A New Approach for Multiple Loads Identification Based on the Segmental Area of the Influence Lines

Abstract

The dynamic responses of bridges under multi moving loads are an essential precursor for their structural health monitoring (SHM). To enable the precise identification of the main moving load(s) among multiple moving loads, this study proposes an improved multi-source dynamic load identification algorithm based on the segmental area of the influence line (SAI). Firstly, the segmental area of the influence line was calculated according to the velocity of loads and the distance between two loads, and then, the moving load could be isolated based on the law of the minimal error combining the base area of the original influence line. In addition, experiments were conducted employing laser displacement sensor systems to acquire structural dynamic responses. The results showed the following for the segmental area of the influence line: (1) identification errors for a single moving load could be controlled within 5%, while an error within 10% was achieved under two moving loads; (2) vehicle displacement identification error remained consistently below 5%; and (3) the proposed algorithm exhibited a speed-insensitive characteristic, enabling effective load identification across varying vehicle speeds. The experimental findings confirm that this method accurately identifies the main moving loads in a small deformation condition and can be extended to similar applications.

1. Introduction

Bridge load identification methods can generally be categorized into two groups. The first is the quasi-static axle load identification method, also referred to as the direct measurement approach [1,2]. Despite its straightforward application, this approach is constrained by low efficiency, high susceptibility to errors, and interference with normal traffic flow. To overcome these drawbacks, researchers have advanced the second category of methods, namely the Moving Force Identification Method for Weigh-In-Motion (WIM) [3,4]. This method relies on the structural responses of the vehicle–bridge interaction system, where inverse force identification algorithms are applied to estimate wheel or axle loads during vehicle passage. Unlike static weighing, dynamic weighing does not require vehicles to stop on weighing devices. Instead, bridge-mounted sensors (e.g., strain gauges, displacement transducers, or acceleration meters) are employed to acquire response data in real time. Combined with advanced techniques such as inverse problem formulations, regularization methods [5], Kalman filtering [6], and wavelet analysis [7], the WIM approach enables efficient, non-contact, and continuous dynamic load identification.

Dynamic weighing-based moving load identification falls within the category of inverse problems in structural mechanics. According to its methodological characteristics, it can be further divided into two main approaches. The first is based on the vibration equations of the vehicle–bridge system, wherein simulated structural responses are compared with measured responses to inversely determine moving loads. The pioneering researchers, Law et al. (1997) [8], established for the first time the theoretical relationship between bridge responses and moving loads based on the time-domain method and the modal superposition method, thereby realizing time-domain identification. Two years later (1999) [9], they further proposed a frequency–time-domain method: the structure was first subjected to modal decomposition to obtain single-mode equations, and then the Fourier transform was applied to derive the explicit analytical relationship between the frequency-domain responses and moving loads, enabling load identification in the frequency domain. Building upon these early studies, subsequent researchers made further improvements to identification algorithms and filtering techniques, aiming to reduce the adverse effects of noise and to enhance the accuracy and robustness of load identification. Li M. et al. [10] proposed an improved fractional Tikhonov method (IF-Tik) for bridge moving load identification, showing higher accuracy and noise resistance than conventional approaches. Wang L. et al. [11] applied a new iterative regularization method for dynamic load identification of stochastic structures, transforming uncertainties into deterministic inverse problems and efficiently evaluating statistical load characteristics. Jiang J. et al. [12,13] developed time-domain algorithms for dynamic load identification in structural systems, employing modal transformation and the implicit Newmark-β method to handle unknown initial conditions and discretize infinite degrees of freedom. Wang L. et al. [14] proposed a distributed dynamic load identification method for aircraft structures with bounded uncertainties, employing Chebyshev polynomials and uncertainty propagation analysis. Li Y. et al. [15] endeavored to identify earthquake ground motion from structural health monitoring data using absolute-coordinate equations and a Kalman filter, validated by numerical and shaking table tests. Li K. et al. [16] proposed a Bayesian framework for distributed dynamic load identification, integrating joint sparsity and spatial correlation priors, with validations confirming accurate noisy-load reconstruction via polynomial fitting. At present, with the advancement of technology, employing neural networks for unknown load identification has become a widely advocated approach. Neural networks demonstrate significant advantages in dynamic load identification: physics-informed and hybrid model–data methods enhance robustness under ill-posedness and noise [17], integration with extended Kalman filtering improves accuracy [18], deep learning architectures (e.g., DRAN and DCNN) enable more efficient and noise-resistant identification [19,20], while NARX networks support nonlinear modeling and vibration control [21]. In addition, convolutional neural networks combined with data augmentation and adaptive optimization can achieve high-accuracy damage identification under limited sample conditions [22]; models that integrate CNN with temporal networks and introduce the SE attention mechanism can enhance the reconstruction accuracy of structural responses under environmental disturbances [23]; the spatiotemporal modeling method based on DCNN-LSTM can accurately predict temperature-induced displacements [24]; and related review studies provide theoretical support and development directions for the application of deep learning in signal reconstruction and damage identification [25]. Overall, these studies indicate that neural networks substantially improve the reliability and engineering applicability of dynamic load identification. Since the first type of identification method essentially inverts load parameters from the structural dynamic response data, it requires high accuracy in sensor measurements. It is highly susceptible to noise interference, often leading to ill-conditioned matrices, where low data precision aggravates the instability of the solution.

