number of deflection measurement points. For from the first i Sample values ​​drawn from the set of probabilities of exceeding limits at various measuring points. A main beam performance evaluation system based on the probability of vehicle-induced deflection exceeding limits includes: The data acquisition and preprocessing module is used to collect measured deflection data of the main beam of the bridge and vehicle monitoring data, complete outlier removal, missing value filling and decoupling of deflection components, and output vehicle-induced deflection component data. The moving block bootstrap self-sampling module is connected to the data acquisition and preprocessing module. It is used to set the single sampling time length, perform block-based self-sampling with replacement on vehicle-induced deflection component data and vehicle monitoring data, and output the self-sampling dataset. The single-point POT over-limit probability calculation module is connected to the moving block bootstrap self-sampling module. It is used to construct a set of candidate thresholds, determine the optimal threshold, fit the generalized Pareto distribution, and calculate the over-limit probability within the single-point self-sampling period. The Bootstrap time-varying probability calculation module is connected to the single-point POT over-limit probability calculation module. It is used to obtain the single-point over-limit probability set through repeated self-sampling, and combined with hierarchical sampling and Monte Carlo simulation, outputs the single-point time-varying over-limit probability and confidence interval. The overall probability calculation module for the main beam is connected to the Bootstrap time-varying probability calculation module. It is used to output the overall time-varying over-limit probability and confidence interval of the main beam based on the series failure criterion, through hierarchical sampling and Monte Carlo simulation. The performance evaluation output module is connected to the overall probability calculation module of the main beam. It is used to output the performance evaluation results and high-risk section information of the main beam based on the probability of the main beam exceeding the limit and the confidence interval, combined with the standard threshold.

[0013] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method and system for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding the limit, which has the following beneficial effects: 1. The moving block bootstrapping (MBB) method is used for self-sampling, which preserves the autocorrelation of time series data and ensures the matching of vehicle data and vehicle-induced deflection data, effectively taking into account the coupling effect of multi-vehicle loads. 2. By combining the POT method and GPD distribution, we focus on the tail characteristics of extreme values, thus avoiding the errors of modeling the entire distribution. The optimal threshold is determined by dual tests of MLR linear fitting and QQ plot goodness of fit, which improves the fitting accuracy of the extreme value distribution. 3. This invention introduces the Bootstrap method and Monte Carlo simulation to quantify the uncertainty of the out-of-limit probability calculation results, providing the most likely value and a 95% confidence interval, making the evaluation results more credible; 4. This invention uses the series failure criterion to calculate the overall over-limit probability of the main beam, which accurately reflects the engineering reality that "over-limit deflection of any section of the main beam will result in overall failure"; 5. This invention can accurately calculate the probability distribution of over-limit deflection caused by time-varying vehicles, quantifying the risk and uncertainty of over-limit deviations. This method not only improves the utilization of monitoring data but also provides theoretical support for optimizing bridge maintenance strategies and enhancing service safety, and has significant technical implications for bridge life prediction and long-term structural safety assessment. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0015] Figure 1 This is a schematic diagram illustrating the principle of the method provided by the present invention; Figure 2 This is a schematic diagram of the original deflection data and the vehicle-induced deflection component curves provided by the present invention, wherein (a) is the original data and (b) is the vehicle-induced deflection component; Figure 3 This is a schematic diagram of the self-sampling results of the vehicle-induced deflection component provided by the present invention; Figure 4 This is a schematic diagram illustrating the relationship between the number of elements in the excess value set and the threshold provided by the present invention; Figure 5 Provided by the present invention Relationship diagram; Figure 6 Provided by the present invention Relationship diagram; Figure 7 Provided by the present invention Relationship diagram; Figure 8 This is a schematic diagram of the u-maximum likelihood relationship provided by the present invention.

[0016] Figure 9 The GPD estimation results with a log maximum likelihood value of 40.759 provided by the present invention are shown in (a) as a QQ plot and (b) as a schematic diagram of data and distribution. Figure 10 The GPD estimation results with a log maximum likelihood value of 6.777 provided by the present invention are shown in (a) as a QQ plot and (b) as a schematic diagram of data and distribution. Figure 11 This is a schematic diagram of the excess value provided by the present invention, wherein (a) represents the threshold, the excess value, and the overall data, and (b) represents a partial schematic diagram; Figure 12 This is a schematic diagram of the AD test results provided by the present invention; Figure 13 Provided by the present invention Statistical results diagram; Figure 14 The probability distribution of exceeding the limit for D-G10-1 within one year provided by this invention; Figure 15The probability distribution of D-G10-1 provided by this invention over 10 years, 25 years, 50 years and 100 years, where a, b, c and d correspond to 10 years, 25 years, 50 years and 100 years respectively; Figure 16 The probability distribution of exceeding the limit of D-G10-1 over different time lengths provided by the present invention; Figure 17 26 sets provided for this invention The number of elements; Figure 18 The most likely probability of exceeding the limit and the 95% confidence interval for the 26 measurement points provided by this invention in one year; Figure 19 The present invention provides an estimate of the probability distribution of the main beam exceeding the limit within one year; Figure 20 The most likely exceedance probability and 95% confidence interval for 26 measurement points provided by this invention within 10 years, 25 years, 50 years and 100 years, where a, b, c and d correspond to 10 years, 25 years, 50 years and 100 years respectively; Figure 21 The present invention provides an estimate of the probability distribution of the main beam exceeding the limit within 10 years, 25 years, 50 years and 100 years, where a, b, c and d correspond to 10 years, 25 years, 50 years and 100 years respectively; Figure 22 This invention provides an estimate of the probability distribution of the main beam exceeding the limit over different time lengths. Detailed Implementation

[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0018] This invention discloses a method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits, such as... Figure 1 As shown, it includes: Collect measured deflection data of the main beam of the bridge and vehicle monitoring data, preprocess the data to obtain vehicle-induced deflection component data; The vehicle-induced deflection component data and vehicle monitoring data are divided into blocks and self-sampled with replacement by setting the single sampling time length. The self-sampled deflection dataset and self-sampled vehicle dataset are then spliced ​​together according to the sampling order. A set of candidate thresholds with equal intervals is constructed. The average excess value corresponding to each threshold is calculated by the mean excess number method. The reasonable lower limit of the threshold is determined by combining the goodness of linear fitting. Within the reasonable threshold range, the optimal threshold is determined by the goodness of fitting test of the QQ plot. The independent extreme values ​​of the excess threshold are extracted and fitted with a generalized Pareto distribution. After passing the Anderson-Darling test, the probability of exceeding the limit of a single measurement point within the self-sampling period is calculated. Repeatedly calculate the out-of-limit probability of a single measurement point within the self-sampling period to obtain the set of out-of-limit probabilities for a single measurement point; set the evaluation time length, simulate the out-of-limit probability of multiple periods through hierarchical sampling, and combine Monte Carlo simulation and Gaussian kernel density estimation to calculate the most likely out-of-limit probability and 95% confidence interval of a single measurement point within the time length. The probability of exceeding the limit is calculated for all m deflection measurement points to obtain the set of probability of exceeding the limit for each measurement point; samples are independently extracted from the set of probability of exceeding the limit for each measurement point through hierarchical sampling, and the probability of exceeding the limit for a single sampling of the main beam is calculated by combining the series failure criterion; the most likely probability of exceeding the limit and the 95% confidence interval of the main beam within the time length L are obtained by Monte Carlo simulation and Gaussian kernel density estimation. Based on the most likely probability of the main beam exceeding the limit and the 95% confidence interval, combined with the deflection threshold specified in the highway bridge technical evaluation standard, the stiffness performance and service safety of the main beam are evaluated, and high-risk sections are identified.

[0019] Specifically, the aim is to use measured deflection data and vehicle data over a period of time as a basis to determine a given vehicle-induced deflection threshold. Under the premise of calculating different time lengths L Internal single deflection measurement point Most likely probability of exceeding the limit 95% confidence interval lower bound Upper limit of the 95% confidence interval The most likely main beam Lower bound of the 95% confidence interval for the probability of exceeding the limit Upper bound of the 95% confidence interval for the probability of exceeding the limit .

