Performance evaluation of cable-stayed bridge expansion joints based on Lasso dimensionality reduction and temperature-displacement-correlation model

Abstract

Expansion joints play a crucial role in accommodating the longitudinal movement of the main beam, which is mainly caused by temperature variation. This paper establishes an accurate model that relates the temperature field of the main beam to the displacement of the expansion joint, enabling reliable performance prediction and early warning of the expansion joint. Firstly, three commonly used methods for characterizing the temperature field of the main beam are introduced, along with their advantages and disadvantages. Secondly, a novel method is proposed using the Lasso algorithm to calculate critical temperatures. The objective is to select temperature channels data that have significant impact on the longitudinal displacement of the main beam. The selected channels data is then linearly weighted based on feature importance to obtain critical temperature. Based on this, a precise relationship model between the main beam temperature and the expansion joint displacement is derived through regression. For the residual term in the model fitting, an expansion joint performance early warning procedure is developed based on the X-bar control chart. Finally, using one-year long-term monitoring data from a newly constructed cable-stayed bridge as an example, the proposed method demonstrates superior capability in predicting the predefined damage of the expansion joint compared to the other two commonly used methods.

Keywords

Bridge structural health monitoring Lasso dimensionality reduction temperature safety evaluation

Introduction

As critical components embedded within the gaps of bridge structures, expansion joints are designed to accommodate the deformations caused by temperature variations, concrete shrinkage and creep, traffic loads, and so on (Chen and Duan, 2014). However, the occurrence of damages in expansion joints is particularly severe due to design and construction deficiencies, inadequate management and maintenance, and vehicle overloading in practical engineering projects (Chang and Lee, 2002; Deng and Li, 2019). In recent years, several cable-stayed bridges or suspension bridges have been reported to have issues with their expansion joints during the operational process. The Jiangyin Yangtze River Bridge with a main span of 1358 meters exhibited severe damage at the connection between the expansion joint and the main beam four years after its opening in 1999 (Guo et al., 2015). Similarly, the Runyang Yangtze River Bridge with a main span of 1490 meters experienced excessive wear of the expansion joint due to the frequent and rapid movement of the main beam shortly after its construction (Sun and Zhang, 2016). The damage and performance degradation of expansion joints have various detrimental effects on bridge structures, including the loss of free expansion capability, restriction of longitudinal movement of the main beam leading to end damage during temperature rise, undulations in the surrounding road surface exacerbating vehicle impacts (Sun and Zhang, 2016). Therefore, real-time monitoring of the operational status of expansion joints, timely warning of their damage and performance degradation, and implementing necessary maintenance measures are of paramount importance to prolong the service life of expansion joints.

Numerous studies have explored the impact mechanism of environmental factors, such as temperature, on the overall behavior of long-span bridges, from the perspectives of theoretical models and empirical data. Zhou et al. (2015, 2020) derived a formula for calculating the vertical deflection of long-span cable-stayed bridges due to temperature through theoretical analysis and validated it using field measurement data from the Shanghai Yangtze River Bridge. Sun et al. (2018) and Shan et al. (2018) have revealed the influence of temperature on the natural frequencies of bridges, considering both theoretical analysis and empirical data. Li et al. (2023) argued that the temperature effects on cable-stayed bridges are complex. They advocated for exploring global 3D temperature effect numerical simulations and optimizing the placement of temperature sensors based on the boundary conditions required by the numerical simulations. Shan et al. (2023a, 2023b) established a global three-dimensional finite element model of the Qingzhou Bridge. They used transient heat conduction analysis to calculate the detailed temperature distribution throughout the entire bridge for all four seasons which found that the horizontal displacement of the bridge towers and the longitudinal displacement of the bridge ends are primarily affected by the average temperature.

