Cable effective length model error-based bridge performance warning method under thermal action

Abstract

The cables of long-span cable-stayed bridges are subjected to substantial tension during long-term service and are more susceptible to corrosion and fatigue failure than concrete structures. Most existing structural health monitoring (SHM) systems do not have monitoring equipment to directly measure cable length, and long-term monitoring of the change in cables is less involved. The displacement response of a bridge is induced by the combination of dynamic effects (wind and highways) and quasi-static effects (temperature). In this paper, the dynamic responses were eliminated by averaging the displacement data for 10 min, and the relationship between temperature and displacement was studied. Based on the monitoring data, the distribution of the thermal field for the bridge was studied and the time variability of the tower displacement was investigated. The correlation was analyzed to study the relationship between the temperature and the tower displacements, the north tower–south tower distance and the tower–girder distances. A strong linear relationship between the temperature and quasi-static responses of the displacements was observed. The thermal expansion coefficient of the effective length of cables was proposed as a quantitative index for long-term cable monitoring. The error in the cable effective length is proposed as the warning index for performance warning research. The results show that the proposed performance warning method can monitor cables and perform warnings when the cable is damaged.

Keywords

structural health monitoring cable-stayed bridge thermal field cable effective length performance warning

Introduction

The bridge cable is the most important component of a bridge and serves to ensure the safety of the bridge. The cables of cable-stayed bridges are subject to long-term environmental corrosion, and accumulative material fatigue, which can easily cause performance degradation (Seo et al., 2015; Yi et al., 2013). With the rapid development of sensing technology, monitoring the safety of bridges and cables through structural health monitoring (SHM) systems has become a trend (Yang et al., 2018a; Zhang et al., 2021; Zhou et al., 2021). The SHM system is equipped with structural monitoring sensors including wind speed, wind direction, temperature, etc., and environmental monitoring sensors including deflection, displacement, acceleration, etc. (Kaur et al., 2017; Pei et al., 2019; Yi et al., 2017). Bridge towers are compressive structures, and bridge cables are tensile structures, so it is essential to study the compressive behavior of concrete structures and the tensile behavior of steel structures (Chan et al., 2021; Huang et al., 2020b). Thermal action is an important external input loading for large civil engineering structures (Hu et al., 2019; Yang et al., 2018b; Zhou et al., 2020). Therefore, it is necessary to study the variability of the temperature-induced quasi-static displacement of bridge cables and towers.

The monitoring system of the bridge has installed numerous air and structural temperature sensors to monitor the temperature changes of the bridge, so it is necessary to study the quantitative values and distribution of thermal action itself. Suzuki et al. (2007) monitored a prestressed concrete box-girder bridge and found that the daily variations in temperature in the pavement and upper beam of the bridge were larger deeper from the surface. Peiretti et al. (2014) analyzed the temperature monitoring data of concrete bridge decks for 4 years, and complemented the definition of maximum and minimum temperatures for bridges in locations with daily temperature variations greater than 10 $°C$ in Eurocode 1. Gottsater et al. (2017) proposed that uneven exposure to solar radiation can cause temperature differences between different parts of bridges, which leads to tensile stresses if the parts cannot move freely. de Battista et al. (2015) carried out research on the thermal performance of the Tamar Suspension Bridge deck and found time delays ranging from 10 to 66 min between the air temperature and the structural temperatures. Liu et al. (2016) summarized the 3-D temperature distribution during the paving process of steel bridge pavement and established the temperature field model of a steel box girder according to the transient temperature field theory. Barsotti and Froli (2000) proposed a new statistical method for the thermal action of concrete bridges, which improves the statistical precision and reliability of concrete segmental box-girder bridges.

The temperature variations will cause the bridge to expand and contract in the longitudinal direction and to bend in the vertical plane, which can cause damage to the bridge. Therefore, it is necessary to study the thermally induced behavior of bridge structures to eliminate the influence of the temperature effect on structural deformation. Lee and Kalkan (2012) carried out analytical research to investigate the vertical and lateral thermal gradients of girders and proposed a second-order curve to evaluate the lateral thermal gradients. Tian et al. (2017) proposed a numerical approach to determine temperature effects on train–bridge-coupled dynamics. The results illustrate that the temperature effect on the bridge displacement is due mainly to the increase in the low-frequency component of excitations. There is a correlation between the temperature and the quasi-static displacement of the bridge, so the bridge can be evaluated by analyzing the influence of temperature on the quasi-static displacement (Barr et al., 2005; Mosavi et al., 2012; Xu et al., 2010).