The second approach adopts a different perspective by emphasizing the overall characteristics of response histories rather than pointwise vibration details. A representative example is the influence line method, which estimates vehicle loads based on global structural response trends under moving actions. Wei Y. et al. [26] developed a low-pass filter method to separate static from dynamic components in bridge responses and identify static influence lines. Numerical verification confirms the method effectively removes dynamic effects. Moslehi T. A. et al. [27] introduced weakened modal shape functions based on Muller-Breslau’s principle, and this method reduces ill-conditioning in moving load identification. Case studies show the approach eliminates singularities and produces smooth, accurate load identification results. Zhu J. et al. [28] proposed a strain influence line method considering transverse load distribution to enhance load identification accuracy. Simulations and experiments demonstrate strong anti-noise capability, with vehicle weight errors mostly below ±10%. Yang J. et al. [29] proposed a vehicle load identification method integrating machine vision and displacement influence lines. Field monitoring validates its feasibility, with gross vehicle weight errors under 25% and advantages in cost-effectiveness and easy deployment. Motivated by overloading-induced bridge failures, Xu W. et al. [30] highlighted the need for real-time vehicle load monitoring. The proposed machine vision-based method serves as a cost-efficient, non-intrusive alternative to traditional pavement-based weigh-in-motion systems.

Despite significant progress, existing studies have predominantly concentrated on single-vehicle scenarios. In practice, however, bridge structures are frequently subjected to simultaneous multiple moving loads, which generate complex structural responses. Methods tailored to single-load cases thus exhibit clear limitations in realistic applications. To address this issue, this study proposes an improved multi-source dynamic load identification algorithm based on the elastic moving load influence line theory. A scaled experimental program involving multiple vehicles was conducted, where displacement responses at critical bridge locations were monitored using laser displacement sensors. The proposed approach is validated through these experiments, demonstrating its capability to accurately identify dynamic loads under multi-vehicle conditions, thereby enhancing the applicability of moving load identification methods in engineering practice.

2. Theory on the Segmental Area of the Influence Lines

When a single moving load is applied to the simply supported beam bridge, the elastic displacement influence line is as depicted in Figure 1. It should be noted that the proposed approach is independent of the type of bridge and boundary conditions due to the characteristics of the influence line method. Based on the fundamental principles of the elastic influence line, the displacement response under the action of the moving load F can be derived as Equation (1).

$$\mathrm{D}\left(\mathrm{x}\right)=\mathrm{F} · \mathsf{\delta }\left(\mathrm{x}\right)$$

(1)

In Equation (1), D(x) denotes the measured displacement response under a moving load, F is the load to be identified, and δ(x) represents the pre-calibrated elastic influence line function corresponding to a unit load. Owing to inevitable measurement errors and load-induced noise, load estimation based on instantaneous measurements may result in considerable inaccuracies. Nevertheless, the integral of the measured response over the entire loading cycle is only marginally affected by noise [31]. Consequently, by comparing the area enclosed by the measured response curve with that of the calibrated influence line, the moving load F can be inversely determined, as formulated in Equation (2), where L denotes the length of the simply supported bridge.

$$\mathrm{F}={\displaystyle \frac{\mathrm{D}\left(\mathrm{x}\right)}{\mathsf{\delta }\left(\mathrm{x}\right)}}={\displaystyle \frac{{\int }_0^{\mathrm{L}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx}}{{\int }_0^{\mathrm{L}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx}}}$$

