A Novel Moving Load Identification Method for Continuous Rigid-Frame Bridges Using a Field-Based Displacement Influence Line

Abstract

This study focuses on a new identification method for moving loads on bridge structures using field-based displacement data from different measurement points on a continuous rigid-frame bridge. A novel approach has been proposed to make use of the area of the absolute field value derived from the displacement influence line of continuous rigid-frame bridges. Considering the potential presence of other nuisance loads (i.e., noise) on the bridge, this method can significantly mitigate the impact of noise by adopting the absolute area method of influence lines. In addition, the new method combines data from various field measurement points to identify the moving loads, which can in turn minimize the influence of measurement errors. To validate the new method, several numerical simulations varying different noises and parameters have been carried out for benchmarking. The results show that our proposed method achieves an outstanding identification accuracy of over 95% for the simulation cases with the disturbance noise amplitude less than 1.0% and the field data with random noise. This new method enables the identification of moving loads on bridges, thereby providing fundamental data for bridge health monitoring and damage detection. This will help improve predictability of the remaining fatigue life of bridge structures.

1. Introduction

Bridge load identification is a critical task in structural health monitoring (SHM), ensuring the safety and longevity of bridge infrastructures [1,2], since many approaches of SHM in practice first need the bridge load information. The accurate determination of moving loads, such as vehicles or trucks, plays a pivotal role in assessing structural performance and detecting potential damage [3,4]. Traditional methods for load identification often rely on dynamic response analysis, weigh-in-motion (WIM) systems, or data-driven approaches [5,6]. However, these techniques may face challenges in cases involving unknown moving loads, complex structural behavior, or noisy measurement data.

In recent years, the influence line (IL) concept has gained significant attention as an effective tool for load identification and damage detection in bridge structures. The influence line represents the structural response at a specific point due to a unit load moving across the bridge, providing valuable insights into load distribution and structural behaviors. Liu et al. [7] proposed a probabilistic frequency-domain approach for IL extraction under unknown moving loads, demonstrating its potential for accurate load estimation. Similarly, Qin et al. [8] and Zhang Q, et al. [9] explored advanced machine learning techniques to enhance moving load identification, highlighting the growing role of artificial intelligence in this field. Yan and Yuen [10] addressed this limitation by developing a method for the online tracking of the influence lines of high-speed railway bridges, utilizing only the dynamic response of passing trains. Unlike traditional approaches, which determine IL through static loading, Zhang et al. [11] proposed a novel technique to identify the rotating influence line (RIL) based on vehicle–bridge interaction (VBI) under stochastic traffic loading conditions. It is based on the assumption that the stiffness has no change in the work life of the bridge and there is no stiffness loss as time goes by.

Several studies have utilized IL-based methods for axle load identification and bridge damage assessment. Qian et al. [12] developed a method for identifying vehicle axle loads using bending moment influence lines, while Pimentel et al. [13] implemented a fiber optic-based WIM system for train load identification. Pourzeynali et al. [14] and Liu et al. [15] further investigated numerical and experimental approaches for moving load identification, emphasizing the importance of dynamic response analysis. Additionally, Kordi and Mahmoudi [16] and Cavadas et al. [17] demonstrated the effectiveness of IL-based techniques in damage detection, leveraging both static and dynamic structural responses. Zhu et al. [18] developed a quasi-static method for identifying structural damage based on a sensor’s line of influence (IL) and an empirical Bayesian threshold estimator and validated the feasibility of the method through numerical models and laboratory tests. Zhou et al. [19] proposed a new method of beam bridge damage identification based on multi-source influence line information fusion, aiming at the problem of poor robustness and accuracy of identification results caused by single measurement point data using the influence line class method, and conducted simulation verification with an actual steel–mixed composite beam bridge.