Performance warning research on bridge damage can be carried out after eliminating the influence of environmental effects. Huang et al. (2020a) studied the temperature and strain data of an asymmetric cable-stayed bridge, and proposed a performance warning method or bridge girders based on long-term monitoring data. Wang et al. (2021) proposed an accurate correlation modeling method based on the multi-rate fusion method. The monitoring data of a cable-stayed bridge were studied, and the proposed method can improve the modeling accuracy. Huang et al. (2018) proposed a new representative temperature and referred to this new representative temperature as the canonically correlated temperature. The proposed method is used to study the performance warning of a bridge expansion joint based on temperature and displacement monitoring data. Yang et al. (2018c) proposed a double threshold deflection estimation method with a specific probability. The girder deflection and temperature monitoring data of a highways and railway cable-stayed bridge were analyzed, and the girder deflection of the bridge was estimated. Yan et al. (2005) proposed a bridge damage detection method based on principal component analysis. The analysis of computer-simulated and laboratory testing data shows that the proposed method can effectively eliminate environmental effects. Zhou et al. (2011) proposed a threshold setting method for structural damage warning based on the auto-associative neural network technique. The validity of the proposed warning threshold setting method was verified by the long-term monitoring data of the cable-stayed Ting Kau Bridge. Zheng et al. (2019, 2021) proposed a bridge influence surface identification method based on empirical mode decomposition to study the deterioration of bridges caused by vehicles during long-term service life. Dervilis et al. (2015) proposed a novel detection method based on a nonlinear manifold strategy. The proposed method can prevent false alarms in bridge performance warnings.

Most current SHM systems do not have sensors that directly measure changes in cable length. In addition, there are few studies on bridge tower deformation and its variability under thermal effects. The bridge tower is generally designed as the axial compressive member. However, when the cable is damaged, it can cause considerable damage bending moments in the tower. Considering the adverse effect of thermal action on bridges, a cable effective length model error-based performance warning method is proposed in this paper. The rest of the paper is structured as follows: First, according to the monitoring data of air and structure temperature, the thermal field and the time and space variability of the towers and mid-span are investigated. Based on the bridge tower and girder displacement data, the variability of the bridge girder and tower displacement are analyzed statistically. Second, the correlation between the temperature and the tower displacement, the north tower–south tower distance and the tower–girder distance are analyzed. Third, the thermal expansion coefficients of the north and south effective lengths are calculated and compared with the design value. Fourth, the capability and validity of the proposed cable effective length performance warning method are verified by the monitoring data. Finally, some detailed conclusions are presented on the research in this paper.

Temperature environment and bridge displacement response

Description of the bridge and health monitoring system

Symmetrical cable-stayed bridge monitoring data were analyzed to investigate the bridge thermal field, bridge tower displacement, and tower–girder distance. Then, correlation modeling and performance warning for air temperature and displacement are carried out. The main portion of the bridge is a 5 span (50.0 + 215.0 + 510.0 + 215.0 + 50.0 m) continuous steel box-girder cable-stayed bridge with two towers and two cable planes (Figure 1). Each side of the span is equipped with an auxiliary pier. The cable tower is a reinforced concrete separated inverted Y-shaped structure and the height of the tower is 181.53 m. The bridge deck pavement adopts double-layer SMA asphalt concrete with a total thickness of 75 mm. The bridge opened to traffic in 2004 and has since suffered corrosion, cracking, and other performance degradation. The monitoring system includes eight types of sensors, which can monitor data such as wind speed, wind direction, temperature, displacement, and acceleration in real time. This paper studies mainly the measured data for the structural temperature, air temperature, and displacement of the bridge main girder and towers.

Figure 1.

Elevation of the bridge and sensor placement (unit: m).

The sensor distributions of the monitored bridge used in this paper are shown in Figure 1 and Table 1. Structural temperature sensors and air temperature sensors are installed at the crossbeam at the lower part of the two towers and bridge main girder. Global position system sensors are installed at the bridge main girder and towers. The detailed positions of the structural temperature sensors, air temperature sensors, and GPS sensors are shown in Figure 1. The serial number and the sampling frequency of the sensors are listed in Table 1. The sampling rate of the GPS and air temperature of the bridge monitoring system is 1 Hz. The sampling rate of the structural temperature sensors is 1/300 Hz.

Table 1.

Sensor displacement and specification.