(2)

When multiple moving loads are simultaneously applied to a beam bridge, it is assumed that all groups of moving loads travel at the same velocity, and that for medium- and large-span bridges, the dynamic effect is relatively small compared with the static load [2,11]. However, for short-span bridges, a dynamic correction model should be additionally considered. In this study, only medium- and large-span bridges are considered for analysis. The theoretical displacement influence line is shown in Figure 1. The displacement derivation formula under a unit moving load is as follows:

$$\mathsf{\delta }\left(\mathrm{x}\right) = \left\{\begin{array}{c}{\mathsf{\delta }}_1\left(\mathrm{x}\right)\\ {\mathsf{\delta }}_1\left(\mathrm{x}\right) + {\mathsf{\delta }}_2\left(\mathrm{x}\right)\cdots \\ {\mathsf{\delta }}_2\left(\mathrm{x}\right)\end{array}\right.\begin{array}{c}\mathrm{x} \in \left[0 ,\left. \mathrm{h}\right)\right.\\ \mathrm{x} \in \left[\mathrm{h} ,\left. \mathrm{L}\right)\right.\\ \mathrm{x} \in \left[\mathrm{L} ,\left. \mathrm{L} + \mathrm{h}\right]\right.\end{array}$$

(3)

where δ₁(x) and δ₂(x) correspond to the influence line functions of the first- and second-unit moving loads in the dual moving load system, respectively, h is the distance between two moving loads and L is the length of the bridge same as Figure 1. Similar to Equation (1), the displacement response under the actual moving loads F₁ and F₂ can be derived as shown in Equation (4).

$$\mathrm{D}\left(\mathrm{x}\right)=\left\{\begin{array}{c}{\mathrm{F}}_1 · \mathsf{\delta }\left(\mathrm{x}\right)\\ {\mathrm{F}}_1 · \mathsf{\delta }\left(\mathrm{x}\right)+{\mathrm{F}}_2 · \mathsf{\delta }(\mathrm{x}-\mathrm{h})\cdots \hspace{1em}\\ {\mathrm{F}}_2 · \mathsf{\delta }(\mathrm{x}-\mathrm{h})\end{array}\right.\begin{array}{c}\mathrm{x} \in \left[0 ,\left. \mathrm{h}\right)\right.\\ \mathrm{x} \in \left[\mathrm{h} ,\left. \mathrm{L}\right)\right.\\ \mathrm{x} \in \left[\mathrm{L} ,\left. \mathrm{L}+\mathrm{h}\right]\right.\end{array}$$

(4)

Analogous to Equation (2), by comparing the area of the measured response curve with that of the calibrated influence line under the moving load, the following relation is obtained:

$$\begin{array}{c}{\int }_0^{\mathrm{h}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx} = {\mathrm{F}}_1 · {\int }_0^{\mathrm{h}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx}\\ {\int }_{\mathrm{h}}^{\mathrm{L}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx} = {\mathrm{F}}_1 · {\int }_{\mathrm{h}}^{\mathrm{L}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx} + {\mathrm{F}}_2 · {\int }_{\mathrm{h}}^{\mathrm{L}}\mathsf{\delta }(\mathrm{x}-\mathrm{h})\mathrm{dx}\cdots \\ {\int }_{\mathrm{L}}^{\mathrm{L} + \mathrm{h}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx} = {\mathrm{F}}_2 · {\int }_{\mathrm{L}}^{\mathrm{L} + \mathrm{h}}\mathsf{\delta }(\mathrm{x}-\mathrm{h})\mathrm{dx}\end{array}\begin{array}{c}\mathrm{x} \in \left[0 ,\left. \mathrm{h}\right)\right.\\ \mathrm{x} \in \left[\mathrm{h} ,\left. \mathrm{L}\right)\right.\\ \mathrm{x} \in \left[\mathrm{L} ,\left. \mathrm{L} + \mathrm{h}\right]\right.\end{array}$$

(5)