In the health monitoring system of a bridge, the longitudinal displacement of the main beam is typically indirectly measured through the displacement of expansion joints. Well-functioning expansion joints should have the capability to allow the main beam to expand and contract freely (Deng et al., 2010). Under normal conditions, various observed data in bridge structures fluctuate within a broad range due to environmental factors such as temperature variations. Field measurements have shown that displacements in expansion joints can be induced by temperature variations, vehicle loads, wind loads, and other factors. Among these factors, temperature variation is the most significant, exhibiting a strong linear correlation (Han et al., 2021; Ni et al., 2007). Researchers such as Ni et al. (2007), Ding and Li (2011), Deng and Li (2019) have developed models to correlate the displacement of expansion joints and temperature in bridge structures, which were further utilized for identifying predefined damages of expansion joints. Huang et al. (2016, 2018) conducted Canonical Correlation Analysis (CCA) to identify the optimal linear combination coefficients for temperature, maximizing the correlation between the transformed temperature and the displacement of expansion joints, which enabled an effective assessment of the performance of the expansion joint. The above-mentioned temperature representation methods essentially involve linear combination or transformation of the original temperature data. However, for large-span bridges with numerous temperature sensors, using all temperature channels data as input would make the constructed correlation model lack clear physical interpretation, making it difficult to explain the uncertainty effects of each input temperature variable on displacement. Additionally, in operational conditions, there are often cases of sensor failures or temporary shutdowns, which compromises the robustness of the algorithm and introduces certain limitations in practical applications. Therefore, it is necessary to explore a more data-efficient and interpretable temperature representation method that uses fewer channels.

This paper presents an enhanced and interpretable model for the main beam temperature-expansion joint displacement relationship, as well as a robust warning procedure for assessing expansion joint performance in cable-stayed bridges. Chapter 2 introduces the data source of this paper. Chapter 3 introduces three commonly used methods for characterizing main beam temperature fields, discussing their advantages and limitations. Chapter 4 proposes a novel approach utilizing the Lasso algorithm to calculate feature temperatures, which selects temperature data that significantly affects the displacement of expansion joint. The selected data is then linearly weighted based on feature importance, yielding critical temperature (T_c). Chapter 5 establishes regression-based correlation models between average temperature, principal component temperature, critical temperature, with expansion joint displacement. Chapter 6 develops a warning procedure for expansion joint performance by analyzing the residuals obtained from model fitting, which incorporates the X-bar control chart. To demonstrate the efficacy of the proposed method, a case study is presented using one-year long-term monitoring data, which indicates that the proposed method outperforms the other two commonly used methods in identifying predefined expansion joint damage. Finally, Chapter 7 provides a comprehensive conclusion.

Data source description

This research utilized the health monitoring data of the Jintang Hantan Bridge, located in Chengdu, Sichuan, China. It is a double-abutment, double-cable-plane cable-stayed bridge with a main span of 430 meters. The bridge was completed and opened to traffic in October 2021, and its structural health monitoring (SHM) system went online in December 2021. Within the SHM system, two tension rope displacement gauges (WY01 and WY02, as shown in Figure 1) were installed at the expansion joints of the western abutment and eastern abutment to monitor the longitudinal displacement of the expansion joints. For the temperature of the main beam, a total of 86 Fiber Bragg Grating temperature sensors (WD01 to WD86) are used. These temperature sensors are installed at 5 sections, including the central section of the side spans, the base of the main abutment, and the central section of the main span. The locations and detailed arrangements of the temperature sensors on the cross-section of the main beam are illustrated in Figures 1 and 2.

Figure 1.

Layouts of expansion joint displacement and main beam temperature sensors of Jintang Hantang Bridge (unit: m).

Figure 2.

Temperature sensors arrangements for the section 3# (sensors numbered from WD37 to WD50).

It is notable that, according to the 2021 and 2022 inspection reports provided by the bridge operation department, the expansion joints at the east and west abutments of Jintang Hantan Bridge were in intact condition. Additionally, based on the analysis of daily monitoring data from June 2022 to July 2023, including displacement, deflection, and the modal frequencies identified by acceleration, it can be observed that the overall structural response shows no anomalies.

In this study, one year of monitoring data spanning from June 2022 to July 2023 was collected. Hourly data from temperature and expansion joint sensors, arranged as shown in Figures 1 and 2, were utilized. After removing outliers and missing values, 6784 data sets were obtained and divided into two stages:

(I) Training stage: The first 10 months of data were used to train the temperature-displacement correlation (TDC) models for the expansion joints and determine the parameters for X-bar control chart analysis.