[0020] Specifically, the method of the present invention is based on the length of time. t Inside, the actual vehicle monitoring data was measured. With deflection data On this basis. Among them, Including the specific time each vehicle enters the bridge. With the number of axles , of which j The specific time and number of axles of each vehicle going onto the bridge were recorded as follows: and ; This refers to the vehicle-induced deflection components obtained after preprocessing the measured deflection data, including a total ofm Each deflection measurement point over time t The monitoring data, with a sampling frequency of H , of which i The monitoring data from each measuring point was recorded as , Includes a series of deflection data and its corresponding collection time , of which a Each deflection data point and its corresponding acquisition time are recorded as follows: and The complete calculation process is as follows: 1) with the first i Taking one deflection measurement point as an example: 1.1) Define the single sampling time length ( ); minimum threshold Maximum threshold Threshold number ( ); resampling time interval Minimum number of excess values ​​and number of fitted values Number of self-sampling times Vehicle-induced deflection exceeding threshold .

[0021] 1.2) In terms of time length t A segment of length is randomly selected from within. The corresponding extraction time Internal deflection and vehicle monitoring data and Repeat this random sampling process for a total of Second-rate( (For rounding up calculation), to obtain indivual and Assemble them in the order they were extracted. indivual ,get Similarly, they are assembled sequentially according to the order in which they were extracted. indivual ,get . Including deflection data and its corresponding collection time Reset For a starting value of 0, with an interval of... , length equal to A set of:

[0022] Right now The Middle a For Its corresponding For the remastered The first in a indivual, This process is called a self-sampling. That is, the first i Each deflection measuring point is based on the length of a single sampling time. Data obtained from a single self-sampling operation; That is, vehicle data is based on the length of a single sampling time. The total number of axles obtained from a single sampling is .

[0023] 1.3) Create a threshold u gather :

[0024] in, total 1 element, arranged in ascending order, with intervals between elements. .

[0025] 1.3.1) with The Middle b Threshold For example, All of the above The deflection data are The corresponding collection time is There are a total of One element. (For) Performing a first-order difference, we obtain ; All of the above The elements in The index in is ,remember The Middle e The elements are There are a total of E Each element. Indicates in The Middle f The first to the second g A subset consisting of -1 elements is denoted as In the interval where the left is closed and the right is open A subset of.

[0026] Pick exist Multiple subsets are selected, and only the maximum value is retained in each subset to form the set of maximum vehicle-induced deflection values. .

[0027] 1.3.2) Calculation The mean, denoted as .remember for .

[0028] 1.4) Refer to steps 1.3.1 to 1.3.2 to calculate. The set of maximum vehicle-induced deflection values ​​corresponding to each threshold, forming the set of maximum vehicle-induced deflection values. and sets .in, and The number of elements is indivual, The Middle k a set and sets The Middle k element Corresponding to The Middle k Threshold Refer to the calculation results in steps 1.3.1 to 1.3.2. If The number of elements is less than At the same time , , Remove the first k Element. Update After removal Number of elements.

[0029] 1.4.1) with The Middle b Threshold For example, take exist subset of , exist subset of . The Middle j Individual and No. j The two pairs match, therefore the least squares method can be used for fitting. and The formula is obtained. Based on this formula and Calculate the prediction set .based on and Calculate goodness of fit .

[0030] 1.5) For The first to the second indivual Refer to step 1.4.1 to calculate and obtain the set. . The CCP The element, where the first element is... k element Corresponding to The Middle k Threshold The result is obtained from step 1.4.1. The index of the largest element in the array is , The Middle One was recorded as ,Pick exist subset of , exist subset of .

[0031] 1.5.1) with The Middle b Threshold For example, The Middle b For There are a total of Each element. Subtract each element from , recorded as Using the maximum likelihood method, estimate The generalized Pareto distribution to which it belongs (GPD) parameters The cumulative integral function is obtained. .Will Sort by elements in ascending order, and you get ; Calculate sets :

[0032] based on and Calculate goodness of fit .

[0033] 1.6) For Each of them Refer to step 1.5.1 to calculate and obtain the set. . The CCP The element, where the first element is... k element Corresponding to The Middle k Threshold The result is calculated according to step 1.5.1. The index of the largest element in the array is , The Middle One was recorded as , There are a total of There are elements. Using the maximum likelihood method, estimate... The generalized Pareto distribution to which it belongs (GPD) parameters The Anderson-Darling test was used to calculate... p Value. If p If the value is less than 0.05, discard the current result; if p If the value is greater than 0.05, calculate the probability that the maximum vehicle-induced deflection exceeds the over-limit threshold. :

[0034] 1.7) Repeat steps 1.2 to 1.6 for a total of Next, we can obtain the set. The set has 100 elements. ( ); for repeating steps 1.2 to 1.6 j The calculation result is... The first in j Each element.

[0035] 1.8) Define the evaluation time length L Number of samplings .

[0036] 1.9) From Random selection with replacement Next, form a new set. There are a total of Each element. The calculation method is as follows:

[0037] The Middle j The elements are denoted as Then the first i Each deflection measurement point over time L Internal over-limit probability for:

[0038] in for The Middle Each element.

[0039] 1.10) Repeat the calculation of step 1.9. Next, get indivual , denoted as set Gaussian kernel density estimation (KDE) was used to establish... The distribution and bandwidth were determined using the Scott method. Based on the KDE estimation results, the method with the highest probability density was used. As the first i Each deflection measurement point over time L Most likely inside The cumulative integral probability of 2.5% is used as the lower bound of the 95% confidence interval for the out-of-limit probability. The cumulative integral probability of 97.5% is used as the upper bound of the 95% confidence interval for the probability of exceeding the limit. .

[0040] 1.11) For different time lengths to be evaluated For each of them L Referring to steps 1.8 to 1.10, we can obtain the... i Deflection measurement points at different time lengths L Within, the most likely probability of exceeding the limit 95% confidence interval lower bound Upper limit of the 95% confidence interval .

[0041] 2) For m For each deflection measurement point, referring to calculation steps 1.1 to 1.10, calculate the data for each measurement point to obtain the set. . Depend on m Composed of sets, the first set i Each sensor corresponds to The Middle i Let there be a set, denoted as . That is, the first i Each deflection measurement point over time L The set of probabilities of exceeding the limit within the inner limit, There is Each element.

[0042] 2.1) Define the number of samples .

[0043] 2.2) For The Middle i a set ,from Randomly select one element from the data, denoted as . .

[0044] 2.3) For The first to the second mFor each set, refer to step 2.2 to calculate and obtain the set. . The CCP m The element, where the first element is... i The element is Therefore, the length of time L The probability of the main beam exceeding the limit based on the series exceedance criterion is: .

[0045] 2.4) Repeat steps 2.2 to 2.3 for a total of Next, we can obtain the set. . The number of elements in the middle is One. Established using Gaussian KDE. The distribution and bandwidth were determined using the Scott method. Based on the Gaussian KDE results, the method with the highest probability density was used. As the main beam, over a period of time L Most likely inside The cumulative integral probability of 2.5% is used as the lower bound of the 95% confidence interval for the out-of-limit probability. The cumulative integral probability of 97.5% is used as the upper bound of the 95% confidence interval for the probability of exceeding the limit. .

[0046] 3) For different evaluation time lengths For each of them L Referring to steps 2.1 to 2.4, the main beam at different time lengths can be obtained. L Most likely inside Lower bound of the 95% confidence interval for the probability of exceeding the limit Upper bound of the 95% confidence interval for the probability of exceeding the limit .

[0047] In general, the above calculation process mainly includes: 1) using the POT method and resampling method to establish the conditional generalized Pareto distribution (GPD) of the maximum vehicle-induced deflection under the condition of exceeding the threshold; based on the time covered by the data... t For vehicles on the bridge, the GPD is converted into the tail of the extreme value distribution of the maximum vehicle-induced deflection to solve the decoupling problem between autocorrelation and extreme value distribution. 2) Using the Bootstrap method, the probability distribution of the maximum vehicle-induced deflection exceeding the limit within the time t covered by the data is established. To solve the problem of confidence assessment of extreme value distributions; 3) Using hierarchical sampling methods, from time... t Probability distribution of maximum deflection caused by the inner vehicle Medium sampling, progressively calculating over longer periods T The probability of exceeding the limit within the time frame; based on multiple measurement points in time. TThe probability distribution of exceeding limits within the time frame is calculated by combining cascaded exceeding limits and hierarchical sampling methods to address the problem of extrapolating time-varying exceeding limits probabilities. The calculation details and basis will be explained in detail later.