Furthermore, let,

$$\left\{\begin{array}{c}{\int }_0^{\mathrm{h}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx} ={\mathrm{D}}_1\\ {\int }_{\mathrm{h}}^{\mathrm{L}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx} ={\mathrm{D}}_2\\ {\int }_{\mathrm{L}}^{\mathrm{L}+\mathrm{h}}\mathrm{D}\left(\mathrm{x}\right)\mathrm{dx} ={\mathrm{D}}_3\end{array}\right.\mathrm{and} \left\{\begin{array}{c}\begin{array}{c}{\int }_0^{\mathrm{h}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx} ={\mathsf{\delta }}_1\\ {\int }_{\mathrm{h}}^{\mathrm{L}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx} ={\mathsf{\delta }}_2\end{array}\\ \begin{array}{c}{\int }_{\mathrm{h}}^{\mathrm{L}}\mathsf{\delta }\left(\mathrm{x}-\mathrm{h}\right)\mathrm{dx} ={\int }_0^{\mathrm{L}-\mathrm{h}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx} ={\mathsf{\delta }}_3\\ {\int }_{\mathrm{L}}^{\mathrm{L}+\mathrm{h}}\mathsf{\delta }\left(\mathrm{x}-\mathrm{h}\right)\mathrm{dx} ={\int }_{\mathrm{L}-\mathrm{h}}^{\mathrm{L}}\mathsf{\delta }\left(\mathrm{x}\right)\mathrm{dx}= {\mathsf{\delta }}_4\end{array}\end{array}\right.$$

Accordingly, Equation (6) can be derived as below,

$$\left\{\begin{array}{c}{\mathrm{D}}_1={\mathrm{F}}_1 · {\mathsf{\delta }}_1\\ {\mathrm{D}}_2={\mathrm{F}}_1 · {\mathsf{\delta }}_2+{\mathrm{F}}_2 · {\mathsf{\delta }}_3\\ {\mathrm{D}}_3={\mathrm{F}}_2 · {\mathsf{\delta }}_4\end{array}\right.$$

(6)

where D₁, D₂, D₃ represent the integrals of the measured displacement responses D(x) over the corresponding intervals, and also, D₁, D₂, D₃ denote the work exerted on the bridge under moving load. While δ₁, δ₂, δ₃, δ₄ denote the integrals of the unit-load elastic displacement influence line function δ(x) over the respective intervals, analogous to the label of D_i, these symbols denote the work produced by the unit force. F₁, F₂ correspond to the two moving loads to be identified. It should be noted that both D_i and δ_j are dependent on the spacing h. Based on Equation (6), the parameters F₁, F₂, and h can be estimated using the nonlinear least-squares method [32].

According to the analysis above, when two or more moving loads pass the bridge at approximately constant relative distance, the proposed segmental area of influence line-based identification approach is, in principle, capable of recognizing multiple moving loads. This is the first trying to isolate the main load from multi loads passing the bridge. In this study, two moving-load scenarios were designed and tested to examine the accuracy of the proposed algorithm in load identification.

3. Experimental Setup

As illustrated in Figure 2a, the experimental model adopted in this study was a 1.2 m long steel simply supported beam. High-precision Japanese Panasonic HG-C1050 laser displacement sensors with a measurement accuracy of 30 μm (Figure 2b) were installed at the 1/4, 1/2, and 3/4 span sections, corresponding to the key control points of bending moment and strain responses. The sampling frequency of the displacement sensors was set to 100 Hz. A Smacq USB-3000 series data acquisition system from Simaco Hua Co., Beijing, China (Figure 2d) was employed to record the displacement responses of the beam as the test vehicle traversed it at a constant speed v. Two types of loading scenarios were designed to simulate practical conditions, namely a single moving load case and a two-load case (Figure 2c). In the single-load case, the cart carried a load of 50 N, while in the two-load case, the loads were 50 N and 19.4 N, respectively, with an actual spacing of 0.25 m between the two carts. Furthermore, to investigate the effect of velocity on identification accuracy, comparative tests were conducted under different speeds for both the single-load and two-load scenarios. The detailed experimental cases are summarized in

Table 1. Summary of test cases.

Case ID	Details		Remarks
	Speed (m/s)	Moving Load (N)
A1	0.046	50	A6 defined as a calibration case
A2	0.052
A3	0.067
A4	0.085
A5	0.096
A6	0.109
B1	0.049	50, 19.4
B2	0.056
B3	0.066
B4	0.072

.