(II) Testing stage: The last 2 months of data were used to evaluate the TDC models and control charts established in the training stage. Since the bridge was intact, the testing data was artificially manipulated to simulate degraded expansion joint performance.

Unless specified, temperatures are measured in degrees (°C), and displacements are measured in millimeters (mm).

Representative methods of main beam temperatures

As mentioned before, the temperature of the main beam is the most critical factor influencing expansion joint displacement, and there is a strong linear correlation between the two. Before constructing the Temperature-Displacement Correlation (TDC) model, the primary task is to identify typical temperature representations from numerous temperature sensors. This chapter discusses several classical temperature representation methods, namely effective temperature, average temperature, and principal component temperature.

Effective temperature T_e

The effective temperature is the weighted average of the temperature distributed on a certain section of the main beam. Let T(x,y) represent the two-dimensional continuous temperature field of a section, the effective temperature is obtained by integrating the two-dimensional continuous temperature field over the cross-sectional area:

T_{e} = \frac{1}{A} \iint T (x, y) d x d y

(1)

Where T_e represents the effective temperature, and A represents the cross-sectional area.

In practice, temperature sensors are usually installed at discrete positions on the section (Figure 2), making it difficult to obtain the precise value of the effective temperature using equation (1). If the section is divided into several sub-regions based on the positions of the temperature measurement points, the effective temperature can be approximated by averaging the temperature values of all sub-regions (Ni et al., 2007):

T_{e} = \sum_{i = 1}^{s} \frac{A_{i}}{A} T_{i}

(2)

where A_i represents the area of the i-th sub-region, T_i represents the temperature value of the i-th sub-region, and s represents the number of temperature sensors or sub-regions.

The expression (2) is a simplified weighted average representation of a two-dimensional temperature field, and its accuracy largely depends on the artificially defined partitioning criteria of the sub-regions to which the sensors belong. Considering the complexity of the actual cross-section of a cable-stayed bridge, such partitioning is often a challenging task with little reward. Therefore, in this study, an effective temperature representation was not adopted. Instead, an average temperature similar to the effective temperature but with lower computational cost was used, as shown in the next section.

Average temperature T_a

The average temperature of the main beam can be obtained by averaging the temperatures of all temperature measurements:

T_{a} = \frac{1}{m} \sum_{i = 1}^{m} T_{i}

(3)

where m represents the number of sensors; considering that the average temperature may vary significantly for each section, this study calculates the average temperature for each of the 5 sections, denoted as WDAVG1 to WDAVG5.

For large-span cable-stayed bridges with numerous sensors, the average temperature representation is an extremely simple and easy-to-implement method, which is frequently used in Chinese standards (T/CECS 529-2018, 2018) and several studies (Ding and Li, 2011; Huang et al., 2018). However, this method is sensitive to sensor failures. More specifically, when certain sensors are turned off or exhibit significant deviations in a particular direction, the average temperature will also exhibit a similar level of deviation.

Principal components of temperature T_pc

Principal Component Analysis (PCA) is a widely used data dimensionality reduction method that maps a high-dimensional and strongly correlated original dataset into a set of low-dimensional uncorrelated variables, which can capture the majority of information from the original dataset (Hastie et al., 2009). In the case of temperature data from multiple measurement points, which often exhibit strong correlations, PCA modeling can be applied to obtain representative temperatures that reflect the main trends (Huang et al., 2018). The calculation of T_pc is as follows:

T_{p c, i} = p_{i}^{T} T

(4)

Where T_p,i represents the i-th temperature principal component, and represents the eigenvector, i.e., the i-th principal direction of the PCA analysis.

The purpose of PCA is to represent the original high-dimensional data with a small number of feature components while capturing the majority of the variance in the original data. Temperature principal components are essentially linear projections of the original temperature data. Although they explain most of the variations in the original data and exhibit strong robustness by being insensitive to abnormal temperature data, they only have statistical significance and lack clear physical interpretations. Moreover, the temperature principal components extracted from the original data may not necessarily have a strong correlation with the displacement of the expansion joint, meaning that the ability to predict the displacement of the expansion joint may not be optimal for the principal component with the maximum variance.

