Research on structural vertical stiffness evaluation of track-specific suspension bridge under the action of the no-load first train

Abstract

The track-specific bridge has the characteristics of a large volume of passenger traffic, narrow transverse width, and large excitation. The long-term and high-frequency operating conditions put forward strict requirements for the safety and comfort of the bridge. Aiming at the problem of achieving fast, accurate, and intelligent evaluation of bridges under the coupling action of trains and temperatures, this paper proposes a method for evaluating the structural vertical stiffness of track-specific suspension bridges under the action of the no-load first train. Firstly, the $f-v-t$ (deformation-velocity-time) data of trains on the bridge are analyzed through linkage numerical simulation, track train transponder, and structural safety monitoring system. Secondly, the CART algorithm establishes a regression model for data correction. The mapping relationship between the modified deformation monitoring value and the theoretical calculation value is analyzed to realize the current structural vertical stiffness evaluation. Finally, it is verified by the world’s largest span self-anchored track-specific suspension bridge. The results show that the theoretical deformation value of the most unfavorable deformation section is 268.55 mm, the actual monitoring value is 232.80 mm, the difference is 35.75 mm, and the relative error is 13.3%. The R² score of the CART regression model is 0.94, and the accuracy of the CART regression model is higher than that of other algorithm regression models. The modified dynamic check coefficient is less than 1, and the ratio of the modified residual deformation value to the maximum deformation value is 7.6%, which meets the specification requirement not exceeding 20%, indicating that the current structural vertical stiffness is in normal operating conditions.

Introduction

With the rapid development of the social economy and the advancement of urbanization, track-specific bridges have developed rapidly, and suspension bridges have been widely used because of their beautiful appearance and lightweight structure. ¹ As a key infrastructure connecting various areas of the city, the structural rigidity of the track bridge directly affects the safety and comfort of train operations. At the same time, the high-frequency and long-term operation of urban rail transit put forward higher requirements for bridge structural vertical stiffness. The existing structural health monitoring system obtains the current health status information of the structure by monitoring load, environment, and structural response. It provides a basis and guidance for maintenance, repair, and management decisions. ² However, due to the characteristics of the track-specific bridges such as large volume of passenger traffic, narrow transverse width, and large excitation, coupled with time-varying coupling effects such as environmental factors and load action, bridges’ overall and local responses are extremely complex.^3,4 This complexity makes the mapping relationship between the actual monitoring value and the theoretical calculation value unclear, and it is difficult to achieve fast, accurate, intelligent evaluation and timely safety warnings based on the safety monitoring system alone.

In recent years, scholars at home and abroad have made useful explorations in the field of bridge structure state assessment. Xiao-gang L et al.^5,6 believes that the structural vertical stiffness of a bridge directly determines its dynamic characteristics, which in turn has an important impact on the traffic safety and running stability of trains crossing the bridge. The permanent deformation of bridge main beams is an important manifestation of structural vertical stiffness. ⁷ To evaluate the vertical stiffness of a bridge and ensure its driving performance, Zheng X-L et al. ⁸ proposed a rapid evaluation method for the vertical stiffness of a long-span bridge based on the relationship between the amplitude of the chord measurement method, the acceleration of the vehicle body, and the coupled vehicle-bridge vibration analysis. Zhang W-M et al. ⁹ proposed a fast and accurate deformation evaluation method for the main beam of a suspension bridge. Based on considering the stiffness of the main beam, an analytical expression for the deformation of the main beam of a suspension bridge under concentrated live load was derived by analogy with the bending moment of a simply supported beam and the shape of the main cable. Track bridges are characterized by clear moving loads. When designing the monitoring system, Hong G et al. ¹⁰ proposed a rapid load test based on no-load train formation in the morning shift to obtain the verification coefficient as an index to evaluate the structural state of the bridge. Zheng-tong G et al. ¹¹ tested methods such as trapezoidal integration in the time domain, Simpson integration in the time domain, and integration in the frequency domain. Through simulation and comparative study, it is found that the time-domain Simpson integral is easy to calculate, and the accuracy meets the requirements. Xiao-hong X et al. ¹² proposed a method based on the Newton–Leibniz formula and Simpson formula, combined with the integral recurrence formula of odd and even sequences, obtained vibration acceleration through acceleration sensors, and then obtained vibration velocity and vibration displacement through integration, which has extremely high calculation accuracy. Machine learning algorithms perform well in detecting structural damage and defects in bridges. Many structural state detection methods based on machine learning algorithms have been proposed and applied in practice.^13,14 The CART algorithm has the advantages of high classification accuracy, simple operation, and fast running speed. Through the classification and regression analysis of the damage data, the damage degree of the bridge can be accurately identified. Chang-sheng X et al. ¹⁵ extracted structural modal features by the additional mass method and used the CART algorithm for damage location and classification, which verified the accuracy and robustness of the method.