In this experimental study, the displacement response data obtained under Case A6—corresponding to a 50 N moving load traveling at a speed of 0.109 m/s—were used to fit the calibration function δ(x). Based on Equations (2) and (5), the load identification theories for single-load and two-load cases were established, respectively. The moving load values were then derived using displacement data collected from the three monitoring points, with their arithmetic mean taken as the identified parameter. Finally, the identified results were compared with the actual loads to evaluate the error, thereby validating the accuracy of the proposed load identification theory in practical applications.

4. Data Analysis

4.1. Analysis of the Original Test Data with Single Load

As shown in Figure 3 and Figure 4, the displacement responses measured at the three monitoring points under a single moving load exhibit pronounced random fluctuations, with Case A6 taken as a representative example.

To reduce the susceptibility of the sensor signals to environmental noise, the Savitzky–Golay polynomial convolution filter [33] was employed. This denoising technique effectively suppresses stochastic disturbances while retaining the essential vibration features, resulting in smoother and more reliable response data.

To further establish a robust mathematical representation of the dynamic responses, Chebyshev polynomial approximation [34] was applied to reconstruct the denoised displacement signals:

$$\mathrm{R} = \sum _{\mathrm{i} =1}^{{\mathrm{N}}_{\mathrm{f}}}{\mathrm{T}}_{\mathrm{i}}\left(\mathrm{t}\right) {\mathrm{C}}_{\mathrm{i}}\left(\mathrm{x}\right) = {\mathrm{R}}_{\mathrm{i}}$$

(7)

where T_i(t) denotes the Chebyshev orthogonal polynomial of order i, N_f the number of expansion terms (smaller than the number of measured data points), C_i(x) the expansion coefficients, and R_i the reconstructed displacement response.

Reconstruction accuracy was quantitatively assessed by the relative percentage error (RPE, Equation (8)), enabling a systematic evaluation of the influence of polynomial order on the fitting precision (Figure 5), where y₀ is the measured displacement response and y_m is the reconstructed response.

$$\mathrm{RPE}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}} \sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{0 , \mathrm{i}}-{\mathrm{y}}_{\mathrm{m} , \mathrm{i}})}^2}$$

(8)

The results show that the reconstructed responses agree well with the measured signals, with errors decreasing and stabilizing beyond the 40th order. At this order, the fitting errors at the three monitoring points were 0.66%, 1.40%, and 1.18%, respectively. Accordingly, the 40th-order Chebyshev polynomial was adopted to balance accuracy and computational efficiency.

A comparison between measured and reconstructed responses under Case A6 is presented in Figure 6.

Since load velocity affects the time span of the recorded responses and, consequently, the integral terms in the identification equations, a time-domain normalization procedure was implemented to eliminate speed effects while preserving waveform geometry (Figure 7).

In this experimental study, Case A6 was adopted as the calibration case, while the other cases served as test cases for load identification. For a single vehicle load of 50 N moving at 0.109 m/s, the normalized time-domain displacement responses processed from the sensor measurements were used as baseline data. By comparing these with the processed responses of other cases and applying the theoretical Equations (2) and (5), the moving loads were effectively identified.

For the calibration Case A6, the integrated areas of the displacement influence lines over the interval [0, 1] were calculated as 0.1793, 0.3508, and 0.2429 at the three monitoring points. The same procedure was applied to the test cases, and the comparison with calibration data enabled accurate identification of single moving loads. As shown in

Table 2. Statistical analysis of individual load identification results across multiple cases.

Case	Monitoring Data Coverage Area			Ratio to Calibration Load			Average Ratio	Calculated Load (N)	Actual Load (N)	Error %	Uncertainty %
	1/4	1/2	3/4
A1	0.1805	0.3505	0.2490	1.01	1.00	1.03	1.01	50.52	50.0	1.04	0.5
A2	0.1752	0.3381	0.2509	0.98	0.96	1.03	0.99	49.56		−0.88	0.4
A3	0.2054	0.3566	0.2470	1.09	1.02	1.02	1.04	52.05		4.11	0.7
A4	0.1883	0.3409	0.2415	1.05	0.97	0.99	1.01	50.27		0.53	0.9
A5	0.1861	0.3498	0.2495	1.04	1.00	1.03	1.02	51.04		2.07	0.3
A6	0.1793	0.3508	0.2429	NA

, the identification errors were controlled within ±5%, confirming the reliability of the proposed method.

4.2. Analysis of the Original Test Data with Two Moving Loads

Figure 8 and Figure 9 present the measured displacement responses at different monitoring points under the excitation of two moving loads (Case B2). Similar to the single-load condition, the responses exhibit pronounced random fluctuations. However, due to the superposition of two moving loads, the displacement signals feature two distinct peaks, reflecting the coupled dynamic effect.