Recent advancements have extended the application of influence lines to distributed sensing and long-gauge strain measurements. Wu et al. [20,21] proposed damage identification methods based on strain envelope lines under moving loads, while Zhu et al. [22] and Yang et al. [23] explored dynamic response processing and strain influence lines for bridge damage assessment. Zhou et al. [24] further introduced an equivalent thrust-influenced line method for tied-arch bridges, showcasing the versatility of IL-based approaches. Yang [25] developed the sensitized long-scale FBG sensor and its standardized production method, studied the moving load identification method and the bridge structural damage identification method based on the long-scale strain influence line, and studied the deflection inversion method based on the long-scale strain.

Despite these advancements, challenges remain in accurately identifying high-speed moving loads and minimizing computational complexity. Moslehi Tabar et al. [26] introduced weakened mode shape functions for high-speed load identification. His method used a surrogate beam model to eliminate the singularities of the solution, and the accuracy is sensitive to the noise. Whereas Sun et al. [27] and Wang et al. [28] focused on strain monitoring and moving train load identification. Furthermore, Sieniawska et al. [29] and Hester et al. [30] investigated structural parameter identification and rotation response analysis, respectively, under moving loads. The above research used various methods to solve the inverse problem of the dynamic, and valuable results were obtained under certain conditions. However, the inverse method for the dynamic cannot solve the sensitive problem of noise. In addition, the dynamic excitation effects on the bridge induced by the moving vehicle were discussed [31,32,33]. This research shows that dynamic effects will be amplified or dampened by the bridge structure itself, depending on the relative vibration frequencies of the bridge and the moving heavy vehicle.

To address these challenges, this study proposes a novel bridge load identification method based on the field-based influence line, integrating data-driven techniques and advanced signal processing. By using the area of the displacement influence line, this proposed approach can eliminate the effect of noise. Also, the present method aims to improve accuracy and computational efficiency, particularly for complex loading scenarios. By leveraging the strengths of existing methods and incorporating innovative algorithms, this research contributes to the advancement of bridge load identification and SHM. Finally, the present method was verified by using the FEM and practical results. The outcomes of this study are expected to enhance the reliability of load estimation and damage detection in bridge engineering applications.

2. Methodology

This study primarily adopts the moving load influence line for the rigid-frame structures. Let the influence line for the continuous rigid-frame bridges be denoted as I(x). The displacement d(x) under the action of a moving load is calculated as shown in Equation (1), where P represents the moving load:

$$d(x)=P·I(x)$$

(1)

According to kinematic principles, the response under impact loads, specifically the maximum displacement amplitude, is primarily dependent on the magnitude of the applied impulse, while the form of the impact load has less influence. Therefore, considering the impact of moving-load noise, the evaluation of the moving load is performed by comparing the area of the influence line with the area of the monitoring data, which is the basic concept of the influence line (IL), as shown in Equation (2) below.

$$d_i={\displaystyle {\int }_0^{L}d}(x)\mathrm{d}x=P·{\displaystyle {\int }_0^{L}I(x)}dx=P·a_i$$

(2)

where d_i is the area of the displacement data measured at the i-th measurement point, and a_i is the area of the i-th point influence line. However, in the case of multi-span continuous beam bridges, as shown in Figure 2, and the influence line, as shown in Figure 4, the values of both d_i and a_i are very small, resulting in great errors in Equation (2). The application of IL will be extended by changing the area of the value as the absolute area, as expressed by the following equation:

$$D_i={\displaystyle {\int }_0^L\left|d(x)\right|}dx={\displaystyle {\int }_0^{L}P·\left|I(x)\right|dx}=P·A_i$$

(3)

where D_i is the area of absolute displacement data measured at the i-th measurement point and A_i is the absolute area of the i-th point influence line. In this case, Equation (4) can be named the method of the absolute area of the influence line (AIL). Considering the noise $\phi (x,t)$, which is the random function of bridge and time, it should be noted that the variable t in the function of $\phi (x,t)$ is not independent, and it is linked to the location of x and can be expressed as t = x/v (v is the velocity). In this case, Equation (4) can be named the method of the absolute area of the influence line (AIL). Considering the noise $\phi (x,t)$, which is the random function of the time and the location of the bridge, Equation (3) can be modified to Equation (4):

$$D_i=P·A_i+\lambda ·{\phi }_i$$

(4)