In summary, the present evaluation method is complicated, difficult, and costly, and requires many monitoring items and a wide range of data, so it is not suitable for the rapid, accurate, and intelligent evaluation of the track-specific bridge. Given this, this paper proposes a method for evaluating the structural vertical stiffness of a track-specific suspension bridge based on the action of the no-load first train. Based on Madis/Civil 2023 finite element software, the theoretical calculation value of the most unfavorable section deformation under the load of the no-load first train is simulated. The mathematical method is used to calculate the spatial relationship between the transponder and the safety monitoring system. The accurate deformation-velocity-time data of the train on the bridge is obtained, and then the regression model is established to obtain the correction coefficient. Finally, the structural vertical stiffness is evaluated by analyzing the relationship between the deformation monitoring value and the theoretical calculation value. Based on the world’s largest span self-anchored track-specific suspension bridge, this study will verify the effectiveness and practicability of this method. Then, it will realize the rapid, efficient, and accurate evaluation of the bridge and improve reliable scientific support for the healthy management of the bridge.

Working principle

Method design

To achieve rapid, accurate, and intelligent evaluation of the structural vertical stiffness of the track-specific suspension bridge, numerical simulations were performed using the no-load first train as the object to determine the most unfavorable cross section. The train track is obtained by the transponder, the functional relationship between velocity and time is established, and the spatial relationship between the transponder and the monitoring point of the safety system is calculated by numerical integration. At the same time, the regression model of the deformation monitoring value of the most unfavorable section and the train track on the bridge is established. The monitoring data is modified, and the mapping relationship between the deformation monitoring value and the theoretical calculation value is analyzed, to realize the vertical stiffness evaluation of the current state of the track-specific suspension bridge. Due to the fast velocity of rail transit trains and the short bridge crossing time, the immediate impact of temperature and wind on bridge deformation in a short time is small. ¹⁶ Therefore, the short-term impact of temperature and wind is not considered, thus simplifying the model and calculation. Figure 1 is the technical route of the evaluation method, and Figure 2 is the schematic diagrams of different sections of the train passing through the bridge.

Figure 1.

Technical route diagram for the stiffness evaluation method.

Figure 2.

The schematic diagrams of different sections of the train passing through the bridge.

Transponder and monitoring point layout

Transponder setup

To prevent signal interference and ensure safe and efficient train operation, the standard of transponder layout in specification ¹⁷ should be followed. The spacing of the transponders in the group is ${\hspace{0.17em}5\hspace{0.17em}}_0^{+0.5}$ m (the spacing of the shunting transponder group is not less than 3m), the number of transponders in the group is not more than 3, the positioning transponder can be single, and the group spacing is not less than 200m (Figure 3).

Figure 3.

Transponder group diagram.

Safety monitoring system measuring point layout

The safety monitoring system can identify the status of the bridge through environmental and load monitoring and structural response monitoring.^18,19 In the monitoring process, the high frequency and long duration of vehicle operation, and complex environmental factors are important sources of excitation. Given the characteristics of the track-specific bridge, such as large volume of passenger traffic, narrow transverse width, large excitation, and high frequency, the monitoring content should comprehensively acquire the on-board load, environmental information, and dynamic parameters. ²⁰ According to the code, ²¹ relevant provisions are made for monitoring items and measuring points of deformation measurement of track bridges. The longitudinal deformation of the beam should be monitored at the midpoint, quarter points, and support points at the beam ends in each span, and the measurement points should be fixed on the beam body.

Analytical principle of spatial relationship between transponder and monitoring point

Establishment of spatial relationship

After defining the installation position of the transponder $Y$ and the deformation measurement point of the most unfavorable section $K_{\mathit{max}}$ of the main beam, the following spatial relationships are established according to the track line mileage direction: The spatial relationship $S_0$ between the position of transponder ${\hspace{0.17em}Y}_0$ closest to the start $K_0$ of the bridge, the spatial relationship $S_{\mathit{max}}$ between the position of transponder $Y_{\mathit{max}}$ closest to the most unfavorable section $K_{\mathit{max}}$ of the main beam, and the spatial relationship $S_z$ between the position of transponder $Y_z$ closest to the end $Kz$ of the bridge. The specific diagram is shown in Figure 4.

Figure 4.

Schematic diagram of spatial relationships.

Spatial relation analysis

The train track (velocity $v$ and time $T$ ) is obtained by the onboard transponder of the first empty train, and the corresponding $T$ - $v$ relationship is established. At the same time, the times $T_0$ , $T_{\mathit{max}}$ , and $T_z$ of the train head passing through the transponder $Y_0$ position, the train center passing through the transponder $Y_{\mathit{max}}$ position, and the train tail passing through the transponder $Y_z$ position are obtained by the ground transponder. Combining the spatial relations $S_0{,S}_{\mathit{max}},S_z$ , and the times $T_0,T_{\mathit{max}},T_z$ of the train passing through the transponder, the Simpson integral method is used to analyze the times $t_0,$ $t_{\mathit{max}}$ , and $t_z$ of the train head passing through $K_0$ , the training center passing through $K_{\mathit{max}}$ , and the train tail passing through $K_z$ .

Simpson's integral method is to divide the integral interval $\left[a,b\right]$ into $n$ equal-width cells, each cell width is $h=\frac{b-a}{n}$ . The interval division points are: $x_0=a,x_1=a+h,x_2=a+2h$ ,…, $x_n=b$ . The function is approximated by a quadratic polynomial over every two cells. On each intercell, the quadratic polynomial is integrated, and the integral values of all intercells are added to obtain Simpson's integral formula

${\int }_a^{b}f\left(x\right)dx\approx \frac{h}{3}\left[f\left(x_0\right)+4{\displaystyle \sum }_{i=1,3,5...}^{n-1}f\left(x_i\right)+2{\displaystyle \sum }_{i=2,4,6...}^{n-2}f\left(x_i\right)+f\left(x_n\right)\right]$

(1)

where $x_0=a,x_n=b,h=\frac{b-a}{n}$ .