In this study, a total of 91 data sets were used, including temperature channel data from 86 temperature sensors (WD1 to WD86) and average temperature data from 5 sections (WDAVG1 to WDAVG5). PCA analysis was performed to extract the top 5 principal components. As shown in Figure 3, the first principal component explains the majority of the data, accounting for 98% of the variance, and the cumulative contribution of the first 2 principal components reaches 99%.

Figure 3.

Distribution of eigenvalues for the temperature principal components.

Calculatin of critical temperature T_C based on Lasso method

The purpose of temperature representation is to serve as input for constructing the TDC model. Due to the non-uniformity of temperature distribution, the temperature load on a structure, being a field load, is not the same for all locations in large structures. In contrast to the temperature representation discussed in Chapter 3, a more direct and reasonable strategy is to use data from multiple temperature sensors as simultaneous inputs and expansion joint displacement as the output, thereby building a correlation model between them. The multivariate linear regression (MLR) model with multiple temperature data points as inputs is represented as follows:

{\hat{D}}_{k} = β_{k, 1} T_{1} + β_{k, 2} T_{2} + \dots + β_{k, p} T_{p} + β_{i, 0}

(5)

where

{\hat{D}}_{k}

represents the estimated displacement of the expansion joint for the k-th (k = 1,2) case, p represents the total number of temperature measurements,

{\overset{⌢}{β}}_{k} = (β_{k, 1}, β_{k, 2}, \dots β_{k, p})

is the coefficient vector for the k-th expansion joint displacement-temperature correlation model obtained through least squares, minimizing the residual sum of squares (RSS).

R S S (β) = \sum_{i = 1}^{N} {[D_{i} - f (T_{i})]}^{2} = \sum_{i = 1}^{N} {[D_{i} - β_{0} - \sum_{j = 1}^{p} T_{i, j} β_{j}]}^{2}

(6)

where N donates the number of training data.

However, among the numerous temperature measurement points, there may exist one or more temperature variables that have poor correlation or exhibit non-linear relationships with the expansion joint displacement. When these variables are fitted using least squares, it greatly increases the complexity of the model and weakens its generalization ability. The Lasso algorithm is an improved method of least squares, and the estimated coefficients are calculated using the following equation (7). It introduces an L1-norm penalty term to the residual sum of squares (RSS), which penalizes irrelevant input variables (e.g., temperatures) to the prediction variable (e.g., displacement) and sets their coefficients to zero, thereby achieving sparsity and facilitating variable selection for dimensionality reduction (Hastie et al., 2009; James et al., 2013).

{\hat{β}}_{Lasso} = \underset{β}{argmin} {\sum_{i = 1}^{N} {(D_{i} - β_{0} - \sum_{j = 1}^{p} T_{i, j} β_{j})}^{2} + λ \sum_{j = 1}^{p} | β_{j} |} = \underset{β}{argmin} {R S S (β) + λ \sum_{j = 1}^{p} | β_{j} |}

(7)

Although both Lasso and PCA have dimensionality reduction effects, the difference lies in Lasso’s ability to remove irrelevant input features for the prediction variable and calculate feature importance based on the different importance of input features for the prediction variable. The selected features are the temperature variables that have the most significant impact on predicting the displacement. On the other hand, the purpose of PCA is to linearly transform input features so that the resulting feature vectors can maximize the representation of the variation (second-order variance) in the original input data. The selected temperature principal components are linear combinations of the original temperature data and do not have specific physical meaning, and their predictive capabilities for displacement remain unknown.

A total of 91 data sets were used, including 86 temperature sensor channel data (WD1 to WD86) and average temperature data from 5 cross-sections (WDAVG1 to WDAVG5). Firstly, the Lasso algorithm was applied to select the features from the original data. The optimal value of λ was determined through 5-fold cross-validation, which involves randomly splitting the original data into five subsets, using four of them as the training set each time, reserving one for validation, and repeating this process five times to obtain the model parameter values corresponding to the minimum average validation error. For the west bridge abutment expansion joint, 6 sets of temperature data were selected, and for the east bridge abutment expansion joint, 8 sets of temperature data were selected, as shown in Figure 3. The critical temperature (T_c) was obtained by weighting the selected data based on feature importance, as shown in equation (8).

