Early studies primarily relied on finite element analysis (FEA) and nonlinear simulations to analyze stress redistribution under seismic or extreme loading scenarios [8–11]. While these methods revealed structural vulnerabilities and mutual deck influence, they are typically limited to short-term simulations and idealized boundary conditions. Moreover, bidirectional loading often induces out-of-phase motion between adjacent girders. For example, Kumar and Nallasivam (2025) [3] employed a finite element (FE) model to show that variations in train speed and damping ratio significantly affect total deformation and equivalent stress amplitude, underscoring the role of stress redistribution and load-path adjustment under dynamic loading.

Current research methodologies often rely on assumptions and simulated experiments that may not fully capture the unique stress dynamics of multilayer systems during actual operations. Existing studies lack continuous long-term tracking of the stress evolution process and fail to focus on locations experiencing significant stress variations. Additionally, the relationship between various influencing factors and stress redistribution in bridge structures remains insufficiently captured, highlighting the need for further research in this area.

Despite extensive FE modeling and the growing adoption of ML approaches, significant knowledge gaps persist. Firstly, most bridge simulations are not supported by in-situ measurements: numerical models are rarely calibrated using long-term monitoring data, leaving their stress predictions largely unverified in practical applications [30].

Second, traditional analyses often treat loading effects as linear and independent; however, nonlinear interactions (e.g., between thermal stress and traffic-induced stress) have not been thoroughly explored by data-driven methods [31].

Third, the response behavior of critical components in double-deck bridges—including main girders, cross-frames, and deck plates—has not been systematically characterized under combined loading. Importantly, no previous study has applied XGBoost or similar models to predict stress redistribution in double-layer bridges, revealing a clear methodological gap. To our knowledge, ensemble machine learning approaches have not yet been used to capture time-series stress patterns in such structures.

Therefore, this study contributes by (i) integrating long-term multi-source monitoring data, (ii) applying XGBoost with SHAP interpretability to identify key drivers of stress redistribution, and (iii) validating ML predictions against calibrated FE models to establish a hybrid data–mechanics framework to quantify stress redistribution.

The Fanhua Avenue Bridge of the Yangtze-Huaihe Water Diversion Project is located in Feixi County, Hefei City, Anhui Province. It is the first dual-purpose road-rail double-deck steel truss arch bridge in Anhui Province. The bridge starts at Zhougongshan Road (Guosan Road), crosses Jianghuai Avenue (Paihe Avenue), the Jianghuai Canal, and the Fangxing Avenue viaduct, and ends at Xiaolongshan Road. As a key municipal bridge in the Jianghuai Communication section of the Yangtze-Huaihe Water Diversion Project, it serves as a critical crossing for Feixi County to access Hefei’s main urban area.

The main bridge comprises a 153 m span double-deck steel truss arch, accommodating both urban roadway and rail transit functions. The upper deck, spanning 47.5 m in width, carries a six-lane urban roadway in both directions as well as a metro line. The lower deck, 56 m wide, accommodates a four-lane roadway in both directions in addition to pedestrian and bicycle facilities.

The main truss employs a flexible steel arch design with an N-type triangular truss configuration incorporating vertical posts. The standard segment length is 9.0 m, with 16 segments in the main span. The truss height measures 11.9 m, and the diagonal bracing has an inclination angle of 53°. The main arch ribs consist of steel box sections with a rise-to-span ratio of 1:4.49. Additionally, five wind bracing systems are incorporated, and the arch axis follows the equation: (1)y=4f(L−x)x2Lwhere L = 128.0 m, f = 28.5 m.

The overall layout of the main bridge and the traffic configuration are illustrated in Fig. 1.

The total weight of the bridge is 11,894 t, distributed as follows: 2630 t for the edge members, 2831 t for the intermediate members, 2472 t for the arch ribs and wind bracing, 3919 t for the deck system, 27 t for the maintenance ladders on the arch ribs, and 15 t for the decorative lighting system. The main truss components, arch rib elements, and transverse beams in the deck system are all constructed using Q345qD-grade steel. The intermediate transverse beams and upper longitudinal connections also employ Q345qD-grade steel.