Monitoring subject	Serial number	Position	Sampling frequency (Hz)	Unit
Structural temperature	WD01	North tower (roof of box girder)	1/300	$°C$
	WD02	South tower (roof of box girder)	1/300	$°C$
	WD03	Mid-span (roof of box girder)	1/300	$°C$
Air temperature	WS01	North tower (inside the beam)	1	$°C$
	WS02	South tower (inside the beam)	1	$°C$
	WS03	Mid-span (on the bridge deck)	1	$°C$
Displacement	GPS01	North tower (top of the tower)	1	$mm$
	GPS02	South tower (top of the tower)	1	$mm$
	GPS03	Mid-span (on the bridge deck)	1	$mm$

Air temperature and structural temperature

Air temperature is an important monitoring data point that affects the displacement of bridge towers, so it is necessary to preprocess the air temperature data. In this paper, the air temperature of the bridge in 2014 was pretreated and the characteristics of air temperature were analyzed. Figure 2 shows the preprocessing results of the bridge mid-span and the north tower data in July and August 2014. Figure 2(a) shows the raw air temperature data at the mid-span (WS03). Obvious outliers are seen in the temperature–time curve. The outlier data are invalid monitoring data caused by sensor failure, so the outlier monitoring data should be eliminated first. The outlier data are processed as follows: First, outliers are detected by comparison with adjacent points and setting thresholds; second, reasonable data are calculated by the interpolation method based on adjacent data; and third, outlier data are replaced with differential data. Noise error still exists after the outlier data caused by sensor faults are removed. By averaging air temperature data for 10 min, the noise error can be eliminated effectively. Figure 2(b) shows the bridge mid-span data after eliminating outliers and 10-min averages. Figure 2(c) and (d) are the raw and revised data of the north tower, respectively. Figure 2(e) and (f) are the raw and revised data of the south tower, respectively. It is worth noting that the air temperature sensors WS1 and WS2 are installed inside the beam of the north and south tower, respectively, and the air temperature sensor WS3 is installed on the mid-span deck of the bridge (Figure 1). The highest air temperatures recorded by sensors WS1 and WS2 were 38.37 $°C$ and 38.63 $°C$ , respectively, which are reasonable air temperatures. The highest air temperature recorded by WS3 is 44.77 $°C$ , which exceeds the reasonable value of air temperature. The maximum temperature of WS3 is caused by its installation position (Figure 1) on the bridge deck, which is affected by high temperature of the bridge deck. The highest temperature of the bridge deck can exceed 60 $°C$ .

Figure 2.

Air temperature data in July and August of 2014 for mid-span girder and south tower: (a) raw data at mid-span, (b) revised data at mid-span, (c) raw data at north tower, (d) revised data at north tower, (e) raw data at south tower, and (f) revised data at south tower.

Figure 3 shows the air temperature data of the bridge north tower, south tower, and mid-span after eliminating outliers and averaging for 10 min. We can conclude from Figure 3 that (1) the north tower and the south tower air temperature–time curves are basically coincident, indicating that the air temperature–time curves of the two towers are symmetrical; (2) the mid-span air temperature–time curve is always higher than the air temperature–time curve of the south tower and the north tower; and (3) the air temperature–time curves of the north tower, south tower, and mid-span showed the same tendency in July and August 2014.

Figure 3.

Air temperature–time curves in July and August of 2014 for towers and mid-span girder.

The monitoring system is equipped with structural temperature sensors in the north tower, south tower, and mid-span to monitor the bridge structural temperature. Figure 4 compares the air temperature and structural temperature of the bridge in July and August 2014. Figure 4 shows that the upper envelopes of the structure temperature are obviously larger than the upper envelopes of the air temperature, and the structure temperature coincides with the lower envelopes of the air temperature.

Figure 4.

Monthly variability of the air and structural temperature at mid-span girder in July and August of 2014.

Figure 5 further analyzes the temperature of the structure on the top and mid-span of the bridge tower from January to November 2014 (sensor failure occurred in December). Figure 5 shows that envelopes of the north and south towers coincide and are lower than the temperature envelopes of the mid-span structure. The difference of envelopes between two towers and mid-span are smaller from January to March 2014 and from September to November 2014, while difference of envelopes is larger from April to October 2014.

Figure 5.

Envelopes of daily maximum values of structure temperatures from January to November 2014.