T_{c, 1} = \sum_{i = 1}^{m} f_{i} \cdot T_{i}

(8)

where T_i represents the feature temperatures selected by Lasso, and f_i represents the normalization coefficient calculated based on feature importance (characterized by the

β

values of the selected feature temperatures).

The selected feature temperatures and their feature importance coefficients for the displacement of the expansion joint (shown in the Pareto chart) are illustrated in Figure 4.

Figure 4.

Feature importance ranking of temperatures selected: West abutment(left); East abutment(right).

The selected feature temperatures are weighted based on their feature importance to obtain the critical temperature T_c, calculated according to equations (9) and (10).

\begin{array}{l} T_{c, 1} = 0.2856 \cdot T_{W D 78} + 0.2315 \cdot T_{W D 36} + 0.1938 \cdot T_{W D 59} + 0.1754 \cdot T_{W D 48} + 0.0683 \cdot T_{W D 31} + 0.0454 \cdot T_{W D 60} \end{array}

(9)

\begin{array}{l} T_{c, 2} = 0.3149 \cdot T_{W D 29} + 0.1674 \cdot T_{W D 14} + 0.1412 \cdot T_{W D 60} + 0.1181 \cdot T_{W D 24} + 0.0867 \cdot T_{W D 63} + 0.0795 \cdot T_{W D 40} + 0.0571 \cdot T_{W D 67} + 0.0351 \cdot T_{W D 57} \end{array}

(10)

For a more general case, the steps for calculating the critical temperature T_c are as follows:

(a) Calculate the residual function by adding the L1 norm to the displacement-temperature relationship described by equation (5).

(b) Use k-fold cross-validation to determine the optimal parameter λ for the Lasso algorithm.

(c) Select the feature temperatures that have the most significant impact on displacement and assess their feature importance.

(d) Calculate the critical temperature T_c by linearly weighting the feature temperatures selected by the Lasso algorithm in the previous step, based on their feature importance, as shown in equation (8).

Establishment of temperature-expansion joint displacement relationship

The key to evaluating the performance of expansion joints lies in establishing an accurate correlation model between the displacement of the expansion joint and temperature. In this chapter, three temperature-displacement correlation relationships models (TDC-AVG, TDC-PCA, TDC-CRI) were established based on average temperature, PCA temperature, and the proposed critical temperature. Linear regression was utilized to derive the mathematical expressions for each model. The goodness of fit was evaluated and residuals were examined.

The fitting residuals (RES) were calculated by equation (11).

R E S_{i} = D_{i} - \hat{D_{i}}

(11)

where

\hat{D_{i}}

represents the displacement predicted by the three models (TDC-AVG, TDC-PCA, TDC-CRI); D_i represents the actual displacement measured; i = 1 or 2 respectively represents the expansion joints located at the west or east bridge tower.

TDC-AVG

A linear regression model was used to establish the displacement-average temperature relationship, as shown in Figure 5 and equations (12) to (13). Due to significant differences in average temperatures among different sections, only the average temperature of the main span section (WDAVG3) is used as a representation of the average temperature, rather than using the average temperature of all sensors. Actually, the goodness of fit calculated from the data used here also validates that utilizing the average temperature of the main span section outperforms using the mean of all sensors or the other four sections. However, for conciseness and coherence, an in-depth discussion is not presented here.

\hat{D_{1}} = - 4.59 T_{a} + 157.86

(12)

\hat{D_{2}} = 5.28 T_{a} - 158.61

(13)

Figure 5.

TDC-AVG model obtained from linear regression: West abutment(left); East abutment(right).

The goodness of fit (R-squared) for equations (12) to (13) is 0.996 and 0.991, respectively.

The distribution of the TDC-AVG model’s fitting residuals, calculated according to equation (11), is shown in Figure 6. According to the principle of least squares, the distribution of the fitting residuals (RES) should follow a normal distribution with a mean of 0.

Figure 6.

Distribution of fitting residuals for the TDC-AVG model: West abutment(left); East abutment(right).

