Bayesian sequence updating method for identifying bridge influence line based on multiple groups of measurement

Abstract

The invention discloses a Bayesian sequence updating method for identifying a bridge influence line based on multiple groups of measurement. The method comprises the following steps: step 1, establishing a prediction error between a single group of measurement response and a bridge influence line identification model; 2, establishing a likelihood function of single-group measurement response, and defining prior distribution of parameters; 3, according to the likelihood function and the prior distribution, deriving posterior distribution of the single-group measurement response; step 4, based on multiple groups of measurement responses, performing iterative updating on posterior distribution; 5, in each updating stage, determining an optimal value of a bridge influence line in the stage, and quantifying posterior uncertainty; 6, in each updating stage, calculating the width of a confidence interval and judging the updating convergence of the Bayesian sequence; if the convergence is satisfied, outputting a bridge influence line; and if not, returning to the step 4 to iteratively update the posterior distribution until the posterior distribution is met. Through the method, the bridge influence line can be accurately identified from multiple groups of measurement responses containing relatively strong random interference.

Description

Technical Field

[0001] The present invention relates to the technical field of engineering structure health monitoring, and in particular to a Bayesian sequence updating method for identifying bridge influence lines based on multiple groups of measurements. Background Art

[0002] With the rapid development of bridge construction, ensuring the structural integrity and safety of serving bridges has become a core issue in civil engineering. The application of advanced structural health monitoring systems has enabled comprehensive acquisition of bridge response measurement data. However, accurately identifying key structural indicators from multiple sets of measurement data is fundamental to conducting scientific safety assessments. As an important static characteristic of bridge structures, bridge influence lines (BILs) possess clear physical meaning and are sensitive to damage. They have been widely used in bridge damage identification, model modification, and bridge weighing systems. Their accurate identification is a prerequisite for these engineering applications. Early BIL estimation methods were primarily based on theoretical assumptions. Due to the lack of measured bridge response data, their identification accuracy and engineering applicability were significantly limited, and they were often confined to numerical simulation studies. With technological advances, research has gradually shifted to data-driven approaches, namely, identifying bridge influence lines through pre-calibrated test vehicle-induced bridge responses. O'Brien's pioneering work established a linear least squares framework based on single-shot measurement data. However, this method is significantly sensitive to random disturbances in the measurement, such as measurement noise, dynamic fluctuations, and random load effects. Although regularization technology can effectively suppress measurement noise interference, it is still difficult to overcome the ill-posedness of the inversion problem when dealing with specific working conditions such as high-stiffness bridges or measurement points in the support area. In response to the influence of dynamic fluctuations in the excitation response of high-speed vehicles on the accuracy of bridge influence line identification, scholars have successively introduced techniques such as variational modal decomposition and nonlinear frequency analysis. However, when the measurement data contains local disturbances caused by random excitation loads, the above methods still have fundamental limitations in eliminating local anomalies that persist in the identification results. In order to expand the applicability of bridge influence line identification methods, innovative methods such as frequency domain analytical models, computer vision technology and physical information neural networks have been systematically introduced.

[0003] After nearly a decade of development, bridge influence line identification has achieved significant progress in accuracy and robustness, but its anti-interference performance remains insufficient. This is because existing methods primarily rely on single-shot measurement data from a uniformly traveling inspection vehicle. However, the actual bridge inspection process is a stochastic system subject to multiple sources of random interference, including but not limited to instrument measurement noise, dynamic fluctuations caused by bridge deck irregularities, and random effects from moving loads other than the inspection vehicle. These random interferences not only reduce the signal-to-noise ratio of a single set of measurement data but also exhibit variability between different measurements. This inherent randomness leads to significant deviations in multiple identified bridge influence lines, severely impacting subsequent engineering applications. While current engineering practices mitigate the effects of interference by strictly limiting inspection vehicle speed to suppress dynamic fluctuations and implementing traffic control to eliminate random loads, these restrictions severely limit the operational flexibility and engineering applicability of bridge influence line identification methods. Therefore, there is an urgent need to develop advanced bridge influence line identification methods with strong anti-interference capabilities to adapt to complex real-world conditions.