Displacement response of bridge tower and girder

The response of bridge tower displacement is induced by a combination of three major types of loadings due to wind, temperature and highways. This investigation focuses on the temperature effect on the bridge, and temperature tends to cause static deformation of the bridge rather than dynamic deformation. Considering the high varying frequency and short duration of dynamic displacement, we eliminated the dynamic effects by averaging the displacement data for 10 min. Figure 6 shows the displacement–time curve of the north and south towers in July 2014. Figure 6 shows that the longitudinal (x-axis) displacement variability tendencies of the north and south towers are opposite. The displacement–time curves fluctuate slightly compared to the temperature–time curves, which is due to the dynamic effects induced by wind or highways.

Figure 6.

Displacement–time curves of north tower and south tower in July 2014.

The opposite displacement of the north and south tower displacement is caused by thermal expansion and cold contraction of the bridge deck. Two expansion joints are symmetrically arranged on both sides of the bridge. When the temperature rises, the bridge deck on the north side and the north tower move to the north expansion joint, while the bridge deck on the south side and the south tower move to the south expansion joint. This is the main cause of the opposite displacement of the north and south towers. The foundation height of the north tower and the south tower is 51.36 m and 59.43 m, respectively (Figure 1). The different heights of the tower foundations are due to the different depths of the river upstream and downstream. The south tower has a greater displacement because the foundation of the south tower is higher than that of the north tower.

The deflection variability of the main girder was studied using mid-span sensors (GPS3). Two GPS sensors were symmetrically installed upstream and downstream of the bridge mid-span, and the average value of the two GPS sensors is taken as the mid-span deflection. Figure 7 shows the displacement–time curves of the longitudinal (x-axis) direction and vertical (z-axis) direction, which shows that the girder deflection fluctuates periodically. It can be seen from Figure 7(c) and Figure 7(d) that the daily variation trends of 15th, 16th, 17th, and 18th are the same, indicating that both longitudinal and vertical displacements have daily variations.

Figure 7.

Displacement–time curves of mid-span in July 2014: (a) longitudinal (x-axis) displacement in July, (b) vertical (z-axis) displacement in July, (c) longitudinal (x-axis) displacement from July 15th to 18th, and (d) vertical (z-axis) displacement from 15 to 18 July.

Statistical relationship between the temperature effect and displacement

Temperature–displacement relationship of bridge tower

The structure temperature sensors are installed in the bridge box girder, so the bridge box girder is related to the deformation of the expansion joint. Figure 4 shows that the structural temperature is lower than the air temperature. The bridge towers and cables are located on the upper part of the bridge and exposed to air, so we used air temperature for modeling research. In addition, the thermal expansion coefficients of bridge cables obtained by air temperature and cable modeling are closer to the design value in the Implementation Procedures for the Performance Warning section (Table 5). Figure 8 shows the comparison between the time variabilities of air temperature and tower displacement from 15 to 18 July. Figure 8(a) shows that there is a strong correlation between the air temperature and bridge longitudinal displacement (x-axis) in the north tower. As presented in Figure 8(b), air temperature is negatively correlated with bridge longitudinal displacement in the south tower. Figure 9 shows the linear regression between the air temperature and tower displacement. Regression analysis can further quantitatively analyze the correlation. The regression results can be evaluated by calculating the correlation coefficient between temperature and tower displacement

R = \frac{\sum (X_{i} - \bar{X}) (Y_{i} - \bar{Y})}{\sqrt{{(X_{i} - \bar{X})}^{2}} \sqrt{{(Y_{i} - \bar{Y})}^{2}}}

(1)

where R is the correlation coefficient between the structural temperature and tower displacement;

X_{i}

and

Y_{i}

are the

i^{th}

air temperature sample and tower displacement sample, respectively; and

\bar{X}

and

\bar{Y}

are the averages of the air temperatures and tower displacement, respectively.

Table 5.

The thermal expansion coefficient of the bridge cable effective length.

Location	Sample number	$R$	$Δ L / Δ T_{e}$ ( $\times 10^{- 4} m / ° C$ )	$L_{0}$ ( $m$ )	Thermal expansion coefficient ( $\times 10^{- 6} / ° C$ )
North cable effective length (North tower–girder distance)	4464	0.92	35.04	285.11	12.29
South cable effective length (South tower–girder distance)	4464	0.87	33.98	285.11	11.92

Figure 8.

Comparison between the time variabilities of air temperature and tower displacement from 15 to 18 July: (a) north tower and (b) south tower.

Figure 9.

Linear regression between air temperature and tower displacement: (a) north tower and (b) south tower.