To solve the spatial relation problem, a cumulative displacement function $D\left(t,T\right)$ is first defined, and the cumulative displacement from $t$ to $T$ is calculated by equation (1).

Cumulative displacement from $t_0$ to $T_0$

$\mathrm{D}\left(t_0,T_0\right)={\int }_{t_0}^{T_0}v\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\approx \frac{h}{3}\left[v\left({\mathrm{t}}_0\right)+4{\displaystyle \sum }_{\mathrm{i}=1,3,5...}^{n-1}\mathrm{v}\left({\mathrm{t}}_i\right)+2{\displaystyle \sum }_{\mathrm{i}=2,4,6...}^{n-2}\mathrm{v}\left({\mathrm{t}}_i\right)+\mathrm{v}\left({\mathrm{T}}_0\right)\right]$

(2)

Cumulative displacement from $t_{\mathit{max}}$ to $T_{\mathit{max}}$

$D\left(t_{\mathit{max}},T_{\mathit{max}}\right)={\int }_{t_{\mathit{max}}}^{T_{\mathit{max}}}v\left(t\right)dt\approx \frac{h}{3}\left[v\left(t_{\mathit{max}}\right)+4{\displaystyle \sum }_{i=1,3,5...}^{n-1}v\left(t_i\right)+2{\displaystyle \sum }_{i=2,4,6...}^{n-2}v\left(t_i\right)+v\left(T_{\mathit{max}}\right)\right]$

(3)

Cumulative displacement from $t_z$ to $T_z$

$D\left(t_{\mathrm{z}},T_{\mathrm{z}}\right)={\int }_{t_z}^{T_z}v\left(t\right)dt\approx \frac{h}{3}\left[v\left(t_z\right)+4{\displaystyle \sum }_{i=1,3,5...}^{n-1}v\left(t_i\right)+2{\displaystyle \sum }_{i=2,4,6...}^{n-2}v\left(t_i\right)+v\left(T_z\right)\right]$

(4)

Create and define the objective function $F\left(t\right)$

$F\left(t\right)=D\left(t,T\right)-S$

(5)

And interval $\left[t_0,T_0\right],\left[t_{\mathit{max}},T_{\mathit{max}}\right]$ , and $\left[t_{\mathrm{z}},T_{\mathrm{z}}\right]$ to establish objective function

$F\left(t_0\right)=D\left(t_0,T_0\right)-S_0$

(6)

$F\left(t_{\mathit{max}}\right)=D\left(t_{\mathit{max}},T_{\mathit{max}}\right)-S_{\mathit{max}}$

(7)

$F\left(t_z\right)=D\left(t_z,T_z\right)-S_z$

(8)

By analyzing (6), (7), and (8), the roots obtained are the time $t_0$ when the head of the train passes through $K_0$ , the time $t_{\mathit{max}}$ when the center of the train passes through $K_{\mathit{max}}$ , and the time $t_{\mathrm{z}}$ when the rear of the train passes through $K_z$ .

Regression model

The monitoring data may be biased due to equipment accuracy and environmental factors, thus affecting the accuracy of the vertical stiffness evaluation. To ensure the accuracy and reliability of the evaluation results, this study will collect the traveling data of the train on the track-specific suspension bridge and the actual deformation monitoring value $f$ of the $K_{\mathit{max}}$ section, and establish the regression model $f\left(t,v\right)$ of the traveling data velocity $v$ and time $t$ for regression analysis of $f\left(t,v\right)$ . Then, the global correction coefficient $\gamma$ of the most unfavorable section deformation of the main beam under the action of the no-load first train is obtained, and the dynamic calibration coefficient, the ratio of residual deformation to maximum deformation, is modified.

Considering that the CART model can explain the complex nonlinear relationship and interaction between deformation and influencing factors and has a high anti-interference ability against outliers or noise, ²² we adopted CART to establish the regression model $f\left(t,v\right)$ for analysis. At the same time, to highlight the reliability and accuracy of CART regression models, we will use the same data set to build linear regression, polynomial regression, support vector regression, gradient boosting regression, K-nearest neighbor regression, and random forest regression models, and compare them with R² scores.

Data extraction and combination

Throughout the entire process of the train crossing the bridge from the front to the rear, the transponder captures the train’s travel time data $t=\left(t_1,t_2,\dots ,t_i\right)$ and the corresponding speed data $v=\left(v_{t1},v_{t2},\dots ,v_{ti}\right)$ within the time range $\left[t_0,t_z\right]$ . Each time point $t_i$ corresponds to a specific speed and deformation value, which is used to describe the train’s position and movement at that moment. The speed data reflects the train’s operational status at different times. The time and speed data are then combined into a feature matrix $X=\left[v,t\right]$ , representing the feature information of each sample. These feature matrices are input into the model to analyze the impact of the train’s operational characteristics on bridge deformation.