[0004] The fusion of multiple sets of measurement data provides an important way to improve the anti-interference performance of bridge influence line identification. In-depth research shows that the static component containing bridge influence line information in multi-source measurements has a time-invariant characteristic, while the amplitude intensity and spatial distribution of random interference show significant randomness. Although the arithmetic average of multiple sets of identification results is a conventional processing method, this method lacks a rigorous theoretical basis. Leng pioneered a multi-set measurement response fusion algorithm based on maximum likelihood estimation within a statistical framework, but this method has inherent difficulties in constructing a probabilistic explanation of the convergence characteristics of bridge influence lines. Therefore, the scientific use of multiple sets of measurement responses within a statistical framework has become a key path to improve the anti-interference performance of bridge influence line identification. Summary of the Invention

[0005] The main technical problem to be solved by the present invention is to provide a Bayesian sequence updating method for identifying bridge influence lines based on multiple sets of measurements, which can accurately identify bridge influence lines from measurement responses containing strong random interference and effectively evaluate the convergence of the identification results.

[0006] In order to solve the above technical problems, the present invention provides a Bayesian sequence updating method for identifying bridge influence lines based on multiple sets of measurements, comprising the following steps:

[0007] Step 1: Establish the prediction error between a single set of measured responses and the bridge influence line identification model;

[0008] Step 2: Establish the likelihood function of a single set of measured responses and define the prior distribution of the parameters in the likelihood function;

[0009] Step 3: deriving the posterior distribution of the single group measurement response based on the likelihood function and the prior distribution;

[0010] Step 4: Based on multiple groups of measurement responses, iteratively update the posterior distribution of the bridge influence line: use the posterior distribution of the i-th group of measurement responses as the prior distribution of the i+1-th group of measurement responses, and refer to step 2 to define the likelihood function of the i+1-th group of measurement responses to derive the posterior distribution of the i+1-th group of measurement responses; where i is an arbitrary positive integer;

[0011] Step 5: At each update stage, determine the optimal value of the bridge influence line at that stage and quantify the posterior uncertainty;

[0012] Step 6: Define the confidence interval width to characterize the posterior uncertainty; in each update stage, calculate the confidence interval width and determine the convergence of the Bayesian sequence update; if the convergence meets the judgment value, output the analytical expression of the bridge influence line; if the convergence does not meet the judgment value, return to step 4 and iteratively update the posterior distribution of the bridge influence line until it meets the judgment value.

[0013] In a preferred embodiment, in step 1, the prediction error is:

[0014] τ=R m -AΩc

[0015] Among them, R m is a single set of measurement responses, A is the vehicle load matrix, Ω is the fitting matrix, and c is the function coefficient vector.

[0016] In a preferred embodiment, in steps 2 and 3, the i-th group of measurement responses The corresponding likelihood function is:

[0017]

[0018] Where σ represents the standard deviation of the prediction error, N g is the number of response sampling points;

[0019] The prior distribution of parameter c is:

[0020]

[0021] Among them, μ 2 is the scale variance that characterizes the change of the function coefficient vector c, N s is the dimension of c;

[0022] Furthermore, the parameter σ 2 and μ 2 The prior distribution of is determined by the conjugate prior principle and is expressed as the inverse gamma distribution:

[0023]

[0024] in, (α0, β0) and (α1, β1) are the hyperparameters of the inverse gamma distribution;

[0025] The i-th group of measurement responses The corresponding posterior distribution is:

[0026]

[0027] In a preferred embodiment, in step 4, the posterior distribution updated after N iterations is:

[0028]

[0029] in,

[0030] In a preferred embodiment, in step 5, the optimal value equation group of the bridge influence line parameters θ=c, σ, μ is:

[0031]

[0032]

[0033] μ 2 =(2(α1+1)+N s ) -1 (||c|| 2 +2β1)

[0034] In a preferred embodiment, in step 5, the convergence of the optimal value equation group is further judged: defining the parameter convergence index Δ θ，k , to characterize the relative difference between the optimal value equations in two adjacent iteration steps; Δ θ，k The calculation formula is:

[0035]

[0036] Among them, θ [k] is the optimal parameter value obtained in the kth iteration, θ [k-1] is the optimal parameter value obtained in the k-1th iteration; k is any positive integer not less than 2;

[0037] When Δ θ，k When it is not greater than the judgment value, it is determined that the convergence of the optimal value equation group meets the requirements.