where ${\phi }_i={\displaystyle {\int }_0^L\phi (x,t)}·I\left(\mathrm{x}\right)\mathrm{d}x$ is the inner product of the noise and influence line, which relates to the distribution of the random noise, and λ is the parameter determined by the type of noise. If the random noise is a type of white noise, the parameter ${\phi }_i$ would be very small or even close to zero. Therefore, using Equation (4) can only obtain the value of identification. In fact, the noise is usually random and unknown; more monitoring data can be used to obtain the identification to mitigate the errors caused by the load noise and the measurement. By combining data from multiple monitoring points, the following equation can be derived:

$$\left(\begin{array}{l}{D}_1\\ D_2\\ \dots \end{array}\right)=P·\left(\begin{array}{l}{A}_1\\ A_2\\ \dots \end{array}\right)+\lambda ·\left(\begin{array}{l}{\phi }_1\\ {\phi }_2\\ \dots \end{array}\right)$$

(5)

When the number of monitoring points is more than two, Equation (5) becomes a system of multiple constraints, and the moving load P and the noise distribution parameter λ can be obtained using the least squares method. In this case, Equation (5) can be named as the method of AIL-Modified. From the discussion above, it can be concluded that the proposed method needs at least 2 measurement points. According to the theory of the least squares method, increasing the quantity of measurement data enhances the accuracy and robustness of the results. As mentioned above, the flowchart of the proposed method was listed in Figure 1 below. As the values of D_i and A_i are calculated from the different measurements, they will not be the same. That is to say, the matrix of Equation (5) will not be illness. It also can be concluded that the method of Equation (5) will not solve the problem of the effects of dynamic amplification.

3. Numerical Simulations

3.1. Simulation Model

3.1.1. The Geometry of the Bridge in the Field

The geometric dimensions and measurement point distribution of the constructed Chenxi Bridge are shown in Figure 2a. This is a large-span, heavy concrete box rigid bridge. The cross-sectional design of this bridge follows a symmetric distribution on both sides of the pier and is divided into six segments, each measuring 90 m. The control cross-section dimensions are illustrated in Figure 2b, with detailed specifications provided in Appendix A. It should be noted that the units in Figure 2a are meters and in Figure 2b are millimeters.

A total of eight measurement points were strategically positioned along the bridge, specifically at the mid-span of each span and at the quarter-span locations of the main span. Taking the geometric center of the bridge as the origin, the coordinates of measurement points P1 through P8 were defined as −225 m, −135 m, −90 m, −45 m, 45 m, 90 m, 135 m, and 225 m, respectively. For the finite element modeling, BEAM188 elements with a trapezoidal cross-section were utilized. The material properties assigned to the model included an elastic modulus of 3.45 × 10¹⁰ Pa and a Poisson’s ratio of 0.24. BEAM188 was chosen, and the static solution module was adopted. In addition, the damping effect and nonlinearity were not considered given the small deformation in the case. The finite element model was developed in ANSYS 21.0, as illustrated in Figure 2c. The boundary of the FEM model is that the middle three piers are fixed and the endpoints of the two ends are supported on vertical constraints (UY). The default settings were accepted.

3.1.2. Validation of the Model on Frequency

The mode shapes of the modal analysis are shown in Figure 3. It can be seen that the first and third modals correspond to the Z direction, which is horizontal, and the second and fourth correspond to the vertical direction. And following that, the modals in the vertical direction are greater. It can be seen that the fifth, sixth, and eighth modals are in the vertical direction; only the seventh modal is in the horizontal direction.

The modal analysis results using the finite element model are presented in

Table 1. The frequency of the simulation and the test (Hz).

Modal Order	1	2	3	4	5	6	7	8
Simulation	0.53	0.53	0.58	0.63	0.79	0.88	1.10	1.29
Measured value	0.52	0.52	0.53	0.68	0.84	0.89	1.14	1.36
Errors	0.01	0.01	0.05	−0.05	−0.05	−0.01	−0.04	−0.07

. A comparison with the measured frequencies reveals that the first two frequencies are almost the same. According to the root mean squared error calculation for the first four frequencies, the values of the RMSE are 3.6% and 4.2% for the eight frequencies, which shows that the two data can be seen as the same and shows the validation of the FEM model.