TDC-PCA

As mentioned in Chapter 1, the first principal component of temperature accounts for over 98% of the explanatory proportion. Therefore, this study employs linear regression to establish the relationship between expansion joint displacement and the first principal component of temperature T_pc, as shown in Figure 7 and Equations (14) to (15).

\hat{D_{1}} = - 5.14 T_{pc} + 80.92

(14)

\hat{D_{2}} = 5.51 T_{pc} - 72.24

(15)

Figure 7.

TDC-PCA model obtained from linear regression: West abutment(left); East abutment(right).

The goodness of fit (R-squared) for equations (14) and (15) is 0.978 and 0.989, respectively.

The distribution of the TDC-PCA model’s fitting residuals, calculated according to equation (11), is shown in Figure 8.

Figure 8.

Distribution of fitting residuals for the TDC-PCA model: West abutment(left); East abutment(right).

TDC-CRI

Similar to methods above, linear regression is employed to establish the relationship between expansion joint displacement and critical temperature. This is depicted in Figure 9 and represented by Equations (16) to (17).

\hat{D_{1}} = - 4.62 T_{c, 1} + 161.88

(16)

\hat{D_{2}} = 5.61 T_{c, 2} - 160.80

(17)

Where

T_{c, 1}

and

T_{c, 2}

are calculated by equations (9) and (10).

Figure 9.

TDC-CRI model obtained from linear regression: West abutment(left); East abutment(right).

The goodness of fit (R-squared) for equations (16) and (17) is 0.996 and 0.997, respectively.

The distribution of the TDC-CRI model’s fitting residuals is shown in Figure 10.

Figure 10.

Distribution of fitting residuals for the TDC-CRI model: West abutment(left); East abutment(right).

In summary, the TDC-CRI model demonstrates the highest fitting accuracy for the expansion joint displacement and temperature data among the three models. Its residuals are constrained to a smaller range of variation. Theoretically, this makes the TDC-CRI model better suited for implementing statistical process control charts to evaluate expansion joint performance, with potential advantages over the other two methods. The subsequent chapter conducts a comparative assessment, applying the three modeled approaches to long-term monitoring data from an actual cable-stayed bridge under simulated damage scenarios. This analysis verifies the effectiveness of the proposed critical temperature method for assessing expansion joint integrity based on field measurements.

Damage evaluation of expansion joint based on x-bar control charts

According to the theory of statistical process control, when the performance of the expansion joint is intact, the residuals of the temperature-displacement correlation model should fluctuate within a small range around the centerline (CL), which is zero, and the variance should remain stable within a small range (Montgomery, 2009). However, when the performance of the expansion joint deteriorates or degrades, the distribution of the fitting errors of the model will undergo significant changes, with the mean often deviating from the centerline and the variance becoming larger. Therefore, the parameters of the control chart for analyzing the residuals of the temperature-displacement correlation model under the intact condition can be set, and the performance of the expansion joint under operational conditions can be evaluated based on the determined control chart.

In this chapter, a new simulation method of expansion joint performance degradation caused by damage was proposed. After that, X-bar control charts determined by the three correlation models (TDC-AVG, TDC-PCA, and TDC-CRI) were eventually obtained, and their effectiveness in detecting the predefined damage condition was evaluated.

Simulation of performance degradation of expansion joints

Referring to the research by (Ding and Li, 2011; Huang et al., 2018), it is assumed that an equal decrease in the displacement of the expansion joint during the testing phase represents a damage condition.

D_{d e} = D - e

(18)

where e represents the assumed displacement decrease, and in this study, which is recommended by (Ding and Li, 2011) to be between 2 mm and 10 mm.

The damage condition indicated by equation (18) can be expressed as a percentage formula in equation (19).

D_{d e} = D - ε D_{a n n u a l}

(19)

where D_annual represents the maximum annual variation of the expansion joint, obtained by subtracting the minimum displacement from the maximum displacement over the year. Based on the measured data, the D_annual values for the expansion joints at the west bridge abutment and east bridge abutment are 226.8 mm and 222.6 mm, respectively. Ɛ represents the assumed damage percentage, which is approximately 3.6% for the assumed 8 mm decrease in displacement for expansion joint in both abutments.