The data processing strategy is consistent with that adopted for the single-load case. Initially, the Savitzky–Golay polynomial convolution filter was employed to suppress noise while retaining essential vibration characteristics. Subsequently, the smoothed signals were approximated using Chebyshev polynomials, as formulated in Equation (7). Figure 10 compares the measured responses with the 40th-order Chebyshev approximation for Case B2, confirming the effectiveness of the fitting approach.

To eliminate discrepancies in time scales between the two-load cases and the calibration case, temporal normalization was applied. Specifically, the passage time of the leading vehicle was used as the reference, and the response curves were normalized to the interval [0, 1 + h₀], where h₀ denotes the normalized inter-vehicle spacing.

In line with the theoretical assumptions, both vehicles were considered to travel at constant velocity, maintaining a fixed spacing h. Thus, h₀ remained invariant. The spacing was determined from the time interval between adjacent peaks, in conjunction with the vehicle speed and bridge span length (L = 1.2 m), as expressed in Equation (9).

$${\mathrm{h}}_0={\displaystyle \frac{\mathrm{h}}{\mathrm{L}}}={\displaystyle \frac{∆\mathrm{t} \mathrm{v}}{\mathrm{L}}}$$

(9)

The calculated inter-vehicle spacings for different cases are summarized in

Table 3. Inter-vehicle distance statistics across multiple cases.

Case	Crest Timing Difference $∆\mathbf{t}$ (s)	Speed (m/s)	Calculate Distance h (m)	Unit Distance h0	Actual Distance (m)	Error %
B1	5.29	0.049	0.259	0.2161	0.25	3.6
B2	4.45	0.056	0.249	0.2071		−0.4
B3	3.95	0.066	0.261	0.2171		4.4
B4	3.56	0.072	0.256	0.2135		2.5

, with relative errors controlled within ±5%.

Using the identified h₀, the response data for all two-load cases were normalized to the time domain [0, 1 + h₀]. Figure 11 illustrates the normalized displacement response for Case B2.

Based on the unit vehicle spacing h₀ obtained from Table 3, the normalized displacement response integrals D₁, D₂, and D₃ were calculated for each test case (B1–B4). Correspondingly, using the h₀ of each case, the normalized displacement response integrals δ₁, δ₂, δ₃, and δ₄ for the reference case (A6) were computed. The statistical results of these calculations are summarized in

Table 4. Interval areas of fitting functions calculated based on h₀ under multiple cases.

Case	Measuring Point Location	Calculated Values
		δ1	δ2	δ3	δ4	D1	D2	D3
B1	1/4	0.0500	0.1343	0.1719	0.0128	0.0529	0.2057	0.0635
	1/2	0.0311	0.3332	0.3213	0.0429	0.0325	0.4582	0.0155
	3/4	0.0159	0.2244	0.1761	0.0635	0.0171	0.3045	0.0237
B2	1/4	0.0444	0.1399	0.1725	0.0122	0.049	0.2171	0.0605
	1/2	0.0292	0.3351	0.3239	0.0403	0.0278	0.4318	0.0152
	3/4	0.0148	0.2253	0.1828	0.0569	0.0155	0.3026	0.0211
B3	1/4	0.0506	0.1336	0.1718	0.0122	0.0549	0.2061	0.062
	1/2	0.0292	0.3351	0.3239	0.0403	0.0329	0.4629	0.0162
	3/4	0.0160	0.2242	0.1752	0.0643	0.0152	0.2776	0.024
B4	1/4	0.0487	0.1356	0.1720	0.0127	0.0468	0.2042	0.0708
	1/2	0.0307	0.3336	0.3219	0.0423	0.0301	0.4615	0.0167
	3/4	0.0157	0.2246	0.1776	0.0619	0.0153	0.2966	0.0261

.

Based on the time-history integrals D₁, D₂, D₃ and δ₁, δ₂, δ₃, δ₄ obtained from Table 4, and in conjunction with the load identification Formulations (5) and (6), the ratios of the two target moving loads to the reference load were computed at each monitoring point. The mean of these ratios was then multiplied by the known reference load (50 N) to yield the estimated values of the target loads. Finally, by comparing the calculated loads with the actual values (50 N and 19.4 N), the identification errors were assessed. The load identification results for each case are presented in

Table 5. Load identification statistics under multiple cases.