3.1.3. The Base of the Influence Line

After the modal analysis, the moving load along the centerline of the bridge was set to obtain the influence line. Based on the simulation models, the displacement influence lines at the eight monitoring points under moving loads were computed, as shown in Figure 4. It can be observed that the influence lines of the continuous beam exhibit both positive and negative values. This confirms that the influence lines in odd-numbered spans are negative, whereas those in even-numbered spans (including the span where the monitoring point is located) are positive. Furthermore, due to the symmetric arrangement of the eight monitoring points, the influence lines also exhibit a symmetric distribution.

3.2. Results with Different Noises

3.2.1. Vehicle Load with Turbulence Noise

Considering the dynamic effects of the noise derived from the car movements, or the so-called vehicle–bridge interaction, the moving load may not remain constant during its passage. In this part, the noise of the moving load is assumed to follow harmonic oscillations. The analysis includes different noise magnitudes and two-point loading configurations (front and rear axles with a 3.0 m distance). The simulation test cases are shown in

Table 2. Simulation cases with different weights.

Case	Speed (m/s)	Load (kN)	Noise	Remark
1	3.0	10	/	Single point
2	3.0	10	5% × sin(5t)
3	5.0	10	10% × sin(5t)
4	3.0	10	10% × sin(10t)	Two points

below. It should be noticed that the vehicle weight was set to 10 kN and the speed of the vehicle set to 3.0 m/s, except for case 3, which was set to 5.0 m/s. The noise was set to the function of time, sin(5t), with a different amplitude, which means the disturbance noise is sin(15x) if the bridge location was taken as the variable.

Figure 5 and Figure 6 present a comparison of the displacements at point 2 (P2) and point 3 (P3) under different noise levels. It can be observed that the impact of noise on the displacement varies with the noise level. The deviation at P2 is larger than that at P3. Overall, the displacement data with added noise fluctuate around the displacement values without noise, and the magnitude of the fluctuation is directly proportional to the noise level.

Figure 7 shows a comparison of the predicted values using different calculation methods under a 5% noise intensity. It can be observed that, under a 5% noise intensity in the moving load itself, all three methods are capable of effectively identifying the moving load with errors less than 5%. It is important to note that the influence line method is somewhat sensitive to the data from the monitoring points, while the absolute value method can, to some extent, mitigate the influence of the monitoring points. And the modified absolute influence line has the best accuracy of identification with a 0.5% RMSE error.

It can be observed that the results from all three methods are very good, achieving accuracy rates of over 98%. The case study results show that, consistent with the findings in He et al. [33], the influence line method (IL) demonstrates high identification accuracy for cases involving only moving loads, and it exhibits strong noise resistance.

3.2.2. Uniform on the Bridge with Random Noise Ranged (−1, 1)

Under the cases of moving-load noise shown in

Table 3. The simulation case with a uniform random distribution load.

Case	Load (kN)	Noise Amplitude	Remarks
5	10	0.5%	The mean of the disturbance noise is zero.
6	10	1.0%

, this study also considers the situation where noise is uniformly distributed across the bridge deck. It is assumed that the noise follows a random distribution within the range of (−1, 1), i.e., the noise is distributed with a zero mean and amplitudes of 1.0% and 0.5% of the moving load magnitude. Given that the total length of the bridge is 360 m, an amplitude of 1.0% means that the overall random load amplitude can reach up to 3.6 times the magnitude of the moving load.

Figure 8 and Figure 9 represent the displacement data at points 1 (P1) and 3 (P3) under cases 5 and 6, respectively. It can be observed that the displacement under zero-mean random loads can be considered as fluctuating around the displacement caused by moving loads, with the fluctuation magnitude being proportional to the noise amplitude. In addition, it can be depicted that the mean value of the responses in three cases is the same, but the higher noise levels lead to stronger fluctuations in the vibration responses.