In actual operational conditions, the performance deterioration of expansion joints is often caused by rainfall accumulation or debris blockage (Chang and Lee, 2002; Sun and Zhang, 2016), leading to a loss of their free expansion capability. The actual degradation of movement capacity of expansion joint is manifested as greater displacement magnitude corresponding to larger decreases in displacement. Additionally, during the operation of the displacement sensor, there may exist data acquisition software misoperation or recalibration that results in an equal offset in the sensor’s reference reading. In such cases, the displacement sensor data would exhibit effects similar to equation (18), without any actual damage. It is necessary to improve or modify equation (18) according to the actual situation.

To address this issue, the present study proposes an unequal displacement decrease simulation for expansion joint damage, modifying the previous equal decrease approach (Ding and Li, 2011). Specifically, when the displacement D exceeds a certain value D₀, damage causes a decrease in expansion joint displacement. The decrease e increases linearly with the displacement, and the linear growth rate k serves as the characterization index for the degree of expansion joint damage. The modification of equation (18) can be represented using equation (20):

D_{d e} = D - e (D)

(20)

where the calculation formula for e(D) is given by equations (21) and (22), for expansion joint in the west abutment and east abutment respectively.

e (D) = {\begin{cases} 0 & D \leq D_{0} \\ k_{1} (D - D_{0}) & D > D_{0} \end{cases}

(21)

where D₀ represents the displacement threshold at which the performance of the expansion joint in the west abutment begins to deteriorate, which is taken as 60 mm.

e (D) = {\begin{cases} 0 & D \geq D_{0} \\ k_{2} (D - D_{0}) & D < D_{0} \end{cases}

(22)

where D₀ represents the displacement threshold at which the performance of the expansion joint in the east abutment begins to deteriorate, which is taken as −60 mm.

The parameter k_i (i = 1,2) represents the degree of degradation of the expansion joint’s performance, with k₁ representing the west abutment and k₂ representing the east abutment. The values defined for different damage condition are as shown in the table below: Table 1

Table 1.

Damage degree definition for different damage state.

Damage condition	k ₁	k ₂
Intact	0	0
Slight damage	0.05	0.05
Severe damage	0.2	0.2

X-bar control charts of three TDC models

The X-bar control chart is a commonly used statistical tool in statistical process control (Montgomery, 2009). It is used to monitor the stability and consistency of continuous data processes and is widely applied in research related to structural health monitoring ((Farrar and Worden, 2012; Fugate et al., 2001; Sohn et al., 2000); It collects a series of sample data, calculates the average value of each sample, and then plots a control chart based on these average values. On the X-bar control chart, a centerline (CL) is typically plotted to represent the target value or average value of the process, along with upper control limit (UCL) and lower control limit (LCL) to indicate the acceptable range of variation for the process. Sample means falling within the control limits are considered as normal random variation, while samples exceeding the control limits may indicate special causes or systemic issues that require further analysis and adjustment.

Unlike individual control charts that directly use raw data, before applying the X-bar control chart, the residuals of the displacement observations need to be grouped into reasonable subgroups of size n. The sample mean within each subgroup is calculated, and a control chart is plotted based on these sample means. Choosing a subgroup size n that is too large can mask drifts present in the mean. In this study, a subgroup size of n = 6 is adopted, indicating that the average value of residuals is taken every 6 hours.

According to the 3-sigma principle, the residuals of the data in normal conditions should remain within three times the standard deviation range of the centerline. The formulas for calculating the centerline and control limits are as follows:

C L = E (X)

(23)

U C L = E (X) + α σ_{X}

(24)

L C L = E (X) - α σ_{X}

(25)

where X represents the subgroup data with size n = 6, CL represents the mean of the subgroup data and α represents the amplification coefficient of the standard deviation, which depends on the predetermined level of statistical significance.

The residuals of the temperature-displacement correlation models often deviate from an ideal normal distribution, as observed in Figures 6, 8 and 10. To address this, kernel density estimation (KDE) is applied to normalize the non-normal residuals and recalculate the control limits at specified significance levels. Lower significance levels yield wider control charts, reducing false alarms but decreasing sensitivity to abnormal displacements. Considering this trade-off, a 0.01 significance level is selected, allowing 1% false alarm rate under intact conditions.