Defining the target variable

In this study, the target variable $y$ represents the actual deformation value $f$ at the $K_{\mathit{max}}$ section. The target variable $y$ is used to describe the deformation of the bridge when the train is running on the suspension bridge. Specifically, as the time $t=\left(t_1,t_2,\dots ,t_i\right)$ and speed $v=\left(v_{t1},v_{t2},\dots ,v_{ti}\right)$ change, the deformation value of the bridge at the $K_{\mathit{max}}$ section $f=\left(f_1,f_2,\dots ,f_i\right)$ also changes accordingly. Therefore, the target variable $y$ represents the monitored deformation value of the bridge under these specific conditions.

Constructing the dataset

The feature matrix is composed of the time and speed data when the train crosses the bridge. For each time point $t_i$ , there is a corresponding speed $v_{ti}$ , so each sample $X_i=\left[t_i,v_{ti}\right]$ contains the train’s operational information at time $t_i$ . The entire feature matrix $X$ includes all time points and speeds. The target variable $y$ is the actual deformation value $f$ corresponding to each time point and speed. Thus, for each sample $X_i$ , the target variable $y_i$ represents the deformation value $f_i$ at time $t_i$ and speed $v_{ti}$ .

The entire dataset $D$ can be expressed as

$D={\left\{\left(X_i{y}_i\right)\right\}}_{i=1}^n={\left\{\left(\left[t_i{v}_{ti}\right],f_i\right)\right\}}_{i=1}^n$

where $n$ is the number of samples collected during the train’s passage from the front to the rear of the bridge.

The bridge deformation data exhibits continuity in both time and speed, and splitting the dataset may disrupt this continuity, affecting the model’s ability to accurately fit the relationship between train operation and deformation. Furthermore, the focus of this method is on fitting and correcting existing data rather than predicting future data. Therefore, using the entire dataset $D$ as both the training set $D_{train}$ and the test set $D_{test}$ maximizes the model’s fitting effectiveness and ensures the correction of potential errors in the data

$D_{train}={\left\{\left(X_i,y_i\right)\right\}}_{i=1}^n$

(9)

$D_{test}={\left\{\left(X_i,y_i\right)\right\}}_{i=1}^n$

(10)

Training the model

The CART regression model was trained using the training set $D_{train}$ . The CART model constructs a decision tree by recursively splitting the dataset. For each node, it selects a feature $X_j$ and a split point $k$ , dividing the dataset into two subsets $D_{left}$ ​ and $D_{right}$

$D_{\mathrm{l}\text{eft}}\left(j,k\right)=\left\{\left(X_i,y_i\right)|X_{ij}\le k\right\}$

(11)

$D_{\text{rig}\text{ht}}\left(j,k\right)=\left\{\left(X_i,y_i\right)|X_{ij}>k\right\}$

(12)

where $X_{ij}$ is the $j-\text{th}$ feature value of the $i-\text{th}$ sample, and $y_i$ is the target variable value of the $i\text{th}$ sample

During the training of the model, the optimal split point is selected by minimizing the weighted mean square error (MSE)

$\left(j^{*},k^{*}\right)=\arg \hspace{0.17em}\underset{j,k}{\min } \left(\frac{\left|D_{\text{left}}\left(j,k\right)\right|}{\left|D\right|}{\text{MSE}}_{\text{left}}+\frac{\left|D_{\text{right}}\left(j,k\right)\right|}{\left|D\right|}{\text{MSE}}_{\text{right}}\right)$

(13)

Among them

${\text{MSE}}_{\text{left}}=\frac{1}{\left|D_{\text{left}}\right|}{\displaystyle \sum }_{\left(X_i,y_i\right)\in D_{\text{left}}} {\left(y_i-{\overline{y}}_{\text{left}}\right)}^2$

${\text{MSE}}_{\text{right}}=\frac{1}{\left|D_{\text{right}}\right|}{\displaystyle \sum }_{\left(X_i,y_i\right)\in D_{\text{right}}} {\left(y_i-{\overline{y}}_{\text{right}}\right)}^2$

${\overline{y}}_{\text{left}}=\frac{1}{\left|D_{\text{left}}\right|}{\displaystyle \sum }_{\left(X_i,y_i\right)\in D_{\text{left}}} y_i$

${\overline{y}}_{\text{right}}=\frac{1}{\left|D_{\text{right}}\right|}{\displaystyle \sum }_{\left(X_i,y_i\right)\in D_{\text{right}}} y_i$

where $\left(j^{*},k^{*}\right)$ is the best splitting point, ${\text{MSE}}_{\text{left}}$ and ${\text{MSE}}_{\text{right}}$ are the mean square error of the left and right child nodes, respectively. ${\overline{y}}_{\text{left}}$ and ${\overline{y}}_{\text{right}}$ subsets were left and right a subset of the target variable of the mean.

The CART model measures the importance of features based on the split contribution of features in the tree

$\text{Feature}\hspace{0.17em}\text{Importance}\left(X_j\right)={\displaystyle \sum }_{m\in M} \mathrm{\Delta }\text{MSE}\left(X_j,m\right)$

(14)

$\mathrm{\Delta }\text{MSE}\left(X_j,m\right)={\text{MSE}}_{\text{parent}}-\left(\frac{\left|D_{\text{left}}\right|}{\left|D\right|}{\text{MSE}}_{\text{left}}+\frac{\left|D_{\text{right}}\right|}{\left|D\right|}{\text{MSE}}_{\text{right}}\right)$

(15)

where $m$ is each node in the decision tree, $M$ is the whole decision tree, and ${\text{MSE}}_{\text{parent}}$ is the mean square error of the parent node of node $m$ .