Figure 10 and Figure 11 compare the identification results using the proposed method and the influence line area method. It can be observed that the identification results of the influence line area method vary significantly across different monitoring points. Additionally, directly using the influence line area method results in larger errors, whereas the absolute value area method effectively reduces the errors.

3.2.3. Uniform Load with Noise Ranged (0, 1)

When there are additional vehicles on the bridge deck, the random loads on the bridge deck can be assumed to be distributed with varying amplitudes within the range of (0, 1). The vehicle load identification results under these conditions are shown in Figure 12 and Figure 13. It can be observed that the identification results using the influence line area method vary significantly across different monitoring points. Furthermore, directly using the influence line area method results in larger errors, whereas the absolute value area method effectively reduces these errors.

3.3. Summary of the Identification Results

Table 4. Simulation cases with a 0.5 mean noise.

Case	Load	Noise Type	Noise Strength	Remark
7	10	Uniform random	0.5%	Mean of disturbance Noise is 0.5
8	10		1.0%

shows the simulation cases with uniform random noise and

Table 5. The identification errors for the different methods.

Case	Noise Details	IL	AIL	AIL-Modified
1	Moving-load noise	1.2%	2.2%	1.8%
2		2.2%	2.4%	2.2%
3		2.3%	1.8%	1.9%
4	Two point (3 m)	1.5%	2.0%	1.5%
5	White noise Mean zero	6.3%	3.2%	2.2%
6		15.3%	2.6%	2.7%
7	White noise Mean 0.5	45%	7.2%	2.3%
8		93%	10.3%	3.2%

summarizes the identified loads under eight cases and three different noise levels. It can be observed that when only moving-load noise is present, the identification accuracy of the influence line area method (IL), the absolute value method (AIL), and the absolute value method with multiple measurement points (AIL-Modified) are the same. When the load spacing is small, but the total load magnitude remains constant, the identification accuracy remains nearly unchanged. However, when there is a uniform load on the bridge, the accuracy of the IL decreases significantly, whereas both the methods of AIL and AIL-Modified can meet engineering requirements.

4. Application in Chenxi Bridge

4.1. Original Data

As for Chenxi Bridge, it has established a displacement and strain monitor system. And at the end of the bridge, a camera was installed. Figure 14a shows the full bridge, and Figure 14b shows the spot where the truck passes the bridge.

The displacement sensors are MIRAN-130, which was produced by the company of MIRAN (Shenzhen City, Cina) and it was shown in Figure 14c, and they were installed under the bridge, as shown in Figure 14d. It should be noted that the MIRAN-130 was calibrated before and after installation. The parameter of the MIRAN-130 was listed in

Table 6. The parameters for the MIRAN-130 displacement sensor.

MIRAN-130	Range	Precision	Temperature	Sample Rate
Value	0–500 mm	0.1% × F.S	−40~85 °C	1–20 Hz

. The raw displacement data can be obtained according to the timestamp in the figure. It should be noted that the range of the sample rate is 1–20 Hz, and, in the application, the sample rate was set to 10 Hz.

In the present study, a subset of data corresponding to the time interval of 21:00:00–21:59:59 from the full selected data was selected, as illustrated in Figure 15. Furthermore, to conduct a more focused analysis, a subset of data from the peak activity period was extracted, which is demarcated by a red rectangle in Figure 15. Then, the measurement data exactly corresponding to the heavy truck passing the bridge shown in Figure 16 were obtained.

4.2. Data Correction

The moving average filter method was used to make the data smoother. This filtering process allows for a detailed examination of the most significant fluctuations within a given time frame. It should be noted that the proposed method does not need the smoothing process to calculate the identification results. After the process of filtering, the smooth data can be obtained, as shown in Figure 17. It can be known from the figure that the total time taken for the truck to pass the bridge was about 26 s.

The peak data indicate that a heavy load crossed the bridge, typically corresponding to a heavy truck, as illustrated in Figure 14b. From Figure 15, the data within the red rectangular region were selected for analysis, with the results presented in Figure 16 and Figure 17. It can be seen that the time taken for the heavy truck to pass the bridge was 26 s and the length of the bridge is 540 m. Thus, it can be deduced that the speed of the truck was 76 km/h, which can be the approximate initial parameter for the identification process.