To ensure the long-term structural health and operational safety of the bridge, an integrated monitoring system has been installed to provide comprehensive monitoring of the entire structure. This system utilizes five types of vibrating wire sensors, strategically placed at key structural locations, covering all critical aspects of the bridge’s operation. The types of sensors and their detailed specifications are provided in Table 1.

Wireless Vibrating Wire Data Logger (6-channel, 6 units), Dynamic Data Acquisition System (1 unit), Communication Gateway (1 unit), totaling 8 units, arranged near the sensors. The monitoring system comprises 48 physical sensors, organized into groups in which each tilt or strain gauge is paired with a corresponding temperature sensor.

Total Station for supplementary measurement of arch ring spatial deformation; Portable Accelerometers for supplementary cable force measurement, primarily relying on on-site measurements by the construction monitoring team.

This monitoring system has continuously collected a variety of structural and environmental parameters since 2022, providing essential data support for in-depth analysis of the bridge’s performance. It helps in identifying key patterns and trends, which are crucial for evaluating the bridge’s health and predicting potential issues.

The bridge is a combined road and rail structure. On 1 May 2024, the southern extension of Hefei Metro Line 4 officially opened. Following the commencement of metro operations, the bridge experienced a significant increase in both static and dynamic loads, leading to notable changes in the load distribution and stress state of the structure. This has resulted in stress redistribution. Such changes introduce new challenges for evaluating the structural performance of the bridge under operational conditions, particularly in long-term safety assessments and adaptability analyses.

To explore these changes, this study selects monitoring data from two distinct periods: before the metro operation in 2023 (without train load) and after the operation in 2024 (with train load). By comparing and analyzing the variations in typical structural responses and stress redistribution characteristics, the study aims to reveal the bridge’s structural adaptability and future tendency under changing load conditions.

XGBoost is used for supervised learning problems, where training data with multiple features is utilized to predict the target variable. The XGBoost model is an ensemble of decision trees, specifically composed of Classification and Regression Trees (CART). In this model, a target value is divided into different leaves, and each leaf is assigned a corresponding score.

CART differs slightly from traditional decision trees, where leaves contain only decision values. In CART, each leaf is associated with a real-valued score, which enhances the model’s capacity to capture weak relationships beyond classification and improves model interpretability. Mathematically, the model can be expressed as: (2)y^i=∑k=1Kfk(xi),fk∈Fwhere K represents the number of trees, and f_k is a function from the function space F defined as: ft(x)=ωq(x),ω∈RT, ω is the score vector assigned to the leaves, T denotes the number of leaves, and q is a function that maps each data point to the corresponding leaf. The function space F consists of all possible CART models. The loss function is defined as follows: (3)objt=∑i=1n[l(yi,y^it−1)+gift(xi)+12hift2(xi)]+ω(ft)+o(x3)where ω (f_k) represents the complexity of the tree, defined as: ω(f)=γT+1/2λ∑j=1Tωj2. Here, T is the number of leaves, γ is the regularization parameter controlling the complexity penalty per leaf, and λ is a parameter that penalizes the squared magnitude of leaf scores ω_j.

The second-order Taylor expansion of the loss function involves its first-and second-order derivatives, denoted by g_i and h_i, respectively. The formulas are: (4)gi=∂y^it−1l(yi,y^it−1) (5)hi=∂y^it−12l(yi,y^it−1)

These terms capture the first-order gradient and second-order curvature (Hessian), which are used to approximate the loss function and optimize the tree structure efficiently.

The final objective function can be simplified into the following form: (6)obj(θ)=∑j=1T[Gjωj+12(Hj+λ)ωj2]+γT

The XGBoost model was trained using the preprocessed feature matrix X ∈ R(n × m), where n denotes the number of samples and m represents the feature dimension, along with the corresponding stress response vector y ∈ Rⁿ. A staged training strategy was adopted, consisting of the following key steps: (1)

Initialization of the Base Learner: The training process begins by constructing an initial decision tree using a greedy algorithm, which serves as the starting point for the boosting iterations.

To further enhance computational efficiency, two key optimization techniques were employed:

Parallel Feature Splitting: Parallel computing was leveraged to accelerate the search for optimal feature split points, significantly reducing training time.