[0038] In a preferred embodiment, in step 5, the quantization matrix of the posterior uncertainty is:

[0039]

[0040] In a preferred embodiment, the calculation formula for the confidence interval width is:

[0041] WCI=2Std

[0042] Where Std is the standard deviation of the identified bridge influence line, and its value is the matrix diagonal elements of .

[0043] In a preferred embodiment, in step 6, the convergence of the Bayesian sequence update is determined by defining an uncertainty convergence index Δ WCI，N , to characterize the relative difference in the width of the confidence interval between two adjacent update stages; Δ WCI，N The calculation formula is:

[0044]

[0045] Among them, WCI [N] is the confidence interval width at the Nth update, WCI [N-1] is the confidence interval width at the N-1th update; N is any positive integer not less than 2;

[0046] When Δ WCI，N When it is not greater than the judgment value, it is determined that the convergence of the confidence interval width meets the requirements.

[0047] In a preferred embodiment, the determination value is 0.01%.

[0048] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0049] The method provided by the present invention cleverly uses the Bayesian sequence update strategy to use the preceding posterior distribution as subsequent prior information to establish an iterative relationship between multiple sets of measurement data, so as to scientifically, accurately and efficiently obtain the optimal solution of the bridge influence line from multiple sets of measurement responses. Secondly, the optimal value of the bridge influence line obtained through multiple iterative updates gradually fits the true value, verifying that the method can accurately identify the bridge influence line from measurement responses containing strong random interference (measurement noise, dynamic disturbance and random load influence), and has strong identification robustness. In addition, the method also scientifically quantifies the posterior uncertainty by defining the width of the confidence interval, effectively evaluating the convergence of the bridge influence line identification process. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 A technical roadmap for the Bayesian sequence updating method described in an embodiment of the present invention;

[0051] Figure 2 Flowchart of the Bayesian iterative update algorithm in an embodiment of the present invention;

[0052] Figure 3Schematic diagram of a three-span continuous beam in Example 1 of an embodiment of the present invention;

[0053] Figure 4 The bridge response signals simulated under different detection vehicle excitations in Example 1 of the embodiment of the present invention, where: (a) measurement signal 1; (b) measurement signal 2; (c) measurement signal 3; (d) measurement signal 4;

[0054] Figure 5 The bridge influence line identification results of a single set of measurement signals in Example 1 of the embodiment of the present invention, where: (a) measurement signal 1; (b) measurement signal 2; (c) measurement signal 3; (d) measurement signal 4;

[0055] Figure 6 The update results of the bridge influence line in Example 1 of the embodiment of the present invention (analysis of anti-measurement noise interference), where: (a) Update stage 1; (b) Update stage 2; (c) Update stage 3; (d) Update stage 4;

[0056] Figure 7 The following is a graph showing the changing trends of RE and WCI values ​​in Example 1 of the embodiment of the present invention (analysis of anti-measurement noise interference);

[0057] Figure 8 The updated results of the bridge influence line in Example 1 (anti-random load interference analysis) of the embodiment of the present invention are as follows: (a) Update stage 1; (b) Update stage 2; (c) Update stage 3; (d) Update stage 4;

[0058] Figure 9 The figure shows the trend of RE and WCI values ​​in Example 1 of the embodiment of the present invention (anti-random load interference analysis);

[0059] Figure 10 The bridge dynamic response signals simulated at different vehicle speeds in Example 1 of the embodiment of the present invention, where: (a) measurement signal 1; (b) measurement signal 2; (c) measurement signal 3; (d) measurement signal 4;

[0060] Figure 11 The updated results of the bridge influence line in Example 1 (anti-dynamic fluctuation interference analysis) of the embodiment of the present invention are as follows: (a) Update stage 1; (b) Update stage 2; (c) Update stage 3; (d) Update stage 4;

[0061] Figure 12 This is a graph showing the changing trends of RE and WCI values ​​in Example 1 of the embodiment of the present invention (analysis of anti-dynamic fluctuation interference);

[0062] Figure 13 This is a real diagram of a three-span continuous box girder bridge in Example 2 of an embodiment of the present invention;