Table 2 shows the correlation coefficients of the north and south towers with air temperature. The data points for calculating the correlation coefficient of air temperature and displacement are 576. As shown in Table 2, the correlation coefficient between the temperature and the displacement of the north tower is 0.89 and the correlation coefficient between the temperature and the displacement of the south tower is −0.92. The displacement of the bridge towers has a strong correlation with the air temperature. In addition, the correlation coefficient of the north tower is positive, and the correlation coefficient of the south tower is negative, which also indicates that the displacements of the north and south towers are symmetrical along the mid-span of the bridge.

Table 2.

Correlation coefficients between air temperature and tower displacement.

$X$	$Y$	Sample number	$R$
Air temperature	Displacement of north tower	576	0.89
Air temperature	Displacement of south tower	576	−0.92

Temperature–displacement relationship of the north tower–south tower distance

According to the analysis of the displacement–time curves of the north tower and south tower in the Temperature Environment and Bridge Displacement Response section (Figure 6), the longitudinal displacements of the north tower and south tower are symmetrical. The distance between the north and south towers changes when the towers are damaged, so this section analyzes the relationship between the north tower–south tower distance and temperature. The north tower–south tower distance is the north tower longitudinal displacement (x-axis) minus the south tower longitudinal displacement (x-axis). The equation of the distance between the north tower and the south tower can be expressed as

D_{n s} = {\sqrt{{(x_{n i} - x_{s i})}^{2} + {(y_{n i} - y_{s i})}^{2} + {(z_{n i} - z_{s i})}^{2}}}^{2}

(2)

where

D_{n s}

is the distance between the north tower and the south tower;

x_{n i}

y_{n i}

and

z_{n i}

are the

i^{th}

longitudinal, lateral and vertical coordinates of the north tower;

x_{s i}

y_{s i}

and

z_{s i}

are the

i^{th}

longitudinal, lateral and vertical coordinates of the south tower.

Figure 10(a) is a comparison between the time variabilities of air temperature and north tower–south tower distance. According to Figure 10(a), an identical changing tendency can be observed between the north tower–south tower distance and the air temperature. Figure 10(b) shows the linear regression result, and Table 3 shows the correlation coefficients of the tower–south tower distance with air temperature. Table 3 shows that the correlation coefficient is 0.96, indicating that there is an obvious linear relationship between the tower–south tower distance and air temperature. According to the results in Figure 10 and Table 3, the distance between the north tower and the south tower increases in the longitudinal direction with increasing temperature.

Figure 10.

Correlation between air temperature and north tower–south tower distance from 15 to 18 July: (a) temperature–time curve and (b) linear regression result.

Table 3.

Correlation coefficients between temperature and north tower–south tower distance.

$X$	$Y$	Sample number	$R$
Air temperature	North tower–South tower distance	576	0.97

Temperature–displacement relationship of the tower–girder distance

The cables are important structures of the bridge and there are no sensors for direct monitoring. The length variation is closely correlated to the mechanical behavior of the cable, and the distance from the top to mid-span is affected mainly by temperature. Therefore, it is of great significance to analyze the variation in the tower–girder distance under temperature action. The tower–girder distance represents the length of the outermost cable of the bridge. The equation of the distance between the girder and the north tower can be expressed as

D_{n g} = \sqrt{{(x_{n i} - x_{g i})}^{2} + {(y_{n i} - y_{g i})}^{2} + {(z_{n i} - z_{g i})}^{2}}

(3)

where

D_{n g}

is the distance between the north tower and the girder;

x_{n i}

y_{n i}

, and

z_{n i}

are the

i^{th}

longitudinal, lateral, and vertical coordinates of the north tower; and

x_{g i}

y_{g i}

, and

z_{g i}

are the

i^{th}

longitudinal, lateral, and vertical coordinates of the girder.

The equation of the distance between the girder and the south tower can be expressed as

D_{s g} = \sqrt{{(x_{s i} - x_{g i})}^{2} + {(y_{s i} - y_{g i})}^{2} + {(z_{s i} - z_{g i})}^{2}}

(4)

where

D_{s g}

is the distance between the south tower and the girder;

x_{s i}

y_{s i}

, and

z_{s i}

are the

i^{th}

longitudinal, lateral, and vertical coordinates of the south tower;

x_{g i}

y_{g i}

, and

z_{g i}

are the

i^{th}

longitudinal, lateral, and vertical coordinates of the girder.