Histogram-Based Approximation: A histogram-based method was employed to approximate feature distributions, efficiently reducing memory requirements while preserving model accuracy.

The final model converged after 302 iterations, achieving a coefficient of determination (R²) of 0.65 on the validation set, demonstrating high predictive accuracy and strong generalization performance.

The experimental environment for this study is Python 3.12 and XGBoost 3.0.4, with a CPU runtime.

To ensure the reliability and consistency of the monitoring data, a multi-step preprocessing workflow was implemented following established procedures reported in previous studies.

First, sensor anomalies were identified using statistical outlier detection and cross-sensor consistency checks, allowing for the removal of spurious spikes caused by transient disturbances [32,33]. Sensor drift was mitigated through periodic baseline recalibration, and hardware faults were addressed by replacing malfunctioning units. Data segments affected by such faults, as identified through diagnostic logs, were excluded from subsequent analyses.

For the raw data obtained from the sensors, secondly, we applied spline interpolation to fill in any missing data points, primarily caused by communication interruptions or temporary sensor downtime. Spline interpolation is a smooth method for imputation that preserves the underlying trends of the data, ensuring continuity and consistency in the dataset.

To enhance data reliability and consistency, imputed values were standardized using z-scores, enabling the identification and removal of potential outliers. Based on the 3σ criterion, Z outliers (0.33% of the dataset) were identified and removed before model training. This normalization process mitigates the influence of extreme values on the subsequent analysis, ensuring that the data is better suited for modeling.

In cases where certain sensors exhibited substantial deviations in their readings, we applied the Yeo-Johnson transformation. This transformation method is effective in addressing issues such as skewness and kurtosis, as it transforms non-normal distributions into a form that approximates a normal distribution. Fig. 2 illustrates the data distributions at three stages: before preprocessing, after outlier removal and interpolation, and after Yeo–Johnson transformation. The transformation effectively reduced skewness and improved the normality of the data.

By implementing these data preprocessing techniques, we were able to significantly improve the quality of the dataset, ensuring that the input data for the subsequent modeling and prediction steps is both accurate and reliable.

The double-deck arch bridge commenced its operation in October 2022, initially bearing only vehicular loads. In May 2024, after the opening of the second-deck rail transit, the upper deck of the bridge began directly bearing the train loads.

Based on the principles of bridge structure and elastoplastic mechanics, it is hypothesized that, when the subway trains pass, due to the concentrated load effect, the upper structure (such as the bridge deck, main beams, or upper arch ribs) primarily bears significant vertical loads and horizontal moments, with the mid-span area being particularly noticeable, exhibiting large dynamic strain amplitudes. Furthermore, the rail deck experiences substantial vertical deformation and tensile–compressive strain in response to the dynamic loading induced by train traffic. Although the lower structure indirectly bears the train loads through the overall bridge, the stresses and strains in both the longitudinal and transverse directions of the bridge are still affected. Notably, at critical load-bearing locations, including the main arches and bearings, stress concentrations arise as the upper-deck load is transferred to the lower structure, leading to localized increases in strain. The regions at the top of the pier and below the arch ribs also experience vertical compressive strains due to the transfer of train loads, significantly enhancing the local strain characteristics.

To extract the strain monitoring data from the same period in 2023 and 2024 (April to June) for the daily 00:00 measurements, the data are plotted on the same graph. Due to their size, comprehensive visualizations of the sensor measurements are included in Appendix A, as shown in Fig. A1.

Further statistical comparison was conducted on the extracted monitoring data from the same time periods, and boxplots were generated for each strain sensor. Considering the distinct load-bearing characteristics of the bridge’s two levels, the strain data were categorized by the first and second decks for separate visualization to investigate the load response characteristics of different structural decks. The result is shown in Fig. 3.

The results indicate that the strain measurements on the first deck showed a slight overall decrease after the rail transit commenced operation, though the number of extreme values increased. Conversely, strain measurements on the second deck revealed a significant impact of the train live load, particularly evident in the longitudinal direction at the 3rd and 4th girders (sensors ZL and B, counting from upstream), where strain values exhibited a sharp increase.