Case	Moving Load	Ratio to Calibration Load			Average Ratio	Calculated Load (N)	Actual Load (N)	Error %	Uncertainty %
		1/4	1/2	3/4
B1	F1	1.06	1.04	1.08	1.06	52.97	50.0	5.95	0.4
	F2	0.37	0.34	0.36	0.36	17.86	19.4	−7.96	0.5
B2	F1	1.10	0.95	1.05	1.03	51.67	50.0	3.34	0.5
	F2	0.36	0.35	0.37	0.36	17.99	19.4	−7.29	0.4
B3	F1	1.09	1.05	0.95	1.03	51.44	50.0	2.88	0.6
	F2	0.36	0.35	0.37	0.36	17.94	19.4	−7.55	0.7
B4	F1	0.96	0.98	0.97	0.97	48.59	50.0	−2.82	0.5
	F2	0.41	0.41	0.42	0.42	20.90	19.4	7.73	0.7

Note: F₁ represents the main load, and F₂ represents the secondary load.

.

As summarized in Table 5, the relative error of the identified loads F₁ and F₂ ranged from 2.82% to 5.95% and 7.29% to 7.96%, respectively. The larger load F₁ was identified with consistently higher accuracy than the smaller load F₂, indicating the robustness of the proposed approach for dominant loads while also revealing its sensitivity to load magnitude in multi-load scenarios. The lower identification accuracy of F₂ may be attributed to the sequential identification process, in which F₁ is identified first and F₂ is subsequently determined based on it. As a result, the identification of F₂ is likely influenced by the error in F₁.

5. Discussion

5.1. Comparison Between Theoretical and Actual Displacement Response

Based on the previously identified moving load values and using the displacement response data from the calibration case (A6) as a reference, the displacement responses for each case were theoretically identified using the moving load identification Formulas (2) and (5). Figure 12 and Figure 13 present the comparison between the theoretical identified displacement response data and the measured results for the A3 (single moving load) and B4 (two moving loads) cases at each monitoring point. The comparison reveals that: For the single moving load case (A3), the theoretical identified displacement responses closely match the measured data, indicating high identification accuracy for this load condition. For the two moving load case (B4), the theoretical identified displacement responses are in good agreement with the measured data, suggesting that the method also achieves acceptable identification accuracy under this dual load condition.

5.2. Relationship Between Identification Results and Load Speed

As shown in Figure 14, an analysis of the correlation between load identification errors and load speed for each case reveals that there is no significant relationship between the identification error and the velocity parameter. This result validates that the influence line-based load identification model exhibits good independence from the velocity parameter.

6. Conclusions

This study presents an improved multi-source dynamic load identification algorithm based on the segmental area of the influence line theory, with its accuracy validated through experimental design. Since Equation (6) is based on the superposition of influence line, the proposed method can applied on the small deformation conditions, which is most common in the real world. Based on the data analysis, the following conclusions can be drawn:

(1) For the single moving load identification case, the identification error is controlled within 5%. For the dual moving load identification case, the identification errors for F₁ and F₂ are controlled within 5% and 10%, respectively. This method can also be extended to cases involving more than two moving loads.

(2) For the dual moving load case, the identification error for vehicle spacing is well-controlled within 5%.

(3) The influence line-based load identification model exhibits velocity insensitivity, achieving good load identification performance under different speed conditions.

Future research may consider integrating deep learning techniques with traditional influence line theory to achieve further advancements in bridge load identification. The present study proposes a conceptual framework for multi loads identification based on elastic influence lines; however, its performance and applicability are still constrained by the assumptions of linear elasticity and the accuracy of influence line segmentation. Additional studies are required to address the identification of irregularly spaced multiple moving loads and to verify the feasibility of the proposed method in bridges of different scales. In recent years, deep learning has demonstrated remarkable capabilities in feature extraction, nonlinear mapping, and time-series prediction, and has been widely applied to missing data recovery, response reconstruction, and damage identification in structural health monitoring. Therefore, integrating deep learning with influence line theory is expected to enable intelligent identification of complex, nonlinear, and uncertain load patterns. Future work will focus on developing hybrid models that couple mechanical mechanisms with neural networks to enhance the robustness, generalization, and real-time performance of bridge load identification systems.