Figure 11(a) is a comparison between the time variabilities of air temperature and north tower–girder distance. Figure 11(b) is a comparison between the time variabilities of air temperature and south tower–girder distance. The north tower–girder distance analysis uses the mean of the north tower (WS01) and mid-span temperatures (WS03), while the south tower-girder distance analysis uses the mean of the south tower (WS02) and mid-span temperatures (WS03). Figure 11(a) and (b) show that the changing tendencies of the tower–girder distance and air temperature are identical. Figure 12 shows the linear regression between air temperature and tower–girder distance, and Table 4 shows the correlation coefficients of the tower–girder distance with air temperature. As shown in Table 4, the correlation coefficients of north tower–girder distance and south tower–girder distance with air temperature are 0.93 and 0.92, respectively. There is an obvious linear relationship between the tower–girder distance and the air temperature, indicating that the tower–girder distance will increase with the temperature.

Figure 11.

Comparison between the time variabilities of air temperature and tower–girder distance from 15 to 18 July: (a) north tower and (b) south tower.

Figure 12.

Linear regression of the correlation between air temperature and tower–girder distance: (a) north tower and (b) south tower.

Table 4.

Correlation coefficients between temperature and tower–girder distances.

$X$	$Y$	Sample number	$R$
Air temperature	North tower–girder distance	576	0.94
Air temperature	South tower–girder distance	576	0.93

Implementation procedures for the performance warning

The bridge performance warning under thermal action consists of two stages: (1) while monitoring data during the offline training stage, the 10-minute average was used to eliminate the dynamic effects (wind and highways), correlation modeling was used to eliminate the quasi-static effect (temperature), and the warning index and thresholds were calculated; and (2) while monitoring data during the online monitoring stage, the bridge cable effective length was monitored under temperature actions.

Offline training stage

The displacement monitoring data obtained under the normal service state under thermal actions are used as the training data set. The process is as follows:

Step 1:

Data preprocessing includes detecting and replacing temperature outliers and eliminating noise errors.

Step 2:

The displacement data are averaged for 10 min to eliminate the dynamic effects (wind and highways).

Step 3:

The correlation model of temperature and cable effective length are established to eliminate the quasi-static effect (temperature).

Step 4:

The residual difference between the measured and estimated values of cable effective length are computed as the warning index.

Step 5:

Warning thresholds are computed and set with training data.

Online monitoring stage

Performance warning is carried out when performance degradation occurs in the bridge. The process is as follows:

Step 1:

Data preprocessing includes detecting and replacing current temperature outliers and eliminating noise errors.

Step 2:

The current displacement data are averaged for 10 min to eliminate the dynamic effects (wind and highways).

Step 3:

A correlation model of the current temperature and cable effective length is established to eliminate the quasi-static effect (temperature).

Step 4:

The residual difference between the current measured and estimated values of cable effective length is computed as the warning index.

Step 5:

A performance warning is carried out when the Shewhart control chart exceeds its threshold. If the threshold is exceeded, go to step 6 for manual inspections. If the threshold is not exceeded, go back to step 1 and continue monitoring the cable effective length

Step 6:

Perform warning and manual inspections.

The concluding flowchart of the cable effective length performance warning method is shown in Figure 13.

Figure 13.

Flowchart for implementing the proposed performance warning method.

Performance warning of cable effective length

Thermal expansion coefficient and cable effective length calculation

In general, most of the existing bridge structure health monitoring systems do not have monitoring equipment to directly measure the length of the cables. Compared with concrete structures, cables are more susceptible to corrosion and fatigue failure. In this paper, based on the monitoring data of cable-stayed bridges, the thermal expansion coefficients of cables are proposed as a quantitative index for long-term cable monitoring. In material mechanics, the thermal expansion coefficient (linear expansion coefficient) of a solid material is defined as the ratio of the increase in length per unit temperature change to the original length. The expression of the thermal expansion coefficient is as follows

α = Δ l / l_{0} Δ t

(5)

where

α

is the thermal expansion coefficient, and

l_{0}

is the original length of the material.

Δ l

and

Δ t

are the length change and the temperature change, respectively. Similarly, the thermal expansion coefficient of bridge cables can be defined as

α_{e} = Δ L / L_{0} Δ T_{e}

(6)

where

α_{e}

is the thermal expansion coefficient of the bridge cable.

L_{0}

is the original length of the bridge cable.

Δ L

and

Δ T_{e}

are the length change and the temperature change of the bridge cable, respectively.

As seen from the bridge structure diagram (Figure 1), the tower–girder distance represents the length of the outermost cable of the bridge. Therefore, the outermost cable can be studied through the tower–girder distance. In this paper, the thermal expansion coefficients of the north cable effective length (north tower–girder distance) and south cable effective length (south tower–girder distance) are analyzed. In this section, correlation analysis and fitting between tower–girder distance and temperature in July 2014 are carried out. Figure 14(a) and (b) show the linear regression of the north cable effective length (north tower–girder distance) and south cable effective length (south tower–girder distance) in July 2014. The fitting equations obtained by linear regression are shown in Figure 14.