The above steps are applied recursively to each subset until a stop condition is reached (i.e., the maximum tree depth is reached or the number of subset samples is less than a certain threshold) to generate leaf nodes. The predicted value of the leaf node is the mean value of the target variable of the corresponding subset of the node

$\widehat{y}={\overline{y}}_{\text{leaf}}=\frac{1}{\left|D_{\text{leaf}}\right|}{\displaystyle \sum }_{\left(X_i,y_i\right)\in D_{\text{lea}\mathrm{f}}} y_i$

(16)

where $\widehat{y}$ is the predicted value of the leaf node, ${\overline{y}}_{\text{leaf}}$ is the average value of the target variable (i.e., the predictor variable) in the leaf node, and $D_{\text{leaf}}$ is the sample set of the leaf node.

Evaluating the model performance

After establishing the CART regression model, the Pearson correlation coefficient R² score was used to evaluate the model performance, representing the model accuracy and data interpretation ability. Its calculation formula is as follows

$R^2=1-\frac{{\displaystyle \sum }_{i=1}^{D_{\text{test}}} {\left(y_i-{\widehat{y}}_i\right)}^2}{{\displaystyle \sum }_{i=1}^{D_{\text{test}}} {\left(y_i-\overline{y}\right)}^2}$

(17)

where $y_i$ is the measured value, ${\widehat{y}}_i$ is the predicted value, $\overline{y}$ is the average of the measured values, and $D_{\text{test}}$ is the number of samples of the test set.

Calculation of correction coefficient

After verifying the accuracy of the regression model, the CART regression model is used to predict the deformation value corresponding to each velocity $v$ and time $T,$ and the correction coefficient $\gamma$ for each data point is calculated

${\gamma }_i=\frac{f\left(v_i,t_i\right)}{f_i}$

(18)

where $f\left(v_i,t_i\right)$ is the predicted value and $f_i$ is the measured value.

Finally, the average of all correction factors is calculated as the final correction factor $\mathrm{\gamma }$

$\gamma =\frac{1}{n}{\displaystyle \sum }_{i=1}^n {\gamma }_i$

(19)

Stiffness evaluation

Dynamic check coefficient

A load test is the most direct and effective method of traditional bridge condition assessment. ²³ The ratio between the measured response and the theoretically calculated response is used to calculate the calibration coefficient to determine whether the bridge is in good condition. ²⁴ However, this method has some problems such as being time-consuming and labor-intensive, causing traffic blocking, and being high cost. Based on the characteristic that the moving load source is clear, this paper presents the dynamic test coefficient as the index of stiffness evaluation. As expressed by $\xi$ , the calculation formula is (20)

$\xi =\frac{∆f}{F}$

(20)

where $∆f$ is the measured elastic deformation value of the measuring point under train load, and $F$ is the theoretical calculated deformation value of the measuring point under train load.

The dynamic test coefficient $\xi$ can evaluates the structural stiffness of the bridge, and the criteria are as follows:

(1) When $\xi$ < 1, it indicates that the current mechanical performance of the bridge is good, and the safety reserve of the structural stiffness is sufficient.

(2) When $\xi$ > 1, it indicates that the mechanical performance of the bridge is poor, and the safety reserve of the structural stiffness is insufficient.

(3) When $\xi$ = 1, it indicates that the mechanical performance of the bridge just meets the design requirements, and the safety reserve of the structural stiffness is not surplus.

Evaluation indicators

According to the monitoring data of the safety monitoring system, the actual deformation monitoring values of $K_{\mathit{max}}$ at $t_0,t_{\mathit{max}}\hspace{0.17em}\text{and}{\hspace{0.17em}t}_z$ are determined, denoted as $f_0,f_{\mathit{max}}$ and $f_z$ , respectively. With $f_0$ as the deformation basis of the main beam, the deformation monitoring value $△f$ of $K_{\mathit{max}}$ section under the same environmental factors (temperature, wind, etc.) is obtained according to equation (21). According to equation (22), the residual deformation monitoring value $f^{\prime }$ of the $K_{\mathit{max}}$ section after and before the passage of the first no-loaded train under the same environmental factors (temperature, wind, etc.) is obtained

$△f=f_{\mathit{max}}-f_0$

(21)

$f^{\prime }=f_z-f_0$

(22)

where $f_0$ is the measured deformation value of $K_{\mathit{max}}$ section at $t_0$ , $f_{\mathit{max}}$ is the measured deformation value of $K_{\mathit{max}}$ section at $t_{\mathit{max}}$ , and $f_z$ is the measured deformation value of $K_{\mathit{max}}$ section at $t_z$ .