[0048] Self-sampling method for deflection time series monitoring data

[0049] Step 1.2, known as the self-sampling process, is the core computational step of the Bootstrap method. Considering the autocorrelation inherent in time series data, the self-sampling of the time series data is performed with a time length... The process is performed in a block-by-block manner, known as the Moving Block Bootstrap (MBB) method. In the MBB method, the duration of each block is fixed. Multiple blocks are extracted with a return value and concatenated in the extraction order to obtain a single self-sampling result. The parameters to be determined in the MBB method include the duration of each block. and the number of sampling times.

[0050] Length of each block The value is based on the following: In bridge health monitoring systems, the vehicle dynamic weighing system (WIM) is typically located on one side of the bridge, while deflection measurement points are distributed across multiple sections of the main beam. Therefore, the actual duration of the vehicle on the bridge is unknown, within a certain time range. It is essential to ensure that the data on vehicles crossing the bridge and the corresponding deflection data match. The reason for this matching will be explained later. Let the bridge length be... X meters, average vehicle speed on the bridge is y If the average time is calculated per second per meter, then the average time for vehicles to cross the bridge is [missing information]. X / y Seconds. The overlap rate between vehicle data and vehicle-induced deflection data depends on whether the deflection generated by the vehicle while driving on the bridge falls within a certain time period. The data was collected within the area. Given that WIM is located on one side of the bridge and there is two-way traffic on the bridge, the deflection measurement point farthest from WIM was measured over a time period of [time value missing]. Of the deflection data collected within seconds, there are ( - / A time interval of ) seconds ensures that the vehicle both passes through the WIM and affects the deflection measurement points. Therefore, the data overlap rate during this time period can be defined as ( - / ) / A coincidence rate greater than 95% indicates that the vehicle data and vehicle-induced deflection data are essentially identical, requiring a time length... ≥20 / .

[0051] The value of the sampling number is based on the fact that the Bootstrap method requires sampling an equal number of new samples as the original samples for statistical calculations. Therefore, the sampling number in step 1.2 is set to [value missing]. This ensures that the self-sampling results are close to and slightly larger than the time coverage length of the original data. t .

[0052] A Method for Calculating the Probability of Exceeding the Maximum Value of Vehicle-Induced Deflection Based on the POT Method

[0053] In steps 1.3 to 1.6, the POT method is mainly used to establish the maximum distribution of vehicle-induced deflection based on GPD; combined with a given vehicle-induced deflection exceeding the limit threshold. The probability that the maximum vehicle-induced deflection exceeds the over-limit threshold was calculated. The POT method used here is based on the second theorem of extremum theory: Let the data X The distribution function is For a given threshold u From the data X Extract all items exceeding the threshold u value Defined as ,in , This is an excess value. Therefore, y The conditional cumulative integral function that it obeys for:

[0054] when Over-distribution Converging to GPD:

[0055] The probability density function of the GPD distribution:

[0056] in, This is an excess value; , is a scale parameter that controls the expansion of the tail; The shape parameter controls the tail thickness; It is a location parameter that controls the starting point of the distribution. The distribution has a thick tail (such as the Pareto distribution). The distribution is an exponential distribution; The distribution of GPD has a truncated tail. The parameters in the GPD distribution can be estimated using maximum likelihood estimation (MLE). .

[0057] Compared to the POT method, another commonly used method for modeling deflection maxima is the interval maxima method. This involves taking the maximum value within a given time interval and fitting it with a generalized extreme value (GEV) distribution. Both the POT and interval maxima methods are commonly used to establish the distribution of vehicle-induced deflection maxima. This invention uses the POT method because: 1) POT is more flexible: It focuses on tail-end characteristics. Since it's difficult to completely separate each vehicle deflection value from the deflection data, it's hard to accurately separate small vehicle-induced deflection maxima. The POT method avoids the inaccuracies of modeling the full distribution of vehicle-induced deflection maxima, only analyzing data exceeding a threshold. 2) It has a solid theoretical foundation: Based on the second theorem of extreme value theory, it provides statistical assurance. Compared to estimating GEV using the extreme values ​​within an interval after taking the interval, the large interference of the interval size selection and the lack of a unified standard for interval size selection lead to subjectivity and instability in the interval maxima method, potentially resulting in significant deviations in the results. 3) When all data exceeding the threshold are independent, the GPD obtained by the POT method is equivalent to the tail of the original distribution and can be used to estimate the probability of tail events in the complete distribution. The POT method focuses on independent and identically distributed data exceeding a certain threshold. u All observations, referred to as excess values, are used to model tail events by statistically analyzing their distribution. Therefore, the key to the POT method lies in choosing a reasonable threshold. u A threshold that is too high may result in too few samples, while one that is too low may introduce non-extreme data. A common approach is to use the Mean Residual Life (MRL) method for threshold selection. For each threshold... u ,calculate :

[0058] Calculate Then, draw A curve graph. For a given... Value, its right side If the curve is linear, then... The value is a reasonable lower threshold. The validity of the mean supernormal method is based on the derivation from the definition of GPD and the properties of expected value, that for a given GPD, when... (Exponential distribution) ;when (Thick-tailed distribution) Approximately linear growth; when (Cutoff distribution) Gradually decrease to 0. Therefore, draw... Follow The change graph, find the one that makes The minimum point at which the graph begins to plateau or grow linearly. Using a reasonable lower limit for the threshold is a reliable processing method. Alternatively, a goodness-of-fit test can be used to assess the reasonableness of the threshold value by examining the degree of matching between the fitted GPD distribution and the fitted data.

[0059] Step 1.3 essentially establishes a set of candidate thresholds. Determine the specific Three parameters are required: minimum threshold Maximum threshold Threshold number ( ). Among them, the present invention will In Sort by size from smallest to largest, then take the last one. The number of values ​​is used as the maximum threshold. Take the last one The values ​​are used as the minimum threshold. ( for (Total number of data points), number of threshold values ​​with equal intervals. =500 thresholds. This method of selection ensures that for the threshold set... any threshold in , Medium greater than the threshold The number of data points is in [ Between these, on the one hand, the threshold set is guaranteed. Threshold in All are relatively large values, all greater than 90%. On the one hand, the data points ensure that the sample size is greater than the maximum likelihood estimate of GPD based on the excess value. This avoids the possibility of insufficient sample size leading to inaccurate estimation of the GPD parameter. Generally, when using maximum likelihood estimation of GPD, the number of data points should not be less than 30; therefore, it is recommended to use a smaller number. =30. Furthermore... It will also be used in other places to ensure the reliability of calculation results.

[0060] In step 1.3, a set of candidate thresholds was established. Afterwards, step 1.3.1 performed resampling calculations. The resampling calculation process can be summarized as follows: Checking whether the divided extreme values ​​of vehicle-induced deflection (the portion greater than a threshold) are continuous in time. If they are continuous in time (the time interval is less than...), then... The value (GPD) represents a fluctuation in vehicle-induced deflection caused by the same axle and exhibits autocorrelation. For fluctuations in vehicle-induced deflection caused by the same axle, only the maximum value is retained. The main reason for resampling is that excess values ​​above the threshold often exhibit autocorrelation. For example, in a fluctuation in vehicle-induced deflection caused by a single vehicle, the portion above the threshold may retain both the maximum and second-largest values ​​of that vehicle's vehicle-induced deflection, leading to information redundancy in the excess values ​​and the data points not being independent. However, the POT method, which models the distribution of extreme values ​​(the portion above the threshold) of vehicle-induced deflection, assumes that the excess values ​​are independent, so autocorrelation would violate this assumption. Therefore, it is necessary to remove autocorrelation from the time series data to make the excess values ​​approximately independent, thereby ensuring the applicability of GPD in the POT method. The specific value depends on the data.

[0061] For the series of candidate thresholds provided in step 1.3, calculate based on step 1.3.2. Then, based on steps 1.4 to 1.5, the result is measured using the R² metric. The linearity on the right side of the curve is considered, and the threshold at which the linearity is maximized is taken as the reasonable lower limit. Then, combining with step 1.5.1, essentially, under the premise of exceeding the reasonable lower limit, the R² index is used to measure the linearity of the estimated GPD distribution and the QQ plot of the excess value corresponding to each threshold. Finally, in step 1.6, the threshold corresponding to the most linear QQ plot is taken as the final threshold, and the corresponding GPD distribution parameters are obtained. The QQ plot essentially compares the quantiles of the data with the quantiles of the theoretical distribution to visually check whether the data conforms to a specific distribution, suitable for small to medium sample sizes. Ideally, the quantiles of the data in the QQ plot are exactly the same as the quantiles of the theoretical distribution; therefore, the R² index can be used to measure the degree of matching between the data quantiles and the theoretical distribution quantiles. The process of selecting the final threshold can be further summarized as ensuring... Assuming the curve is linear on the right side, the threshold that best matches the data with the fitted distribution is the final threshold. Ultimately, the fitted distribution and data still need to pass the A / D test before the calculation results at the corresponding threshold are accepted.