The dataset was primarily constructed using data from 26 strain gauges placed at the mid-span of the bridge. The sensor placement and naming conventions are as described in Table 1 from the previous section.

Key variables, including force direction, beam location, and other relevant parameters, were extracted from the sensor data together with their corresponding stress measurements to construct the dataset. The structure of the dataset and the definitions of the variations are illustrated in Fig. 4a and Table 2. In this dataset, Deck, beam position, stress direction, and train operation are categorical variables, while the remaining variables are continuous. Specifically, two time-related variables need to be declared. Time difference represents the cumulative number of days since the commencement of metro operation. This variable captures long-term structural adaptation and gradual stress redistribution patterns that evolve with operational aging and environmental cycles.

Variable Hour of the day accounts for diurnal variations in temperature, traffic flow, and other environmental factors that can induce cyclical stress fluctuations in bridge components. To enable processing by the XGBoost model, categorical variables were first converted into numerical values through appropriate encoding techniques. This preprocessing step ensures that the model can efficiently handle both categorical and continuous data, making it suitable for subsequent analysis and prediction.

Model Parameter Selection

In this study, the XGBoost regression model was selected as the core predictive algorithm [34]. The hyperparameters related to tree complexity, such as the number of estimators and maximum tree depth, were systematically tuned following a standard protocol outlined in prior work [35]. The model parameters in this study were fine-tuned through grid search and cross-validation, with the optimal configuration determined as follows in Fig. 4:

n_estimators: 1000 (Number of boosting trees, with the optimal number of iterations confirmed using early stopping)

max_depth: 10 (Maximum depth of individual trees, balancing model complexity and the risk of overfitting)

learning_rate: 0.1 (Learning rate, controlling the step size in gradient boosting)

random_state: 42 (Random seed, ensuring the reproducibility of the experiments)

This parameter combination demonstrated the best generalization performance on the validation set. The joint tuning of the learning rate and tree depth effectively mitigated overfitting in the high-dimensional feature space, improving the model’s robustness and predictive accuracy.

The model, based on the regularized gradient boosting framework [36], effectively captures the nonlinear relationships between stress responses and multidimensional features. The comparative results are summarized in Table 3. XGBoost achieved the most balanced and robust performance across all subsets, with coefficients of determination. By contrast, the RF model achieved a relatively high R² on the training set (0.71) but exhibited substantial performance deterioration on the validation and test sets, with markedly elevated RMSE (Root Mean Square Error) values, indicating pronounced overfitting. SVM (Support Vector Machine) performed poorly, with negative R² values across all subsets, implying that it failed to learn any meaningful relationships and performed worse than a naive mean predictor. Similarly, ARIMA and MLR (Multiple Linear Regression) yielded R² values close to 0.1 and RMSE values exceeding 22,000, confirming that purely linear models are insufficient for representing the complex structural dynamics considered here.

Finally, to quantify this pattern during bridge operation, a new index is introduced. Combining previous conclusions about spatial changes during the redistribution period and FE analysis yields a transverse load distribution of mid:edge = 2:1 under representative operating conditions. We use this as a reference for field data. (7)SRI(t)=2σmid(t)/σedge(t)−1where:

σmid(t)=median{σB2(t),σB3(t)} is the median stress of the middle beams (B2 and B3) at time t.

σedge(t)=median{σB1(t),σB4(t)} is the median stress of the edge beams (B1 and B4) at time t.

SRI quantifies the deviation of the measured mid-to-edge stress ratio from the FE-derived baseline of 2:1. Its overall trend can be summarized as: (8)SRI={≈0,stress distribution is consistent with the FE>0,stress shifts toward the middle beams<0,stress returns toward the edge beams

Engineering analysis indicates that, within the range of stress amplitudes observed in our monitoring, the sensitivity of the SRI to random noise is approximately on the order of 10⁻³. Variations smaller than |{SRI}| < 0.003 are considered indistinguishable from measurement noise. This index offers a quantitative metric for stress redistribution between the midspan and edge beams, enabling the assessment of the bridge’s load transfer characteristics.