X-bar control charts are established for the three correlation models under slight and severe damage scenarios, as presented in Figures 11, 12, 13, 14, 15, and 16. The charts facilitate comparison of damage detection effectiveness. All three models provide certain damage warning capabilities. Specifically, the TDC-CRI model produces narrower control limits versus the other two models. For the west abutment expansion joint, all models effectively detect both slight and severe damage, with more out-of-limit points for TDC-CRI indicating higher sensitivity. For the east abutment, TDC-CRI successfully detects even slight damage while the other two models only alarm under severe conditions due to fewer outliers.

Figure 11.

X-bar Control Charts of the TDC-AVG Model for expansion in west abutment: (a) k₁ = 0.05; (b) k₁ = 0.2.

Figure 12.

X-bar Control Charts of the TDC-AVG Model for expansion in east abutment: (a) k₂ = 0.05; (b) k₂ = 0.2.

Figure 13.

X-bar Control Charts of the TDC-PCA Model for expansion in west abutment: (a) k₁ = 0.05; (b) k₁ = 0.2.

Figure 14.

X-bar Control Charts of the TDC-PCA Model for expansion in west abutment: (a) k₂ = 0.05; (b) k₂ = 0.2.

Figure 15.

X-bar Control Charts of the TDC-CRI Model for expansion in west abutment: (a) k₁ = 0.05; (b) k₁ = 0.2.

Figure 16.

X-bar Control Charts of the TDC-CRI Model for expansion in east abutment: (a) k₂ = 0.05; (b) k₂ = 0.2.

In summary, control charts based on the proposed critical temperature maintain higher damage identification rates for both abutments across damage levels, which demonstrates the superiority of the critical temperature method compared to conventional temperature representations for structural health monitoring and data-driven anomaly detection.

Conclusion

This paper proposes a new method for calculating the critical temperature of the main beam based on Lasso dimensionality reduction. A correlation model between expansion joint displacement and critical temperature is also constructed. A modified unequal displacement decrease simulation method is further proposed to simulate different damage severity levels of expansion joints. The performance evaluation of expansion joints is then implemented by establishing X-bar control charts for the residuals of the correlation model under simulated damage conditions. The main conclusions are summarized as follows:

1. A novel temperature characterization approach is proposed to calculate the critical temperature of the main beam using Lasso dimensionality reduction. This provides an effective way to represent the temperature field affecting the displacement of expansion joints. The proposed method is verified on one year of monitoring data from an actual large-span cable-stayed bridge, demonstrating its applicability in practical engineering.

2. The proposed unequal displacement decrease simulation can reflect different damage severity levels of expansion joints, which aligns better with real-world situations compared to equal decrease simulation. The performance assessment integrating this method with different temperature-displacement correlation models validates its superiority in simulating practical damage conditions.

3. The established X-bar control charts based on the proposed critical temperature show higher sensitivity in detecting predefined displacement drops under both slight and severe damage scenarios, exhibiting better performance than charts using traditional temperature representations. This proves the effectiveness of the proposed critical temperature in bridge health monitoring applications.

In summary, the proposed critical temperature calculation and unequal displacement decrease simulation methods provide valuable references for performance evaluation of bridge expansion joints. The techniques established in this study lay a foundation for future research and facilitate the application of advanced analytics in civil infrastructure asset management.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work was supported by the Guangxi Science and Technology Plan Project of China (No. AA21077011).

ORCID iD

Tugang Xiao

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Performance evaluation of cable-stayed bridge expansion joints based on Lasso dimensionality reduction and temperature-displacement-correlation model

Abstract

Keywords

Introduction

Data source description

Representative methods of main beam temperatures

Effective temperature Te

Average temperature Ta

Principal components of temperature Tpc

Calculatin of critical temperature T C based on Lasso method

Establishment of temperature-expansion joint displacement relationship

TDC-AVG

TDC-PCA

TDC-CRI

Damage evaluation of expansion joint based on x-bar control charts

Simulation of performance degradation of expansion joints

X-bar control charts of three TDC models

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Effective temperature T_e

Average temperature T_a

Principal components of temperature T_pc

Calculatin of critical temperature T_C based on Lasso method