[0062] In steps 1.4 to 1.5, step 1.4 eliminates cases where the excess value is insufficient after resampling. In order to ensure that the sample size is greater than the maximum likelihood estimate of GPD based on the excess value in the subsequent case, and ensure The calculated mean is representative. Furthermore, in step 1.5, to ensure... R The effectiveness of indicator 2 was only calculated for the first two indicators. indivual u , ensuring that the calculations were performedR The number of data points for the two indicators is greater than In summary The role of parameters in the calculation, this study suggests The empirical value is 30.

[0063] In step 1.6, the final threshold was determined to be... Excess value set The set contains a total of Each element is estimated using the maximum likelihood method. GPD parameters to which it belongs Then, provided that the AD test can be passed, it can be based on the given... Combined with GPD distribution parameters , and The extreme value of the vehicle-induced deflection was calculated. probability Specifically, it is easy to see that the number of vehicle-induced deflection fluctuations should be consistent with the number of axles on the bridge. Therefore, the sample coverage time can be used as a reference. Number of axles on the bridge As an approximation of the quantity of vehicle-induced deflection fluctuations. Less than The cumulative integral probability of a portion is:

[0064] therefore:

[0065]

[0066]

[0067] in, This refers to the threshold for vehicle-induced deflection. In existing research, the threshold for deflection is often derived from three methods: 1) a given probability threshold (95% confidence interval) from the statistical distribution of the data. ±3 (etc.). 2) Obtained from the specifications. The Technical Evaluation Standard for Highway Bridges stipulates that when the maximum deflection at mid-span of a cable-stayed bridge steel box girder is greater than 1 / 600 of the calculated span, the technical evaluation scale is 4. 3) Determined through the most unfavorable working condition using finite element virtual loading. In this invention, based on the deflection limit of scale 4 in the specifications, a deflection threshold is set. For bridges in service, if the maximum vehicle-induced deflection exceeds this allowable value, the deflection-based rating is downgraded to level 4, meaning the bridge's deflection exceeds the assessed range of good structural integrity, indicating a decline in structural performance. Therefore, exceeding this threshold can be considered a significant degradation in bridge performance.

[0068] A method for calculating the most probable out-of-limit probability and the confidence interval of the out-of-limit probability based on Bootstrap.

[0069] The bootstrap method is a nonparametric self-sampling technique that generates a new sample set by repeatedly sampling (with replacement). This method can well approximate the statistical characteristics of the population with large samples. Its core theories are: 1) Nonparametric assumptions: The bootstrap method does not require knowledge of the data distribution (a nonparametric method), only assuming that the data are independent and identically distributed. 2) Asymptotic consistency: With a large sample size... As time progresses, the results of the Bootstrap method gradually approximate the true distribution. The Bootstrap method needs to satisfy the following assumptions: Assumption 1: The finite sample represents the population. The Bootstrap method assumes that the original finite sample can represent the distribution characteristics of the population. Therefore, by performing repeated sampling with replacement, the process of drawing a sample from the population can be approximated.

[0070] Assumption 2: Independence of self-sampling. In the Bootstrap self-sampling process, each sample is independent. This independence allows the application of the law of large numbers in statistics, especially with large samples.

[0071] The classic steps of the Bootstrap method are: 1) From the original sample Extraction with replacement n These samples form a new sample set. X 2) Regarding X Calculate the statistics of interest (such as maximum, mean, median, etc.). 3) Repeat the above process. B This yields the distribution of the statistics.

[0072] Step 1.7 essentially uses the Bootstrap method to establish the sample coverage duration. Set of probabilities of exceeding the maximum value of the deflection caused by the inner vehicle . The distribution can be approximated as Distribution of the probability of exceeding the maximum value of the deflection caused by the inner vehicle.

[0073] The evaluation time length is defined in step 1.8. L Then, step 1.9 calculated the duration. L Probability of exceeding the maximum value of the deflection caused by the inner vehicle Essentially, step 1.9 assumes duration. L Depend on Composed of several cycles, the duration of a single cycle The probability of exceeding the limit in each period follows a certain order. The distribution, therefore, can be determined by... Random sampling with replacement Next, simulate the probability of exceeding the limit in each cycle. Let the... j The sampled results are Then the final probability of exceeding the limit is:

[0074] That is to say, in Within each cycle, each duration The probability that the time limit is not exceeded is the duration. L The probability of not exceeding the limit. Because... It may not be an integer, therefore for Round up to , The decimal part is Furthermore, since computer calculations may suffer from precision loss due to the continuous multiplication of extremely small numbers, the above calculation formula is transformed to convert continuous multiplication into addition.

[0075] In summary, step 1.9 can be summarized as... sampling After that, a duration is obtained. L Probability of exceeding the maximum value of the deflection caused by the inner vehicle This calculation process takes into account the length of time. The variability of the probability of the maximum deflection caused by the inner vehicle exceeding the limit.

[0076] Building upon step 1.9, step 1.10 is essentially a Monte Carlo sampling method, approximating the distribution followed by the random variable through repeated sampling. Specifically, it involves the number of sampling steps defined in step 1.8. Through repeated calculation steps 1.9, a total of Next time, you can eventually obtain indivual This constitutes the duration. L Set of probabilities of exceeding the maximum value of the deflection caused by the inner vehicle Then use KDE, based on Estimate The distribution of the set was determined; and based on the maximum probability density value, the corresponding values ​​of the cumulative integral probability of 2.5% and 97.5% in the KDE distribution, these were used as the most likely out-of-limit probability and the upper and lower bounds of the 95% confidence interval for a single sensor. In this calculation, the KDE was used to analyze the set. The bandwidth was estimated based on the distribution it follows, and the Scott rule was chosen as the basis for bandwidth selection. The reason for using KDE and selecting the Scott method based on bandwidth values ​​is as follows: 1) KDE is a nonparametric method for estimating data distribution. It does not rely on any specific probability distribution assumptions. Regardless of whether the data is unimodal, multimodal, skewed, or has other complex structures, KDE can reflect the true distribution of the data well. The Gaussian kernel is a commonly used symmetric kernel that can smooth the data distribution, generating a smooth probability density function. This means it can remove noise and outliers from the data, providing a smooth and continuous probability density curve.

[0077] 2) The Scott rule calculates the bandwidth based on the standard deviation and sample size, which can effectively balance the accuracy and bias of KDE estimation results, helping to avoid oversmoothing or overfitting and ensuring the accuracy of the estimation results. When dealing with large sample sets, the Scott rule can effectively reduce the subjectivity caused by manually selecting the bandwidth and enhance the generalizability of the estimation.

[0078] In step 1.11, for different given evaluation time lengths L Based on the calculation results in step 1.7, simply repeat steps 1.8 to 1.10 to obtain the corresponding most likely overlimit probability and the upper and lower bounds of the 95% confidence interval.

[0079] A Performance Evaluation Method for Main Girder Based on Series Exceedance Criteria

[0080] Step 2 is essentially a calculation for different measuring points, that is, at the same time interval. L Next, by calculating steps 1 to 1.10 once for each measuring point, the duration of each measuring point can be obtained. L Let the set of probabilities of the maximum deflection exceeding the limit caused by the vehicle be denoted as the first. i The set of out-of-limit probabilities for each sensor is: Then, in steps 2.1 to 2.3, the series exceedance criterion is used to calculate the exceedance probability of the entire main beam from the exceedance probability calculated from a single deflection measuring point. The series exceedance criterion means that if any deflection of any of the deflection monitoring sensors exceeds the limit, it represents that the system exceeds the limit. It is worth noting that in this invention, the calculations for different measuring points are performed multiple times through independent self-sampling, that is, each self-sampling result is derived from multiple deflection data at different times. Therefore, the probability of exceeding the maximum vehicle-induced deflection value at each measurement point is independent of each other. The reason for not performing synchronous self-sampling at different measurement points will be explained later.