[0063] Figure 14This is a simplified diagram of the BRT bridge health monitoring system in Example 2 of an embodiment of the present invention;

[0064] Figure 15 The strain response of the bridge at 1 / 2 span in Example 2 of the embodiment of the present invention;

[0065] Figure 16 The strain response of the bridge at 1 / 4 span in Example 2 of the embodiment of the present invention;

[0066] Figure 17 This is a real diagram of vehicles intersecting on a BRT bridge in Example 2 of an embodiment of the present invention;

[0067] Figure 18 This is the updated result of the strain influence line at the half span of the bridge in Example 2 of the embodiment of the present invention;

[0068] Figure 19 This is the updated result of the strain influence line at the 1 / 4 span of the bridge in Example 2 of the embodiment of the present invention;

[0069] Figure 20 This is a graph showing the changing trends of RE and WCI values ​​at half the span of the bridge in Example 2 of the embodiment of the present invention;

[0070] Figure 21 This is a trend diagram of the RE and WCI values ​​at the 1 / 4 span of the bridge in Example 2 of the embodiment of the present invention. DETAILED DESCRIPTION

[0071] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention; it is obvious that the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0072] In the description of the present invention, it should be noted that the terms "upper," "lower," "inner," "outer," "top / bottom," and the like, indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings and are intended solely to facilitate and simplify the description of the present invention. They are not intended to indicate or imply that the devices or components referred to must have, be constructed, or operate in a specific orientation, and therefore should not be construed as limitations on the present invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0073] In the description of the present invention, it should be noted that, unless otherwise clearly stipulated and limited, the terms "installed", "provided with", "set / connected", "connected", etc. should be understood in a broad sense. For example, "connection" can be a wall-mounted connection, a detachable connection, or an integral connection. It can be a mechanical connection or an electrical connection. It can be a direct connection or an indirect connection through an intermediate medium. It can be the internal connection of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances.

[0074] like Figures 1 to 21 As shown, the embodiment of the present invention provides a Bayesian sequence updating method for identifying bridge influence lines based on multiple sets of measurements. Figure 1 、 Figure 2 , the method mainly includes the following steps:

[0075] Step 1: Establish the prediction error between a single set of measured responses and the model.

[0076] Assume that the test vehicle is idling along a fixed route on the bridge. The vehicle is equivalent to several concentrated loads with fixed spacing. The static response of the bridge can be expressed by the following linear equation:

[0077]

[0078] Among them, R s is the static response vector of a certain section of the bridge; A is the load matrix constructed from vehicle load information; Represents the bridge influence line coefficient vector. Use shape function to fit the bridge influence line:

[0079]

[0080] Where Ω is the fitting matrix and c is the function coefficient vector. The bridge influence line identification model is embedded in the Bayesian framework to establish the measurement response R m and the static response R of the characterization model s The prediction error τ between τ and τ can be shown as follows:

[0081] τ=R m -AΩc (3)

[0082] Step 2: Determine the likelihood function and define the parameter prior distribution.

[0083] Assume that the i-th group measurement response The corresponding prediction error τ obeys a Gaussian distribution with zero mean, and the likelihood function can be expressed as:

[0084]

[0085] Where σ represents the standard deviation of the prediction error, N g is the number of response sampling points; in addition, the prior distribution of parameter c is expressed as follows:

[0086]

[0087] Where μ 2 is the scale variance that characterizes the change of the function coefficient vector c, N s is the dimension of c, and the parameter σ 2 and μ 2 The prior distribution of is determined by the conjugate prior principle and is expressed as an inverse gamma distribution:

[0088]

[0089] in, (α0, β0) and (α1, β1) are the hyperparameters of the inverse gamma distribution;

[0090] Step S3: Derive the posterior distribution of the single set of measurement responses.

[0091] Multiplying the above likelihood function with the prior distribution gives the posterior distribution of the i-th group of measurement responses:

[0092]

[0093] Step 4: Update the posterior distribution based on multiple sets of measured responses.