Figure 14.

Linear regression of the correlation between air temperature and cable effective length in July 2014: (a) north tower and (b) south tower.

Table 5 shows the thermal expansion coefficient of the bridge cables. According to the bridge structure diagram (Figure 1), the original length $L_{0}$ of the cable can be calculated as 285.11 m. $Δ L / Δ T_{e}$ is the coefficient of the first order term in the fitting equation. According to equation (6), the thermal expansion coefficients of the north and south cable effective lengths can be calculated as $12.29 \times 10^{- 6} / ° C$ and $11.92 \times 10^{- 6} / ° C$ , respectively. The thermal expansion coefficient of the bridge strand cables is $12 \times 10^{- 6} / ° C$ . The calculated thermal expansion coefficients and the design value are close to each other, which indirectly proves the correctness of the regression model between tower–girder distance and air temperature. The experimental results show that the thermal expansion coefficient of metal materials varies with temperature, but the variation range is very weak in engineering practice. Therefore, the thermal expansion coefficient can be defined as the health index of the bridge cable. When the thermal expansion coefficient changes greatly in a short time, the cable can be considered to have structural damage and corresponding detection and maintenance measures should be taken.

Warning validity for the performance degradation of cable effective length

The damage identification of long span cable-stayed bridges is carried out by using a cable effective length-temperature model. The residual difference between the measured and predicted values of the converted length of the stayed cable is defined as the structural anomaly index

e = L_{m} - L_{e}

(7)

where

e

is the error of cable effective length-temperature modeling.

L_{m}

is the measured value of cable effective length.

L_{e}

is the estimated value of cable effective length.

This paper simulates the performance degradation of the cable by increasing the deflection in the testing phase. The expression of cable performance degradation is as follows

e_{d e} = e + ε L_{0}

(8)

where

e_{d e}

is the error of the displacement after damage.

e

is the error of cable effective length-temperature modeling.

ε

is the coefficient that adjusts the degree of cable effective length damage, and

L_{0}

is the original length of cable effective length.

In this paper, six damage cases were set to simulate the performance degradation of the cable effective length. Equation (8) was used to simulate six types of performance degradation for the north and south cable effective lengths. Table 6 shows six detailed degrees of degradation of cable effective length. Case 1 in Table 6 is the normal state of the north cable effective length. Case 2 to Case 6 in Table 6 represent a range of 0.5% to 2.5% damage in the north cable effective length. Case 7 in Table 6 is the normal state of the south cable effective length. Case 8 to Case 12 in Table 6 represent a range of 0.5% to 2.5% damage in the south cable effective length.

Table 6.

Performance warning cases of bridge cable effective length.

Case number	North cable effective length (%)	Case number	South cable effective length (%)
Case 1	$ε = 0$	Case 7	$ε = 0$
Case 2	$ε = 0.5$	Case 8	$ε = 0.5$
Case 3	$ε = 1 .0$	Case 9	$ε = 1 .0$
Case 4	$ε = 1.5$	Case 10	$ε = 1.5$
Case 5	$ε = 2 .0$	Case 11	$ε = 2 .0$
Case 6	$ε = 2.5$	Case 12	$ε = 2.5$

The errors of cable effective length were calculated for performance warning research in July 2014, which were 4464 sample numbers. The first 26 days (3744 sample numbers) were used as the training data and the subsequent 5 days (720 sample numbers) were used as the testing data. Three significance levels of 0.05, 0.01 and 0.003 were set to study the effects of the significance levels on cable effective length performance warning. The warning rate was the percentage of the warning sample numbers and the testing sample numbers.

Table 7 and Table 8 illustrate the warning numbers and warning rates for the performance degradation of the north and south cable effective lengths, respectively. Table 7 and Table 8 show that the warning number and warning rate increase with the significance level. As shown in Table 7, the warning rate of the north cable effective length was 100% in case 5, indicating that more than 2.0% of the damage could be detected. As presented in Table 8, the warning rate of the south cable effective length also reached 100% in Case 11, indicating that 2.0% deformation could also be detected.

Table 7.

Warning numbers and warning rates for the performance degradation of the north cable.