[0081] Therefore, based on the series exceedance criterion, the exceedance probability of the entire main beam can be calculated from the exceedance probability obtained from a single deflection measurement point. For the first... i One sensor, from Randomly select one element from the data, denoted as . That is, the firsti The probability of exceeding limits for each sensor. After sampling the probability set of exceeding limits for each sensor once, the probability of exceeding limits for the main beam under the series exceeding limit criterion is calculated. The calculation formula is:

[0082] Step 2.4 is essentially a Monte Carlo sampling method, which calculates a series of durations by repeatedly performing steps 2.2 to 2.3. L Probability of lower main beam exceeding limits , forming a set Similar to step 1.10, use KDE to establish a distribution estimate, and then provide the main beam over time. L The most likely probability of exceeding the limit and the upper and lower bounds of the 95% confidence interval.

[0083] Step 3 is essentially about different durations L Based on the calculation in step 2, repeating steps 2.1 to 2.4 will yield the main beam at different time intervals. L The most likely probability of exceeding the limit and the upper and lower bounds of the 95% confidence interval.

[0084] In summary, based on time length t Vehicle monitoring data measured internally With deflection data It enables different time lengths L Inside, a single deflection measuring point Most likely probability of exceeding the limit 95% confidence interval lower bound Upper limit of the 95% confidence interval The most likely main beam Lower bound of the 95% confidence interval for the probability of exceeding the limit Upper bound of the 95% confidence interval for the probability of exceeding the limit The calculation will be performed using a specific object as an example to demonstrate the calculation process and explain the values ​​of each parameter during the calculation.

[0085] The following example uses a double-tower, single-cable-stayed bridge. This bridge has three spans and four lanes, with lanes 12 and 34 having different vehicle orientations. Lane 14 is on the outer side of the bridge deck, while lanes 23 are on the inner side. Vehicle data for this bridge was collected through a dynamic weighing system at the bridgehead where pile number 5 is located, recording the speed, weight, and number of axles for each passing vehicle. Subsequent research uses deflection field monitoring data and vehicle monitoring data from January 16th to January 22nd, 2020, as an example. This time period has been verified, and the data quality is high.

[0086] Data Overview and Preprocessing Methods

[0087] To provide a data foundation for subsequent calculations and avoid interference from outliers in the analysis results, preprocessing of vehicle and deflection monitoring data is necessary. The preprocessing steps include three parts: anomaly identification and removal, missing data imputation, and decoupling of deflection data. During anomaly identification and removal, data that remained unchanged for more than 10 minutes, data exceeding the sensor's measurement range, and segments inconsistent with strain were removed using the strain-deflection consistency evaluation method proposed in Chapter 3. After processing, the total missing rate of monitoring data from 26 deflection sensors from January 16th to January 22nd, 2020, was 0.103%; the D-G8-2 sensor had the highest missing rate at 0.309%. During missing data imputation, since the missing data in the deflection data were all random points, not non-random or continuous, and the overall missing rate was low, linear interpolation could be used to imput the few missing data points. During the deflection data decoupling process, after extracting the temperature-induced deflection component from the deflection data, the vehicle-induced dynamic deflection component and the vehicle-induced static deflection component are retained by subtracting the temperature-induced deflection component from the original data, and this is used as the vehicle-induced deflection result. Because the deflection decoupling method involves a sliding window averaging of historical data, half an hour of data could not be decoupled. The final retained data is from 00:30 on January 16, 2020, totaling 6 days, 23 hours, and 30 minutes. The final result is 26 data points with a length of 6030000 (=7...). twenty four 60 60 10Hz–30 60 The deflection monitoring data (10Hz), taking the data measured by D-G10-1 as an example, the raw data is as follows: Figure 2 As shown in Figure a, the calculated vehicle-induced deflection components are as follows: Figure 2 As shown in b.

[0088] Calculation results of time-varying over-limit probability at a single deflection measuring point

[0089] Referring to step 1, the vehicle-induced deflection at the D-G10-1 measuring point is analyzed as an example. The most likely probability of exceeding the limit and the 95% confidence interval of the probability of exceeding the limit are calculated at different times. The same calculation process is used for other measuring point data.

[0090] Step 1.2 mainly involves self-sampling calculations, utilizing the single sampling time length mentioned in Step 1.1. The bridge is 808 meters long. Data collected by the dynamic weighing system at the bridgehead shows that the average vehicle speed on the bridge from January 16th to January 22nd, 2020 was 15.773 m / s. To ensure that the overlap rate between the vehicle data and the vehicle causing the vehicle-induced deflection component is greater than 95%, the minimum time length is 20 seconds. / =20 × 808 / 15.773 = 1024.508 seconds. (Take...) =1024.5 seconds (the deflection data sampling frequency is 10Hz, so t needs to be rounded to 0.1), the length of a single sampling data is The number of samplings is Next, sample coverage time after sampling 603430.5 seconds. After determining the length of a single sampling segment and the number of samplings, in Figure 2 In the vehicle-induced deflection component shown in b, from the 0th data point to... Randomly select one data point from among the n data points, and record the result as the nth data point. i The first point. Extract the first... i Point 1 to the 2nd A data segment of 10245 points was collected as a single sampling result, and the number of axles corresponding to the time range covered by each data segment was recorded. This sampling process was repeated 589 times, resulting in 589 data segments of length 10245 and corresponding axle counts. These 589 data segments were then concatenated according to the sampling order to obtain the final sampling result. and Sampling results middle There are a total of 6,034,305 data points, and the total number of axles. Sampling results like Figure 3 As shown.

[0091] Step 1.3 mainly aims to determine a series of thresholds to be selected, which uses the minimum threshold mentioned in step 1.1. Maximum threshold Threshold number ( ).Will In After sorting from smallest to largest, take the 30th value from the end as the maximum threshold. Take the 10% from the bottom 6,034,305 values ​​were used as the minimum threshold. , take the threshold number at equal intervals A threshold. Step 1.3.1 mainly involves resampling calculation, using the resampling time interval mentioned in step 1.1. Here It is set to 1 second because Seconds, i.e., coverage duration Within this range, an average of one vehicle-induced deflection fluctuation occurs every 1.708 seconds. Therefore, The value should be less than 1.708 seconds to avoid incorrectly deleting the maximum vehicle-induced deflection caused by different axles. Furthermore, due to the unavoidable presence of low-amplitude, high-frequency noise interference in the measured data, Take the sampling time interval of adjacent data Seconds can cause data to wrap around the threshold u The repetitive phenomenon caused the threshold after resampling to be affected. u An accumulation of excess value has occurred nearby. Therefore, a reasonable... The value should be greater than 0.1 seconds and less than 1.708 seconds; here it is taken as... =1 second. In determining... After setting the threshold to 1 second, step 1.4, referring to steps 1.3.1 and 1.3.2, calculates the corresponding set of maximum vehicle-induced deflection values ​​for each of the 500 candidate thresholds determined in step 1.3. and The set of the maximum values ​​of vehicle-induced deflection is obtained. and sets After the calculation in step 1.4 is completed, It was updated to 395, i.e., set , 、{ Each of the following contains 395 elements. Before the update various sets The number of elements and the threshold set { Threshold in} Relationship such as Figure 4 As shown in the figure, the red dashed line represents the case where the number of elements in the excess value set is equal to 30; the set after elimination... various With threshold set { Threshold in} Relationships such as Figure 5 As shown.

[0092] Steps 1.4.1 to 1.5 are essentially threshold selection. Step 1.5 calculates the resulting set. There are a total of 365 ( ) elements, set With set { The relationship between the 1st to the 365th elements in} is as follows: Figure 6 As shown. Figure 6 The red dashed line in the middle corresponds to the set. maximum value Its corresponding The value is 6.340, which is a reasonable threshold. u lower limit .gather{ There are 120 elements greater than 6.340 in the subset. , Each has 120 elements.

[0093] Steps 1.5.1 and 1.6 essentially estimate the performance of the GPD distribution based on the maximum likelihood method to determine the final threshold. The set calculated in step 1.6... There are 120 elements in the set. With sets Relationship such as Figure 7 As shown. Furthermore, in step 1.6, the set... 120 thresholds Both used the maximum likelihood method to estimate GPD parameters to which it belongs There are a total of 120 GPD distributions. These 120 sets of GPD parameters correspond to 120 log-maximum likelihood values, and are compared with the threshold set. Relationship such as Figure 8 As shown. Figure 7 and Figure 8 The red dashed line in the middle corresponds to the set. maximum value When, its corresponding threshold .