[0094] The posterior distribution of the response of the i-th group of measurements is As the i+1th group of measurement responses The prior distribution of is as shown in formula (8), and the i+1th group of measurement responses is defined according to formula (4) Likelihood function of :

[0095]

[0096] Combine the prior distribution of formula (8) and the likelihood function of formula (9) Multiplying together gives the updated posterior distribution:

[0097]

[0098] After N updates, the posterior distribution can be expressed as:

[0099]

[0100] in, Substituting the likelihood function and prior distribution into equation (11), we can obtain the analytical expression of the posterior distribution:

[0101]

[0102] Step 5: At each update stage, derive the optimal value equations for the bridge influence line parameters (MPVs) and perform a convergence check. Once convergence is achieved, quantify the a posteriori uncertainty of the bridge influence line.

[0103] Taking the negative logarithm of equation (12), we get the objective function L of the bridge influence line:

[0104]

[0105] Minimize the above equation to obtain the first-order derivative of L with respect to c, σ, and μ. Set the first-order derivatives of c, σ, and μ to 0 and obtain the optimal value equation system θ = c, σ, μ:

[0106]

[0107] μ 2 =(2(α1+1)+N s ) -1 (||c|| 2 +2β1) (16)

[0108] It should be understood that T is the transpose operator of the matrix. The optimal value equation set θ = c, σ, μ is a set of vectors, and its convergence judgment method is: define the parameter convergence index Δ θ，k , to characterize the relative difference between the optimal value equations (vectors) in two adjacent iteration steps. θ，k The calculation formula is:

[0109]

[0110] Among them, θ [k] is the optimal parameter value obtained in the kth iteration, θ [k-1] is the optimal parameter value obtained in the k-1th iteration; k is any positive integer not less than 2.

[0111] If the parameter convergence index Δ θ，k is not greater than the judgment value, and the convergence is satisfied, then the optimal value equation group is output and the quantization matrix of the posterior uncertainty of the bridge influence line is calculated; if Δ θ，k If the value is greater than the judgment value, the convergence is not satisfied, and the parameters in the optimal value equation group are iteratively updated until the convergence is satisfied. In this embodiment, the judgment value is 0.01%.

[0112] The process of calculating the quantization matrix of the posterior uncertainty is as follows: take the second-order derivative of L and obtain the Hessian matrix with respect to c:

[0113]

[0114] Hessian matrix of the optimal value MPVs of the bridge influence line With H cc The relationship between can be expressed as:

[0115]

[0116] And Invert to get the quantized matrix of posterior uncertainty:

[0117]

[0118] Step 6: In each update phase, the width of the confidence interval (WCI) is calculated and the convergence of the Bayesian sequence update is judged. Once convergence is satisfied, the analytical formula of the bridge influence line is output.

[0119] The upper and lower limits of the bridge influence line identification results can be defined as:

[0120] UB,LB=MPVs±Std (21)

[0121] Where UB is the upper limit of the bridge influence line, LB is the lower limit of the bridge influence line, and Std is the standard deviation of the identified bridge influence line, which is a matrix The diagonal elements of . Now define the confidence interval width to characterize the posterior uncertainty, and its calculation formula is:

[0122] WCI=UB-LB=2Std (22)

[0123] The convergence judgment method of Bayesian sequence update is: define the uncertainty convergence index Δ WCI,N , to characterize the relative difference in confidence interval width between two adjacent update stages; Δ WCI ,N The calculation formula is:

[0124]

[0125] Among them, WCI [N] is the confidence interval width at the Nth update, WCI [N-1] is the confidence interval width at the N-1th update, and N has the same meaning as in formula (11).

[0126] If the uncertainty convergence index Δ WCI，N If the uncertainty convergence index Δ WCI,N If the convergence is not satisfied, the process returns to step 4 and iteratively updates the posterior distribution of the bridge influence line until the convergence is satisfied. In this embodiment, the convergence is not satisfied.

[0127] The anti-interference ability of the above method is verified through two specific practical examples below.

[0128] Example 1: Influence Line Identification of a Three-Span Concrete Continuous Beam

[0129] like Figure 3 As shown, the span of each span of the continuous beam is L1 = 10 meters, and the cross section is rectangular with a cross-sectional size of 1 meter (height) × 0.5 meters (width). The parameters of the beam material include the elastic modulus E = 3.25 × 10 4 MPa and density ρ = 2.25 × 10 3 kg / m 3 Based on the MATLAB programming platform, the beam body is uniformly discretized into 300 beam units (unit length 0.1m) along the longitudinal direction, and the plane beam unit finite element model is established. i and wheelbase D j The configured inspection vehicle (specific parameters are shown in Table 1) was driven across the bridge at a constant speed of 20 km / h for numerical simulation analysis.