Considering the error of the measured value obtained by the monitoring system, the dynamic calibration coefficient and the ratio of the residual deformation value to the maximum deformation value required by the specification are revised. ²¹ The evaluation criteria are as follows: If Equations (23) and (24) are met at the same time, it indicates that the structural stiffness is in a normal state; otherwise, the safety monitoring system should be relied on for structural stiffness early warning, and the disposal suggestions are put forward after analysis

$\gamma ·\xi =\gamma ·\frac{∆f}{F}\le 1$

(23)

$\frac{\gamma ·f^{\prime }}{f_{\mathit{max}}}=\frac{\gamma ·\left(f_z-f_0\right)}{f_{\mathit{max}}}\le 20\%$

(24)

Application analysis

Project overview

Egongyan track bridge is the world’s largest span self-anchored track-specific suspension bridge. Based on it, this paper will verify the feasibility of the proposed vertical stiffness evaluation method for the track-specific suspension bridge. The span arrangement of the bridge is: 50 + 210+600 + 210 + 50 = 1120 m, and the cross-section arrangement is: 2.25 (cable area, air nozzles) + 0.25 (railing) + 2.35 (sidewalk) + 0.9 (anti-collision, isolation belt) + 10.5 (track boundary) + 0.9 (anti-collision, isolation belt) + 2.35 (sidewalk) + 0.25 (railing) + 2.25 (cable area, air nozzles) = 22.0 m. The main beam is a composite structure of steel and concrete beams. The main structure of the steel beam is Q420qD, and the prefabricated bridge panel is made of C55 concrete. The main pier is a reinforced concrete structure. The bridge adopts high-strength parallel steel wire bundles as double main cables, and the main cable spacing is 19.5 m. The suspender is parallel, using zinc–aluminum alloy coating Φ7 parallel steel wire material. The anchor section and anchor span section of the main cable are made of prestressed concrete box beams, and the rest of the cable is made of steel box beams. Figure 5 show the bridge layout.

Figure 5.

Egongyan track bridge layout.

Load condition of the no-load first train

According to the engineering design, the maximum driving speed is 80 km/h, the wheelbase composition of one car of the as train is (2200 + 11200+2200) mm, and the axle spacing between two cars is 4400 mm. According to the traffic volume management plan of the Chongqing track transit circle line, a 6-car train formation will be considered soon. The main parameters of vehicle size are shown in Table 1, and the schematic diagram of the train formation is shown in Figure 6.

Table 1.

As train size main parameters.

Vehicle type	Body length /mm	Body width /mm	Vehicle distance /mm
As	19300	3000	13400

Figure 6.

Train formation (6 sections) schematic diagram.

According to the characteristic of clear load action of the no-load first train, the load condition of the empty train is used as the moving load of the Madis/Civil 2023 finite element model to calculate the theoretical deformation value of the no-load first train. According to the specification, ²⁵ the no-loaded weight of Mc, Mp, and M compartments is 21363 kg, 20617 kg, and 21360 kg, respectively. Each carriage has 4 axles. The single axle weight P1 of the Mc carriage is 54.08 kN; the single axle weight P2 of the Mp compartment is 53.40 kN; the single axle weight P3 of the M car is 51.54 kN. The load diagram of the no-load train is shown in Figure 7.

Figure 7.

Load model of no-load train (6 sections) (unit: m).

Numerical simulation

Calculation model

The finite element model of the Egongyan track bridge was established by Madis/Civil 2023. The model was divided into 937 nodes and 924 elements (674 beam elements and 250 tension elements). The main beam and bridge tower are simulated by beam elements, and the main cable and boom are simulated by tension truss elements. The boundary and constraint conditions of the finite element model are set as follows: a fixed support is set at the bottom of the bridge tower; the main beam and the main cable anchor point, the main beam and the lower edge of the boom, and the main cable and the main tower are rigidly connected. The elastic connection between the main beam and the support of the bridge tower is simulated, and the longitudinal direction of the bridge is unconstrained. The main beam at the auxiliary pier and junction pier adopts general support to release longitudinal constraints. The material characteristic parameters of each major component are shown in Table 2, and the moving vehicle load is defined according to Figure 7. The impact coefficient and dynamic coefficient are defined according to specification [25]. Figure 8 is the finite element model of the track-specific suspension bridge.

Table 2.

Main component characteristic parameters.

Component name	Materials	Unit weight (kN/mm3)	Modulus of elasticity (kN/mm2)	Poisson’s ratio
King beam	C55 concrete	2.5 × 10−8	35.5	0.2
	Q420qD	7.85 × 10−8	206	0.31
Main tower	C55 concrete	2.5 × 10−8	35.5	0.2
	C60 concrete	2.5 × 10−8	36.0	0.2
Main cable	1860 MPa high-strength steel wire	7.85 × 10−8	200	0.3
Derrick	1770 MPa high-strength steel wire	7.85 × 10−8	205	0.3

Figure 8.

The finite element model of the track-specific suspension bridge.

Calculating the effect

The spatial deformation of the whole bridge is analyzed by loading the no-load train. The results show that the maximum downward deflection of the main beam occurs at the 3/4 section of the middle span, and the deformation value is 268.55 mm. The maximum upward deflection occurs at the 1/4 position of the main span, and the deformation value is 92.27 mm. Therefore, it is determined that the most unfavorable section of the main beam deformation is the 3/4 section of the middle span of the main beam, and the theoretical calculation deformation value is 268.55 mm. The deformation of the main bridge under the action of trains is shown in Figure 9. The displacement of the main beam under the action of the train is shown in Figure 10.