[0094] Depend on Figure 8 It can be seen that, At its maximum, the log-maximum likelihood value was not the maximum value, but only -119.921, while the actual log-maximum likelihood value was -13.652. The reason for choosing the QQ plot R2 metric instead of the maximum logarithmic natural value is that the magnitude of the log-maximum likelihood value is limited by the characteristics of the data being fitted, and log-maximum likelihood values ​​corresponding to different data sets are not comparable. Specifically, Figure 9 When the maximum log-likelihood value is -13.652 ( A comparison chart of QQ charts and data distribution; Figure 10 When the log-maximum likelihood value is -119.921 ( A comparison chart of QQ charts and data distribution. Figure 9 and Figure 10 It can be seen that when the log-maximum likelihood value is large, the excess value set has 32 elements. The estimated GPD matches well for most small values ​​in the QQ graph, but a large deviation occurs at the tail, which leads to a more inaccurate estimate of the excess probability extrapolated to the limit event; while In larger cases, the excess value set contains 125 elements. Although the log-maximum likelihood is smaller, the better GPD at the tail fits the data better and is more accurate in extrapolating the excess probability of extreme events.

[0095] In summary, with To ultimately select the threshold, the corresponding set of excess values ​​is as follows: Figure 11 As shown. Figure 11 The data within the blue box in 'a' is expanded as follows Figure 11 b, as a partial schematic diagram. (By...) Figure 11 As can be seen from b, the data points exceeding the threshold in the overall data are continuous. Through the aforementioned resampling process, it can be effectively ensured that the set of excess values ​​consists of the peak values ​​of each vehicle-induced deflection fluctuation, and each excess value is independent of the others.

[0096] Referring to step 1.6, based on The corresponding set of excess values, in the GPD obtained using maximum likelihood estimation, has the following parameters: location parameters Scale parameters Shape parameters Its distribution pattern is as follows Figure 9 As shown in b. It is worth noting that the shape parameters at this point... This indicates that the GPD has a truncated property, meaning that the GPD support region is finite and the distribution is restricted to a finite interval. Between ] . The upper bound of the distribution is This indicates that the maximum value of the extreme vehicle-induced deflection exists (5.133 cm). Further, the Anderson-Darling test (AD test) is needed to verify the fit between the data and the distribution. The AD test is commonly used to determine whether multiple observations originate from a single population. Compared to another commonly used test, the Kolmogorov-Smirnov test (KS test), it is more sensitive to tail data and is particularly suitable for extreme value analysis, testing whether the data matches a specified GPD distribution. The AD test establishes a pair of hypotheses: 1) Null hypothesis: The data follows a specified distribution; 2) Alternative hypothesis: The data does not follow a specified distribution. Based on this, the AD statistic is calculated as follows:

[0097] in n The sample size is the number of elements in the excess value set. i It refers to the ranking of the data (all data should be arranged from smallest to largest). This is the cumulative probability density function of the GPD distribution. After calculating the AD statistic based on the excess value and the GPD distribution, the fitted GPD needs to be simulated and sampled 1000 times. The sample size for each sampling is equal to the number of elements in the excess value set, and the AD statistic for each sampling result is calculated. Among these 1000 statistics, the number of those less than the AD statistic based on the excess value and GPD distribution, divided by 1000, is the significance level index for the AD test. pValue. The confidence interval is 0.95, i.e. p When the value is greater than 0.05, the null hypothesis is not rejected, meaning the data can be considered consistent with the distribution. Ultimately, The corresponding set of excess values, and the GPD obtained using maximum likelihood estimation ( , , The results of the AD test are as follows: Figure 12 As shown, where p The value is 0.656, which is greater than 0.05, so the excess value set can be considered to conform to the specified GPD.

[0098] In summary, the results are based on 589 random sampling combinations at measurement point D-G10-1. Since the AD test was passed, it can be considered a valid sampling calculation; if the AD test was not passed, the current sampling calculation is considered invalid, and its result is not retained. Combined with the parameters provided by the aforementioned calculations: number of axles... =353345, Number of elements in the excess value set =125, GPD cumulative integral function ( , , Threshold The sample coverage time was 603430.5 seconds. The span of the section where the D-G10-1 measuring point is located is 392 meters. Therefore, the exceedance threshold is 392 meters / 600 = 65.33 centimeters, and the corresponding exceedance portion is 65.33 - 6.503 = 58.827 centimeters. As mentioned above, At this time, GPD exhibits truncation property, and its distribution is restricted to the finite element interval. The value is between 58.827 and 5.133, meaning the cumulative integral probability is greater than 5.133. The probability of the maximum vehicle-induced deflection exceeding the limit within the sample coverage period of 603430.5 seconds. .

[0099] In the Bootstrap method, sampling 1000 times typically yields an acceptable result. Referring to step 1.7, the final set is obtained. Of these, 14.9% of the 1000 calculations failed the A / D test, therefore the final set... The number of elements is 851. For these 851 elements... Statistical results are as follows Figure 13 As shown, there are 705 of them. All equal to 0, maximum equal .

[0100] In step 1.8, the evaluation time length is temporarily taken.L Taken as 1 year (i.e., 31,536,000 seconds, equal to 52.2612 times 603,430.5 seconds); number of samples Let the value be 1000. Referring to steps 1.9 to 1.10, the probability of exceeding the limit for 1000 measuring points D-G10-1 within one year can be obtained. The distribution of this probability of exceeding the limit is similar to the KDE results. Figure 14 As shown. Figure 14 The red area represents the 95% confidence interval, with the lower bound being the probability of exceeding the limit of the cumulative integral probability in the KDE at 2.5%. The corresponding reliability is 6.292; the upper bound is the out-of-limit probability of a cumulative integral probability of 97.5% in KDE. The corresponding reliability is 5.116; the most likely probability of exceeding the limit is... The corresponding reliability is 5.631. At this point, the distribution of the probability of exceeding the limit is an asymmetric distribution, and the most likely probability of exceeding the limit is close to the lower bound of the 95% confidence interval.

[0101] Table 1. Calculation results of D-G10-1 over 10, 25, 50, and 100 years.

[0102] Taking 10 years, 25 years, 50 years, and 100 years as examples, based on the calculation results in step 1.7, the evaluation time length in step 1.8 is used... L Change to these 5 time lengths, number of samples All values ​​are set to 1000. The probability distribution of exceeding limits and the KDE results for measuring point D-G10-1 over these five time periods are calculated. The calculated results of the probability distribution of exceeding limits and the KDE are as follows: Figure 15 As shown, the most likely out-of-limit probability and the upper and lower bounds of the 95% confidence interval are shown in Table 1. Figure 14 and Figure 15 It can be seen that, regarding the length of time L When considering five scenarios—1 year, 10 years, 25 years, 50 years, and 100 years—the KDE-estimated distribution matches the distribution of the out-of-limit probability well. Therefore, using the maximum probability density value of the KDE-estimated distribution as the most probable out-of-limit probability is feasible. The distribution of the out-of-limit probability varies with the length of time. L As the value increases, it gradually changes from an asymmetric distribution to a symmetric distribution; at 100 years, the most likely probability of exceeding the limit is... There is a 95% confidence that the probability of exceeding the limit will be within 100 years. to between.

[0103] Furthermore, based on the calculation results in step 1.7 and referring to step 1.11, the evaluation time length is taken. L The time periods are 100, including 1 year, 2 years, 3 years, ..., 100 years.L Number of samplings All values ​​are set to 1000. The probability distribution of exceeding the limit and the KDE results for measuring point D-G10-1 at different time periods are calculated. The most likely probability of exceeding the limit and the 95% confidence interval of the probability of exceeding the limit are also calculated. The final calculation results are as follows: Figure 16 As shown. By Figure 16 It can be seen that the most likely out-of-limit probability and the upper and lower bounds of the 95% confidence interval for each year all increase linearly with the year. The linear fitting formula for the most likely out-of-limit probability versus time is as follows: Figure 16 The formula shown in blue represents the goodness of fit. The formula for the linear fit between the upper and lower bounds of the 95% confidence interval and time is as follows: Figure 16 The formulas shown in green and yellow are as follows: The values ​​are 0.999 and 0.997, respectively. The main reason for the slightly lower linear fit of the 95% confidence interval is that the error is more significant when the time period is less than 20 years. Therefore, in subsequent calculations, the time length can be used. L For any two scenarios with a time horizon greater than 20 years, calculate the most likely probability of exceeding the limit at each measuring point. Then, by using linear fitting, the resulting linear function can be used to estimate the time horizon of a single measuring point. L The most likely probability of exceeding the limit and the confidence interval for any period greater than 20 years.