[0130] Table 1. Vehicle information of inspection vehicle

[0131]

[0132]

[0133] For conditions where the measurement signal-to-noise ratio (SNR) is significantly reduced when the bridge structure is stiff or the measurement points are located near the supports, this study verified the method's ability to resist measurement noise interference by superimposing Gaussian white noise with a signal-to-noise ratio (SNR) of 5dB to 10dB on the bridge response signal. Figure 4 The simulated bridge response signals under different detection vehicle excitations are shown in Table 2. The specific configuration parameters of the multiple measurement signals are detailed in Table 2. First, the bridge influence line is identified based on a single measurement point data set. The identification results are shown in Figure 2. Figure 5 As shown. The research results show that there are statistically significant deviations in the influence lines of bridges under various identification conditions, and the fitting effect between a single group of influence lines and the theoretical true value does not reach the ideal level, making it difficult to meet the basic accuracy requirements of engineering applications. This phenomenon reveals the inherent defects of existing methods in processing multiple groups of measurement scenarios, namely, insufficient utilization of measurement signals leads to a lack of robustness in the inversion results. Subsequently, based on Figure 1 The algorithm framework shown carries out the bridge influence line update calculation in stages to obtain the MPVs and uncertainty quantification index WCI at each iterative stage. Figure 6 The updated trajectory of the bridge influence line during the first four iterations is shown in Table 3. Figure 7The numerical evolution characteristics of WCI and RE during the entire updating process are summarized respectively. Among them, RE is defined as the optimal value of the bridge influence line and the true value The relative deviation is calculated as follows:

[0134]

[0135] Depend on Figure 6 From the updating process, we can see that as the number of iterations increases, the optimal value of the bridge influence line and the true value The fitting effect shows a gradual improvement. The data in Table 3 show that the RE index gradually decreases from 12.58% in the initial stage to 4.11%, which effectively verifies that the proposed method can still achieve high-precision identification of influence lines under high noise background. Figure 7 As can be seen, the WCI and RE show highly consistent trends throughout the update process, and both tend to converge stably after the fourth update. This demonstrates that monitoring the convergence characteristics of the WCI can effectively assess the stability of the bridge influence line inversion process. When the WCI reaches the convergence threshold, the corresponding MPVs are closest to the theoretical true value.

[0136] Table 2. Parameter information of multiple measurement signal groups (measurement noise immunity analysis)

[0137]

[0138] Table 3. WCI and RE values ​​(measurement noise immunity analysis)

[0139]

[0140] In the actual engineering application of bridge influence line identification, in order to maximize the bridge traffic efficiency, usually only the lane where the detection vehicle is located is closed, and the other lanes still maintain normal traffic flow. Under this working condition, the detection vehicle shown in Table 2 is traveling at a low speed in the closed lane, and at the same time there are high-speed vehicles in the open lane, which induces local fluctuations in the dynamic response of the bridge. In order to verify the robustness of the method to random load interference, this study constructs multiple groups of measurement signals containing random load effects for simulation analysis. The random disturbance application position and measurement system parameter configuration are detailed in Table 4. Based on Figure 4 The algorithm framework is used to calculate the bridge influence line update, and the MPVs and WCI evolution processes are obtained as follows: Figure 8 and as shown in Table 5. As the number of iterations increases, the RE of the bridge influence line MPVs gradually decreases from 11.45% in the initial stage to 3.02%. After four update iterations, the algorithm successfully obtains high-precision bridge influence line MPVs, fully demonstrating the strong robustness of the proposed method to random load interference. Figure 9The results further show that the WCI and RE indicators exhibit highly consistent trends throughout the update cycle, with the WCI value reaching convergence after the fourth update. This phenomenon further confirms that uncertainty quantification indicators can effectively evaluate the convergence of the bridge influence line identification process.