4.3. Identification for Test Data

Figure 18 shows the identification results with the measurement data and

Table 7. The identification results for the different methods (ton).

Method	P1	P2	P3	P4	P5	P6	P7	P8
IL	42.7	43.0	39.2	42.4	42.2	38.3	37.9	41.1
AIL	42.6	42.3	44.6	41.1	42.9	42.8	42.7	43.8
AIL-Modified	43.1
Measured value	44.6

presents the identification results of the three methods applied to the measured data. As shown, the accuracy of the IL method depends on the location of the data points. It should be noted that there is a mass meter at the end of the bridge, so the weight of the truck can be obtained directly. In general, accuracy is proportional to the data range. The maximum RMSE errors for the IL and AIL methods at different points are 15.2% and 10.3%, respectively. It can be seen that the AIL method is better than IL. In contrast, the error of the AIL-Modified method is 3.4%, which is significantly lower compared to the other two methods. From the results of Table 7, it can be seen that the AIL-Modified method has a better performance compared to the other two methods.

5. Discussions

From the analysis of influence line theory, it can be concluded that when there is uniformly distributed load noise on the bridge deck, the monitored displacement data essentially represent the combined effect of both the moving load and the uniformly distributed load. As shown in Equation (4), if the bridge deck load is approximately uniformly distributed, Equation (2) can be simplified to Equation (6).

$$d_s(x)=P·I(x)+{\displaystyle {\int }_0^{L}f(t)·I(x)dx}=P·I(x)+f(t)·A$$

(6)

Here, f(t) represents the uniformly distributed load value, which is the function of time, and A represents the influence line area. By summing all data points on both sides of Equation (6), Equation (7) can be obtained.

$$A_s=P·A+f·L·A$$

(7)

It can be observed that, in this case, using the influence line area results in the combined estimation of the moving load and bridge deck load, leading to significant errors. However, by using absolute area integration, as shown in Equation (8), it is evident that the displacement influence line has both positive and negative values in a multi-span continuous bridge, resulting in the absolute influence area $\left|A\right|$ being significantly larger compared to the direct influence area A. For instance, in Figure 2, the $\left|A\right|$ of measurement point 2 is 16.3 times of A. As for the other measurement point, the minimum ratio of $\left|A\right|$ over A is 15.4. The moving load can still be identified when the bridge deck load is relatively small ($f·L<<P$).

$${\left|A\right|}_s\approx P\left|A\right|+f·L·A$$

(8)

6. Conclusions

This study modifies the influence line area method by integrating its absolute values while accounting for the distribution of uniformly distributed loads on the bridge, which can mitigate the effect of noise. A moving load identification method that considers uniformly distributed load noise is derived. In practice, the displacement data when the truck or heavy vehicle passes the bridge are relatively easy to obtain. By using Equation (7) or Equation (5), the vehicle load will be identified. The following conclusions are drawn from the simulation analysis validation:

(1)

For cases with noise subject to the moving load: The three methods, including IL, AIL, and AIL-Modified, can all identify moving loads effectively. The wheelbase and moving speed have little impact on the identification methods.

(2)

The IL method is highly sensitive to the location of monitoring points, while the AIL and AIL-Modified methods are less dependent on the location of the monitoring points.

(3)

For cases with uniformly distributed load noise subject to the bridge, the IL method becomes nearly unreliable. The AIL method performs within acceptable error margins, and the AIL-Modified method provides more accurate identification results.

(4)

The application of the proposed method to the bridge demonstrates its feasibility and potential for implementation in similar bridge structures. However, the effectiveness of this method is highly dependent on the accuracy of input data. Therefore, further research is necessary to assess its applicability to other types of bridges. And the location of the displacement sensors also has a significant effect.

(5)

The accurate moving load identification is fundamentally crucial to fatigue life models that help to assess the remaining life of bridge structures. This underpins the predictive and preventative maintenance regime that is cost-effective and eco-friendly.