[0104] Calculation results of the probability of main beam exceeding limits

[0105] Taking the D-G10-1 measuring point as an example, the specific calculation process and results of calculation steps 1.1 to 1.11 are shown. Referring to step 2, the calculations of steps 1.1 to 1.7 are performed for all 26 deflection measuring points, where the values ​​of each parameter are consistent with those described in section 5.3.2: single sampling time length. Both were 1024.5 seconds; After sorting from smallest to largest, take the 30th value from the end as the maximum threshold. 10% from the bottom 6,034,305 values ​​were used as the minimum threshold. Threshold number Set the value to 500; resampling time interval Take 1 second; Minimum number of excess values ​​and number of fitted values Take 30; Number of self-sampling times Take 1000; Vehicle-induced deflection exceeding threshold Values ​​were taken from 1 / 600 of the span length of each sensor, resulting in a total of 26 sets. This represents the duration of 26 sensors. The possible probability of exceeding the limit within 603430.5 seconds. Because the probability of each measurement point's self-sampling results passing the A / D test is different, and the number of elements in each set is different, there are 26 sets. The number of elements is as follows Figure 17 As shown. By Figure 17 It can be seen that, in Assuming all values ​​are 1000, the sensor with the lowest probability of passing the AD test in its self-sampling results is D-G16-2, ultimately retaining only 576 out-of-limit probabilities. Overall, the final set of out-of-limit probabilities for each measurement point... The number of elements is over 500, fully describing all possible scenarios of exceeding the limit probability at each measuring point within 603430.5 seconds. It is assumed that synchronous self-sampling is performed on multiple deflection measuring points, i.e., data from multiple deflection measuring points are collected simultaneously. The corresponding portions are extracted and repeated multiple times to form self-sampling results of multiple deflection measurement points synchronized in time. The probability that all self-sampling results of measurement points pass the A / D test is... Figure 17 The sum of the probabilities of all the test points passing the test is only 0.00288.

[0106] In step 1.8, the evaluation time length is temporarily taken. L Taken as 1 year (i.e., 31,536,000 seconds, equal to 52.2612 times 603,430.5 seconds); number of samples Let the value be 1000. Referring to steps 1.9 to 1.10, the most likely probability of exceeding the limit and the 95% confidence interval of the probability of exceeding the limit for each measuring point within one year can be obtained. The calculation results are as follows: Figure 18 As shown. Figure 18 In the diagram, the horizontal axis represents the main beam section number; blue corresponds to deflection sensor number 1 within that section, and orange corresponds to deflection sensor number 2. Figure 18 It can be seen that the measurement point with the highest probability of exceeding the limit is D-G13-2, with a probability of exceeding the limit of... The measurement point with the largest 95% confidence interval is D-G17-1, and the confidence interval is [ The probability of exceeding limits varies within the same cross-section, mainly due to the different traffic flow conditions in lanes 12 and 34, as well as localized damage on the main beam. According to the bridge inspection report, a total of 136 structural defects were found in three steel box girder inspections conducted between 2017 and 2019, including 64 fatigue cracks larger than 20mm and 51 cracks larger than 30mm. The cracks were primarily located on the main beams near piers 4, 1, 3, and 2, from most frequent to least frequent. Piers 3 and 4 are the boundaries of the spans containing D-G17-1 and D-G17-2, and D-G16-1 and D-G16-2, respectively. The measuring point with the highest probability of exceeding limits, D-G13-2, is adjacent to pier 3. The deflection sensors near pier 1, from closest to furthest, are D-G3-1 and D-G3-2, and D-G4-1 and D-G4-2. The probability of exceeding the limit for D-G16-1 and D-G17-1 within sections G16 and G17 is greater than that within other spans. D-G13-1 has the highest probability of exceeding the limit. This shows that the evaluation results of the probability of exceeding the limit correspond to the bridge inspection results. The main beams near piers 3 and 4 are relatively more dangerous, corresponding to more main beam cracks. Special attention should be paid to this area during subsequent service.

[0107] Further, define If the value is 1000, referring to steps 2.1 to 2.4, the probability of exceeding the limit for 1000 main beams within one year can be obtained. The distribution of this probability of exceeding the limit is similar to the KDE result. Figure 19 As shown, Figure 19 The red area represents the 95% confidence interval, with the lower bound being the probability of exceeding the limit of the cumulative integral probability in the KDE at 2.5%. The corresponding reliability is 4.935; the upper bound is the out-of-limit probability of a cumulative integral probability of 97.5% in KDE. The corresponding reliability is 4.742; the most likely probability of exceeding the limit is... The corresponding reliability is 4.855. At this point, the distribution of the probability of exceeding the limit is an asymmetric distribution, and the most likely probability of exceeding the limit is close to the lower bound of the 95% confidence interval.

[0108] Returning to the 26 sets calculated based on steps 1.1 to 1.7 The evaluation time length L The sampling period was changed to four scenarios: 10 years, 25 years, 50 years, and 100 years, with a total of [number of sampling times]. Taking 1000 as the value, and referring to steps 1.9 to 1.10, the most likely probability of exceeding the limit and the 95% confidence interval of the probability of exceeding the limit for each measuring point under these four years can be obtained. The calculation results are as follows: Figure 20 As shown in Table 2, the measurement points for the maximum out-of-limit probability and the maximum confidence interval are as follows. Table 2 shows that the measurement points for the maximum out-of-limit probability vary with different time lengths. LThe time intervals are all D-G13-2, and the maximum confidence intervals are all D-G17-1; from Figure 18 and Figure 20 It can be seen that the confidence interval widens more slowly than the probability of exceeding the limit increases, resulting in a longer time interval. L As the value increases, the width of the confidence interval decreases as a proportion of the most likely probability of exceeding the limit, which is reflected in... Figure 18 and Figure 20 The length of time is represented by the middle. L The larger the value, the narrower the confidence interval appears; when the design life of 100 years is reached, the probability of exceeding the limit for D-G13-2 is... The confidence interval for D-G17-1 is [ ].

[0109] Table 2. Calculation results for 26 measuring points over 10, 25, 50, and 100 years.

[0110] Further, define If the value is 1000, referring to steps 2.1 to 2.4, the probability of exceeding the limit for 1000 main beams within 10, 25, 50, and 100 years can be obtained. The distribution of this probability of exceeding the limit is similar to the KDE results. Figure 21 As shown, the most likely out-of-limit probability and the upper and lower bounds of the 95% confidence interval are shown in Table 3. Figure 19 and Figure 21 It can be seen that, regarding the length of time L When considering five scenarios—1 year, 10 years, 25 years, 50 years, and 100 years—the KDE-estimated distribution matches the distribution of the out-of-limit probability well. Therefore, using the maximum probability density value of the KDE-estimated distribution as the most probable out-of-limit probability is feasible. The distribution of the out-of-limit probability varies with the length of time. L As the value increases, it gradually changes from an asymmetric distribution to a symmetric distribution; at 100 years, the most likely probability of exceeding the limit is... The corresponding reliability is 3.816; there is 95% confidence that the probability of exceeding the limit in 100 years is... to Comparing Tables 2 and 3, it can be seen that the most likely probability of the main beam exceeding the limit at 100 years is on the same order of magnitude as the probability of D-G13-2, which has the highest probability of exceeding the limit among all deflection measurement points. The most likely probability of the main beam exceeding the limit is about 5.5 times that of D-G13-2.

[0111] Table 3. Calculation results of the main beam over 10, 25, 50, and 100 years.