[0141] Table 4. Parameter information of multiple measurement signal groups (analysis of anti-interference performance under random loads)

[0142]

[0143]

[0144] Table 5. WCI and RE values ​​(random load impact immunity analysis)

[0145]

[0146] Numerical simulations were conducted under Class B road roughness conditions and with vehicle speeds ranging from 40 to 60 km / h to verify the robustness of the proposed method to dynamic fluctuations caused by high-speed vehicles. A measurement noise with a signal-to-noise ratio of 15 dB was introduced into the measurement response. The dynamic response signal characteristics and the corresponding bridge influence line update process are shown in Figure 2. Figure 10 and Figure 11 As shown, the evolution of relevant error indicators is detailed in Figure 12 Table 6. Affected by dynamic fluctuations, the bridge influence line identification results in the first update phase exhibit significant discrete characteristics, with an RE reaching 14.03%. Due to the random nature of dynamic disturbances in each measurement, bridge influence line identification based on single measurement data inevitably leads to large deviations in the inversion results. When the proposed method is used to process multi-source measurement data, the fit between the optimal value and the true value of the bridge influence line gradually improves with the increase in the number of updates. In the sixth update phase, the RE dropped to 4.48%, while the WCI index trend maintained synchronous convergence characteristics with the bridge influence line update process. These results collectively demonstrate that this method can not only effectively resist the influence of dynamic disturbance components on the identification process, but also scientifically evaluate the convergence characteristics of the bridge influence line update results through quantitative uncertainty indicators within a Bayesian framework.

[0147] Table 6. WCI and RE values ​​(dynamic disturbance immunity analysis)

[0148]

[0149]

[0150] In this verification, the results of the continuous beam numerical example show that this method can accurately identify bridge influence lines from measurement responses containing strong random interference (measurement noise, dynamic disturbances and random load effects) when scientifically and efficiently processing multiple sets of measurement signals, overcoming the shortcomings of traditional bridge influence line identification technology based on a single set of measurement signals. In addition, this method can evaluate the convergence of the identification results using uncertainty indicators.

[0151] Example 2: Influence Line Identification of a Three-Span Continuous Box Girder Bridge

[0152] Xiamen Bus Rapid Transit System (BRT) provides exclusive closed lanes for BRT vehicles. The test bridge of this institute is a three-span continuous box girder bridge with a span of L2 = 30m for each span. The actual bridge photo is as follows: Figure 13 The bridge only allows certain types of buses to pass through, and the load form is fixed type, with axle loads of 3762kg and 7819kg respectively, and a wheelbase of 6m. The architecture of the bridge health monitoring system is shown in the figure below. Figure 14 As shown in Figure 1, the hardware includes: strain sensors arranged at the 1 / 4, 1 / 2 and 3 / 4 sections of the mid-span, and a data acquisition system and industrial computer installed at the bottom of the pier (see the detailed hardware layout for details). Figure 14 The monitoring data is uploaded to the cloud through an industrial computer, and users can use the "Bridge Intelligent Monitoring Hosting Platform" to remotely view the response data in real time. In order to verify the anti-interference performance of the proposed method, the bridge strain response data segment on April 13, 2023 was extracted from the monitoring system for analysis. The measurement data of some periods are as follows: Figure 15 and 16 As shown. The measurement signal 5 is triggered by the vehicle intersection condition (such as Figure 17 As shown in Figure 2, this data will increase the RE during the bridge influence line update process and is therefore not included in the bridge influence line inversion calculation. Ultimately, the bridge influence line is identified using the response data of an unloaded BRT vehicle crossing the bridge at a constant speed. The reference value of the bridge influence line is determined using a deterministic method.

[0153] The method provided by the present invention is used to sequentially update the bridge influence line MPVs and uncertainty index WCI. The results are as follows: Figures 18 to 21 As shown in the figure, after four iterations, the bridge influence lines at the half-span and quarter-span locations show excellent fit with the reference values. As the number of updates increases, MPVs gradually converges to more accurate inversion results, validating the method's ability to scientifically and efficiently process multiple sets of measurement signals to improve the interference resistance of bridge influence line identification. Figure 21Table 7 further demonstrates that during the update process, the RE and WCI indicators maintain a synchronized trend. The convergence characteristics of the uncertainty indicators can be used to evaluate the convergence of the bridge influence line update state. This characteristic allows the quality of bridge influence line identification to be determined in practical projects without relying solely on the true baseline.