Case number	$ω =$ 0.05		$ω$ = 0.01		$ω$ = 0.003
Case number	Alam number	Alam rate (%)	Alam number	Alam rate (%)	Alam number	Alam rate (%)
Case 1	14	1.94	4	0.56	0	0
Case 2	314	43.61	170	23.61	91	12.64
Case 3	678	94.17	597	82.92	507	70.42
Case 4	718	99.72	714	99.17	707	98.19
Case 5	720	100	720	100	719	99.86
Case 6	720	100	720	100	720	100

Table 8.

Warning numbers and warning rates for the performance degradation of south cable.

Case number	$ω$ = 0.05		$ω$ = 0.01		$ω$ = 0.003
Case number	Alam number	Alam rate (%)	Alam number	Alam rate (%)	Alam number	Alam rate (%)
Case 7	12	1.67	7	0.97	2	0.28
Case 8	214	29.72	79	10.97	37	5.14
Case 9	598	83.06	437	60.69	315	43.75
Case 10	714	99.17	706	98.056	674	93.61
Case 11	720	100	718	99.72	716	99.44
Case 12	720	100	720	100	720	100

Figure 15 and Figure 16 show the performance warning results of the north and south cable effective lengths with a significance level of 0.05. The thresholds are calculated from the training data. Figure 15(a) shows that Case 1 is the normal state of the north cable effective lengths. Figure 15(b)–(d) show that the sample numbers of the north cable effective length exceeding the control threshold are increasing. The same result can be obtained for the south cable effective length from Figure 16(b)–(d). As presented in Figure 15(e) and Figure 16(e), all testing data in Case 5 and Case 11 exceed the threshold and the warning rate reached 100%.

Figure 15.

Performance warning results of the north cable effective length: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, and (d) Case 6.

Figure 16.

Performance warning results of the south cable effective length: (a) Case 7, (b) Case 8, (c) Case 9, (d) Case 10, (e) Case 11, and (d) Case 12.

Conclusions

In this paper, the monitoring data of a symmetrical cable-stayed bridge under temperature action were investigated to reveal the relationship between the temperature and displacement. Based on the temperature and displacement data provided by the health monitoring system, the correlation analyses were carried out between the temperature and the tower displacement, the north tower–south tower distance and the tower–girder distance. Then the bridge performance warning was studied using the cable effective length. The conclusions of this study are drawn as follows:

(1) The displacement of towers and girder caused by wind and highways is a kind of high frequency structural dynamic response. However, the displacement of towers and girder caused by temperature is a kind of low-frequency quasi-static structural response. A data preprocessing method is proposed to calculate the 10-minute average of the displacement, and the results show that dynamic effects (wind and highways) and outlier data are eliminated simultaneously.

(2) WS1 and WS2 are installed inside the beam of the north and south tower, while WS3 is installed on the bridge deck. The air temperature recorded by WS3 exceeds the reasonable value due to the influence of high temperature on the bridge deck. Therefore, this paper suggests that the air temperature sensor of the monitoring system should be installed inside the beam rather than the bridge deck to avoid being affected by the high temperature of the bridge deck.

(3) The height of the south tower foundation is greater than that of the north tower foundation, resulting in a greater displacement of the south tower than the north tower. During the long-term service of the bridge, the longitudinal displacement of the south tower is longer than that of the north tower for a long time, which will lead to the destruction of the south expansion joint more easily. The influence of tower foundation height should be considered in the safety design of bridge expansion joints.

(4) In July 2014, the correlation coefficient between the north cable effective length and temperature was 0.92, and that between the south cable effective length and temperature was 0.87. Temperature is the main factor affecting the change of bridge cable length. Therefore, the thermal effects should be considered in the safety design of strand cables.

(5) The outermost cable effective lengths are represented by the tower–girder distances. The thermal expansion coefficients of the north and south cable effective lengths are 12.29×10−6/°C and 11.92×10−6/°C, respectively, which are consistent with the design value of 12×10−6/°C. The proposed thermal expansion coefficients of the cable effective length can be used as a quantitative index for long-term monitoring.

(6) The error of cable effective length is proposed as the warning index for performance warning research. Six damage cases were set up and the control chart was used for performance warning. The warning rates of the north and south cable effective lengths reached 100% in Case 5 and Case 11, respectively. The proposed cable effective length warning method can successfully detect displacement damage with a severity of 2.0% occurring at both the bridge north and south cable effective lengths.

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 jointly supported by the National Natural Science Foundation of China (Grant Nos. 51978128, 52078102), and the LiaoNing Revitalization Talents Program (Grant No. XLYC1802035).

ORCID iD

Ting-Hua Yi

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