[0112] Furthermore, based on the 26 sets calculated in steps 1.1 to 1.7 ,definition The value is 1000. Refer to step 3 and determine the evaluation time length. L The time periods are 100, including 1 year, 2 years, 3 years, ..., 100 years. L The probability distribution of the main beam exceeding the limit at different time points and the KDE results were calculated. The most likely probability of exceeding the limit and the 95% confidence interval for the probability of exceeding the limit were also calculated. The final calculation results are as follows: Figure 22 As shown. By Figure 22 It can be seen that the most likely out-of-limit probability and the upper and lower bounds of the 95% confidence interval for each year increase linearly with the year. The linear fitting formula for the most likely out-of-limit probability and the time length is as follows: Figure 16 The formula shown in blue represents the goodness of fit. The linear fitting formula for the upper and lower bounds of the 95% confidence interval and the time length is as follows: Figure 16 The formulas shown in green and yellow are as follows: All Using only the evaluation time length L The linear fitting results are used to estimate the most likely exceedance probability and confidence interval for 1 year and 100 years. The estimated results are compared with the calculated results. Both are 0.9999. Therefore, in subsequent calculations, only the time length needs to be calculated. L For any L The probability of exceeding the limit and the confidence interval for each possible value can be estimated by linear regression.

[0113] A main beam performance evaluation system based on the probability of vehicle-induced deflection exceeding limits includes: The data acquisition and preprocessing module is used to collect measured deflection data of the main beam of the bridge and vehicle monitoring data, complete outlier removal, missing value filling and decoupling of deflection components, and output vehicle-induced deflection component data. The moving block bootstrap self-sampling module is connected to the data acquisition and preprocessing module. It is used to set the single sampling time length, perform block-based self-sampling with replacement on vehicle-induced deflection component data and vehicle monitoring data, and output the self-sampling dataset. The single-point POT over-limit probability calculation module is connected to the moving block bootstrap self-sampling module. It is used to construct a set of candidate thresholds, determine the optimal threshold, fit the generalized Pareto distribution, and calculate the over-limit probability within the single-point self-sampling period. The Bootstrap time-varying probability calculation module is connected to the single-point POT over-limit probability calculation module. It is used to obtain the single-point over-limit probability set through repeated self-sampling, and combined with hierarchical sampling and Monte Carlo simulation, outputs the single-point time-varying over-limit probability and confidence interval. The overall probability calculation module for the main beam is connected to the Bootstrap time-varying probability calculation module. It is used to output the overall time-varying over-limit probability and confidence interval of the main beam based on the series failure criterion, through hierarchical sampling and Monte Carlo simulation. The performance evaluation output module is connected to the overall probability calculation module of the main beam. It is used to output the performance evaluation results and high-risk section information of the main beam based on the probability of the main beam exceeding the limit and the confidence interval, combined with the standard threshold.

[0114] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0115] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits, characterized in that, include: Collect measured deflection data of the main beam of the bridge and vehicle monitoring data, preprocess the data to obtain vehicle-induced deflection component data; The vehicle-induced deflection component data and vehicle monitoring data are divided into blocks and self-sampled with replacement by setting the single sampling time length. The self-sampled deflection dataset and self-sampled vehicle dataset are then spliced ​​together according to the sampling order. A set of candidate thresholds with equal intervals is constructed. The average excess value corresponding to each threshold is calculated by the mean excess number method. The reasonable lower limit of the threshold is determined by combining the goodness of linear fitting. Within the reasonable threshold range, the optimal threshold is determined by the goodness of fitting test of the QQ plot. The independent extreme values ​​of the excess threshold are extracted and fitted with a generalized Pareto distribution. After passing the Anderson-Darling test, the probability of exceeding the limit of a single measurement point within the self-sampling period is calculated. Repeatedly calculate the out-of-limit probability of a single measurement point within the self-sampling period to obtain the set of out-of-limit probabilities for a single measurement point; set the evaluation time length, simulate the out-of-limit probability of multiple periods through hierarchical sampling, and combine Monte Carlo simulation and Gaussian kernel density estimation to calculate the most likely out-of-limit probability and 95% confidence interval of a single measurement point within the time length. The probability of exceeding the limit is calculated for all m deflection measurement points to obtain the set of probability of exceeding the limit for each measurement point; samples are independently extracted from the set of probability of exceeding the limit for each measurement point through hierarchical sampling, and the probability of exceeding the limit for a single sampling of the main beam is calculated by combining the series failure criterion; the most likely probability of exceeding the limit and the 95% confidence interval of the main beam within the time length L are obtained by Monte Carlo simulation and Gaussian kernel density estimation. Based on the most likely probability of the main beam exceeding the limit and the 95% confidence interval, combined with the deflection threshold specified in the highway bridge technical evaluation standard, the stiffness performance and service safety of the main beam are evaluated, and high-risk sections are identified.

2. The method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits as described in claim 1, characterized in that, The preprocessing described in step S1 specifically includes: Data is subjected to outlier removal, missing value imputation, and decoupling of deflection components; Specifically, the deflection component decoupling involves extracting the temperature-induced deflection component from the measured deflection data, subtracting the temperature-induced deflection component from the original deflection data, and obtaining vehicle-induced deflection component data that includes vehicle-induced dynamic deflection and vehicle-induced static deflection.

3. The method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits as described in claim 1, characterized in that, The single sampling time length t s The values ​​of satisfy: Where X is the total length of the bridge and y is the average speed of vehicles on the bridge, to ensure that the overlap rate between vehicle data and vehicle-induced deflection data is ≥95%; the number of self-sampling times is... Where t is the total duration of the original data, This is for rounding up.

4. The method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits as described in claim 2, characterized in that, The construction of the equally spaced candidate threshold set {u} specifically involves: dividing the self-sampled deflection dataset... D ttsi Sort the data in ascending order and take the last one. n min The number of values ​​is used as the maximum threshold. u max Take the last one The values ​​are used as the minimum threshold. u min Where H is the sampling frequency; in u min and u max Multiple thresholds are selected at equal intervals to form a set of candidate thresholds.

5. The method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits as described in claim 3, characterized in that, The formula for calculating the out-of-limit probability of a single measurement point within the self-sampling period is: in, Optimal threshold The corresponding number of independent extrema exceeding the threshold, This represents the total number of axles within the self-sampling period. The vehicle's deflection exceeded the threshold. Let be the cumulative distribution function of the generalized Pareto distribution. For shape parameters, For position parameters, This is the scale parameter.

6. The method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits as described in claim 5, characterized in that, The calculation of the probability of exceeding the limit for a single measurement point within a time length L is described. The formula is: in, The number of hierarchical samplings. To extract from the set of probabilities beyond the limit P f The Middle j The sample values ​​drawn this time. for P f The Middle Each element.

7. The method for evaluating the performance of a main beam based on the probability of vehicle-induced deflection exceeding limits as described in claim 1, characterized in that, The probability of the main beam exceeding the limit in a single sampling under the series failure criterion The formula is: Where m is the total number of deflection measurement points. For from the first i The sample value drawn from the set of out-of-limit probabilities of each measurement point.

8. A main beam performance evaluation system based on the probability of vehicle-induced deflection exceeding limits, characterized in that, The main beam performance evaluation method based on the probability of vehicle-induced deflection exceeding the limit, as described in any one of claims 1-7, includes: The data acquisition and preprocessing module is used to collect measured deflection data of the main beam of the bridge and vehicle monitoring data, complete outlier removal, missing value filling and decoupling of deflection components, and output vehicle-induced deflection component data. The moving block self-sampling module, connected to the data acquisition and preprocessing module, is used to set the single sampling time length, perform block-based self-sampling with replacement on vehicle-induced deflection component data and vehicle monitoring data, and output the self-sampling dataset. The single-point POT over-limit probability calculation module is connected to the moving block bootstrap self-sampling module. It is used to construct a set of candidate thresholds, determine the optimal threshold, fit the generalized Pareto distribution, and calculate the over-limit probability within the single-point self-sampling period. The Bootstrap time-varying probability calculation module is connected to the single-point POT over-limit probability calculation module. It is used to obtain the single-point over-limit probability set through repeated self-sampling, and combined with hierarchical sampling and Monte Carlo simulation, outputs the single-point time-varying over-limit probability and confidence interval. The overall probability calculation module for the main beam is connected to the Bootstrap time-varying probability calculation module. It is used to output the overall time-varying over-limit probability and confidence interval of the main beam based on the series failure criterion, through hierarchical sampling and Monte Carlo simulation. The performance evaluation output module is connected to the overall probability calculation module of the main beam. It is used to output the performance evaluation results and high-risk section information of the main beam based on the probability of the main beam exceeding the limit and the confidence interval, combined with the standard threshold.

Images