Figure 9.

The main bridge deforms under the action of the train.

Figure 10.

Displacement of the main beam under the action of the train.

Deformation monitoring system

The long-term safety monitoring system of the Egongyan track bridge is mainly composed of a sensor subsystem, a data acquisition and transmission subsystem, a data processing and analysis subsystem, a data storage and management subsystem, and an early warning subsystem. The monitoring contents include load and environment monitoring, static and dynamic response monitoring of the whole structure, and local response monitoring of the structure, among which the deformation monitoring of the main beam belongs to the static and dynamic response monitoring of the whole structure.

The method proposed in this study will be verified by obtaining the vertical deformation data of the main beams by the Leica GNSS Deformation System. The test equipment mainly includes an observation pier, power supply system, RS232 communication equipment, AS10 antenna equipment, GMX902 GG receiver, and Leica GNSS Spider software. The measuring points are arranged at the top of the P13 tower, the top of the P14 tower, the outer side of the upper and lower walkway railings of the bridge deck at the 1/4 position of the main span, the outer side of the upper and lower walkway railings of the bridge deck at the middle of the main span, and the outer side of the upper and lower walkway railings of the bridge deck at the 3/4 position of the main span. Based on the calculation results, the verification utilizes the measuring point located at the quarter-point near the P14 pier, highlighted in red in Figure 12. The measurement performance and accuracy of the Leica GNSS Deformation System are shown in Table 3. Figures 11 show the measurement points’ longitudinal layout and mid-span cross-section layout. The actual photo of the monitoring point can be seen in Figure 12.

Table 3.

The performance and accuracy of the GNSS system.

Measurement performance		Accuracy
Sampling rate		1 Hz
Static mode of post-processing		Levels: 3 mm + 0.5 ppm
		Vertical: 5 mm + 0.5 ppm
Real-time dynamic positioning of RTK	Single baseline (<30 km)	Levels: 8 mm + 1 ppm
		Vertical: 15 mm + 1 ppm
	Network RTK	Levels: 8 mm + 0.5 ppm
		Vertical: 15 mm + 0.5 ppm

Figure 11.

Schematic diagram of the location of the most unfavorable cross-section monitoring points.

Figure 12.

The actual photo of the monitoring point.

Method verification

Spatial relationship analysis

According to the specifications and actual operation, the layout of the ground transponder of the world’s largest span self-anchored track-specific suspension bridge is determined: The point $Y_0$ ​ is located 15.02 m from the P14 pile, $Y_{\mathit{max}}$ ​ is 248.26 m from the P14 pile, and $Y_Z$ is 54.76 m from the P13 pile. Combined with the GNSS measuring point of the most unfavorable section and the starting and finishing sections of the suspension bridge, the spatial relationships $S_0$ , $S_{\mathit{max}}$ , and $S_z$ are established according to the upstream train direction, and the specific values are shown in Table 4. Figure 13 shows the spatial relationship between the transponder and the special section.

Table 4.

Spatial relationship (unit: m).

Geometric relations	$S_0$	$S_{\mathit{max}}$	$S_z$
Date	194.98	98.26	264.76

Figure 13.

Spatial relationship diagram between transponder and special section (unit: mm).

The data of the transponder on June 13, 2024, was selected for verification, and the running track (velocity $v$ and time $T$ ) of the first empty train between Haixia Road Station and Xijiawan Station was obtained through the on-board equipment of the transponder, and the relative relationship between $V-T$ was established. Meanwhile, $T_0{,T}_{\mathit{max}},$ and $T_z$ are determined to be 5:41:03 $,$ 5:41:15 $,$ and 5:41:36, respectively. Figure 14 shows the relationship between travel time and velocity of the no-load first train.

Figure 14.

Track of the no-load first train.

Given that $T_0$ , $T_{\mathit{max}}$ , $T_Z$ and $S_0$ , $S_{\mathit{max}}$ , $S_{\mathrm{Z}}$ are combined with formulas (6)–(8), the roots $t_0$ , $t_{\mathit{max}}$ , and $t_z$ are solved, and the results are shown in Table 5.

Table 5.

Train transponder time.

	$t_0$	$t_{\mathit{max}}$	$t_z$
Time	5:40:53	5:41:15	5:41:54

The measured value of the monitoring system

The vertical displacement monitoring data of the GNSS upstream 3/4 section of the world’s largest span self-anchored track-specific suspension bridge safety testing system on June 13, 2024, were obtained, as shown in Figure 15.

Figure 15.

Monitoring system data.

According to the monitoring data, the actual deformation monitoring values $f_0,f_{\mathit{max}}$ and $f_z$ of the most adverse deformation section of the main beam at $t_0,t_{\mathit{max}},$ and ${\hspace{0.17em}t}_{\mathit{max}\hspace{0.17em}}$ are determined, and the displacement values are -11.80 mm $,$ ‐232.80 mm $,$ and 5.20 mm $,$ respectively.

It can be obtained from the data that the theoretical value of vertical deformation of the most unfavorable section of the main beam is 268.55 mm, the actual monitoring value is 232.80 mm, the difference is 35.75 mm, and the relative error is 13.3%. The calculated calibration coefficient is 0.89, it is preliminarily judged that the bridge structure is in good condition and the performance meets expectations.