[0154] The above description is only a preferred embodiment of the present invention and does not limit the patent scope of the present invention. Any technical equivalent transformation made using the contents of the present invention specification shall fall within the protection scope of the present invention.

Claims

1. A Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements, characterized by: The following steps are involved: Step 1: Establish the prediction error between a single set of measured responses and the bridge influence line identification model; Step 2: Establish the likelihood function of a single set of measured responses and define the prior distribution of the parameters in the likelihood function; Step 3: deriving the posterior distribution of the single group measurement response based on the likelihood function and the prior distribution; Step 4: Based on multiple groups of measurement responses, iteratively update the posterior distribution of the bridge influence line: use the posterior distribution of the i-th group of measurement responses as the prior distribution of the i+1-th group of measurement responses, and refer to step 2 to define the likelihood function of the i+1-th group of measurement responses to derive the posterior distribution of the i+1-th group of measurement responses; where i is an arbitrary positive integer; Step 5: At each update stage, determine the optimal value of the bridge influence line at that stage and quantify the posterior uncertainty; Step 6: Define the confidence interval width to characterize the posterior uncertainty; in each update stage, calculate the confidence interval width and determine the convergence of the Bayesian sequence update; if the convergence meets the judgment value, output the analytical expression of the bridge influence line; if the convergence does not meet the judgment value, return to step 4 and iteratively update the posterior distribution of the bridge influence line until it meets the judgment value.

2. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 1, characterized in that: In step 1, the prediction error is: τ=R m -AΩc Among them, R m is a single set of measurement responses, A is the vehicle load matrix, Ω is the fitting matrix, and c is the function coefficient vector.

3. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 2, characterized in that: In steps 2 and 3, the i-th group measures the response The corresponding likelihood function is: Where σ represents the standard deviation of the prediction error, N g is the number of response sampling points; The prior distribution of parameter c is: Among them, μ 2 is the scale variance that characterizes the change of the function coefficient vector c, N s is the dimension of c; Furthermore, the parameter σ 2 and μ 2 The prior distribution of is determined by the conjugate prior principle and is expressed as the inverse gamma distribution: in, (α0, β0) and (α1, β1) are the hyperparameters of the inverse gamma distribution; The i-th group of measurement responses The corresponding posterior distribution is:

4. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 3, characterized in that: In step 4, the posterior distribution updated after N iterations is: in, 5. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 4, characterized in that: In step 5, the optimal value equations of the bridge influence line parameters θ = c, σ, μ are: m 2 =(2(α1+1)+N s ) -1 (||c|| 2 +2β1).

6. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 5, characterized in that: In step 5, the convergence of the optimal value equation group is judged: the parameter convergence index Δ is defined θ，k , to characterize the relative difference between the optimal value equations of two adjacent iteration steps; Δ θ，k The calculation formula is: Among them, θ [k] is the optimal parameter value obtained in the kth iteration, θ [k-1] is the optimal parameter value obtained in the k-1th iteration; k is any positive integer not less than 2; When Δ θ，k When it is not greater than the judgment value, it is determined that the convergence of the optimal value equation group meets the requirements.

7. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 5, characterized in that: In step 5, the quantization matrix of the posterior uncertainty is:

8. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 7, characterized in that: The calculation formula for the confidence interval width is: WCI=2Std Where Std is the standard deviation of the identified bridge influence line, and its value is the matrix diagonal elements of .

9. The Bayesian sequential updating method for identifying bridge influence lines based on multiple sets of measurements according to claim 8, characterized in that: In step 6, the convergence judgment method of Bayesian sequence update is: define the uncertainty convergence index Δ WCI，N , to characterize the relative difference in the width of the confidence interval between two adjacent update stages; Δ WCI，N The calculation formula is: Among them, WCI [N] is the confidence interval width at the Nth update, WCI [N-1] is the confidence interval width at the N-1th update; N is any positive integer not less than 2; When Δ WCI，N When it is not greater than the judgment value, it is determined that the convergence of the confidence interval width meets the requirements.

10. A Bayesian sequence updating method for identifying bridge influence lines based on multiple sets of measurements according to any one of claims 6 or 9, characterized in that: The determination value is 0.01%.

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