Regression model

First, the operation time of the empty train on the bridge was determined to be from 5:40:53 to 5:41:54 on June 13, 2024, for a total of 62 seconds. Starting at 5:40:53, a sample set $X_i={\left[t_i,v_{ti}\right]}_{i=1}^{62}$ was constructed based on the corresponding speed data obtained from the transponder. Using the monitoring data from Figure 15, the section monitoring data $D={\left\{\left(X_i{y}_i\right)\right\}}_{i=1}^{62}={\left\{\left(\left[t_i{v}_{ti}\right],f_i\right)\right\}}_{i=1}^{62}$ for the no-load first train’s operation on the bridge was extracted, and a dataset with a total of 62 samples was established.

The CART regression model $f\left(t,v\right)$ was established using MATLAB R2023a software. Meanwhile, regression models of Linear, Polynomial, Support Vector Regression (SVR), Gradient Boosting, K-nearest neighbors (KNN), and Random Forest were established using data set $D$ . Figure 16 shows the CART regression model, and Figure 17 show the contrast regression models. Figure 18 shows the R² comparison plot of the regression models.

Figure 16.

CART regression model.

Figure 17.

The contrast regression models.

Figure 18.

Regression model R² comparison graph.

Stiffness evaluation

After the R² evaluation of model reliability, the correction coefficients of each point were calculated according to equation (18), and the results are shown in Figure 19. Therefore, the correction coefficient of the CART regression model calculated according to equation (19) is 1.04.

Figure 19.

Correction coefficients for each point.

Under the same environmental factors (temperature, wind, etc.), the deformation monitoring value $△f$ of the most unfavorable section of the main beam deformation under the action of the no-load first train and the residual deformation monitoring value $f^{\prime }$ after and before the passage of the no-load first train are calculated by equations (20) and (21). The calculation result shows that $△f=221.00\text{mm}$ , $f^{\prime }=17.00\text{mm}$ .

According to the calculated as $\gamma ·\xi =\gamma ·\frac{∆f}{F}=0.86\le 1$ or less and $\frac{\gamma ·f^{\prime }}{f_{\mathit{max}}}=\frac{\gamma ·\left(f_z-f_0\right)}{f_{\mathit{max}}}=7.6\%\le 20\%$ .The results show that the deformation of the main beam meets the requirements and the vertical stiffness of the structure is in a normal state.

Conclusions and prospects

Conclusions

In this paper, a method for evaluating the structural vertical stiffness of a track-specific suspension bridge is proposed based on the action of the no-load first train, and the effectiveness and feasibility of the method are verified on the world’s largest span self-anchored track suspension bridge. The conclusion is as follows:

(1) Through the spatial relationship established between the transponder and the measurement point arrangement of the safety monitoring system, combined with the transponder travel track (velocity $v$ and time $T$ ) and Simpson’s integral method, the specific time of the train to each section of the bridge can be accurately calculated.

(2) The $f-v-t$ data set was built with the monitoring system data and transponder driving data, and the CART regression model was constructed with it. Through this model, the global correction coefficient $\gamma$ of the most unfavorable section deformation of the main beam under the action of the no-load first train is further calculated to correct the dynamic calibration coefficient and the ratio of the residual deformation value to the maximum deformation value.

(3) Based on the characteristic that the moving load source is clear, the dynamic test coefficient $\xi$ is proposed as the standard of stiffness evaluation, and $\gamma ·\xi$ and $\frac{\gamma ·f^{\prime }}{f_{\mathit{max}}}$ are proposed as the evaluation indexes of the structural stiffness evaluation method proposed in this paper.

(4) Relying on the Egongyan track bridge, through Madis/Civil 2023 analysis, the theoretical calculation value of the deformation of the most unfavorable section under the moving load of the no-load train is 268.55 mm, and the measured value of the monitoring system is 232.80 mm, the difference is 35.75 mm, and the relative error is 13.3%. The R² score of the CART regression model is 0.94, and the accuracy is higher than that of the six comparison models. The correction coefficient calculated based on the CART regression model is 1.04. The modified dynamic check coefficient is 0.85, and the ratio of residual deformation value to maximum deformation value is 7.6%. The results show that the deformation of the main beam meets the requirements of the code, and the vertical stiffness of the structure is in a normal state.

Prospects

(1) While there is a correlation between deformation and stiffness, this study uses vertical deformation to reflect vertical stiffness. In future research, more comprehensive indicators will be considered.

(2) The initial intention of this study was to conduct a stiffness evaluation based on the no-loaded train, as its load conditions are clearly defined. Since there is only one no-loaded train passing each day without affecting traffic, different speed tests and their impact on bridge stiffness were not considered. We will consider this aspect in our next research phase as a separate study topic.

(3) This study proposed a stiffness evaluation method for rail bridges based on the first no-loaded train and achieved certain results. However, the study did not fully consider the dynamic amplification effect caused by track irregularities. Although the train’s speed over the bridge ranged between 55 km/h and 70 km/s, track irregularities still have some impact. In future research, we plan to combine numerical simulations with field tests to develop a more complex vehicle-bridge coupling model and systematically analyze the dynamic amplification effect caused by track irregularities.