A cointegration approach for cable anomaly warning based on structural health monitoring data: An application to cable-stayed bridges

Abstract

For long-span cable-stayed bridges, cables are one of the most important components to resist various actions. With the application of structural health monitoring technique, real-time recording of cable forces is achieved, and hence, the warning system on cable anomaly established. However, it is still difficult and there are challenges to conduct the warning system effectively, especially due to the phenomena of false alarm or omission. A practical reason is the warning index’s sensitivity to the ambient environment. Temperature variations, for instance, usually disturb the force-based cable anomaly warning and result in the false evaluation of structural condition. In view of eliminating the effects of environmental temperature, cointegration, a statistical concept from econometrics, is employed in cable anomaly warning studies. An approach that extracts warning index by linear combination of two non-stationary time series using the cointegration algorithm is developed in order to produce a more stationary cointegrated residual series (warning index series). The calculated stationary relationship between two time series is insensitive to the influence of environmental temperature and is capable of cable anomaly warning. Specifically, the framework of the cable anomaly warning system is first proposed. Subsequently, time-series test methods are introduced to check the non-stationary order and calculate the cointegration parameters of measured cable forces and environmental temperature. The computed cointegrated residual series is fed into statistical analysis as a warning index and the procedure of cable anomaly warning under the influence of environmental temperature is illustrated in detail. Finally, a case study for a cable-stayed bridge is demonstrated with results and discussions.

Keywords

anomaly warning cable force cable-stayed bridge cointegration structural health monitoring

Introduction

Long-span cable-stayed bridges are usually located in critical positions in transport network, such as great rivers or canyons (Mehrabi, 2016). They play important roles in regional communication and development (Liu et al., 2016). Cables are one of the most important components for cable-stayed bridges to resist various actions in complex operational environment. Meanwhile, cables in bridge are facing threats resulting from their own performance degradations and aggressive operational environments (Xu et al., 2019b); cable anomalies may cause serious consequences for the whole bridge.

In view of the operational safety of cable-stayed bridges, it is essential to monitor cable forces measurements that directly reflect the structural conditions and hence study anomaly warning methodology (Liu et al., 2015). Structural health monitoring (SHM) system has gained wide attention from the engineering communities and has been installed in many long-span cable-stayed bridges to constantly monitoring the real-time situation of bridges (Aktan et al., 2000). Measurements of the cable-stayed bridge including cable force and environmental variation can be recorded. Many researchers have attempted to detect anomalies of structural components based on SHM data (Doebling et al., 1998; Huang et al., 2018; Ostachowicz and Krawczuk, 1991). The data-driven anomaly warning methods for long-span cable-stayed bridges have been also studied. Ding et al. (2010) described a multi-stage scheme for structural anomaly warning using the measured dynamic responses from SHM system, the usefulness of the warning method is examined on the Runyang Cable-Stayed Bridge using 236 days of health monitoring data. In addition, a certain number of model-driven approaches to the damage identification in cables have been developed based on vibration measures in the past decades (Degrauwe et al., 2009; Lepidi et al., 2009; Treyssède, 2018; Ubertini, 2014). The physical–mathematical demonstration for damage identification through changes in cable frequencies has been investigated (Lepidi et al., 2007). Xu et al. (2019b) proposed an analytical method for the characteristics of cables with broken wire based on the microstructural mechanical model. The temperature effects on cable response have also been discussed (Lepidi and Gattulli, 2012).

One of the most fundamental problems in anomaly warning is that of removing the environmental variation-induced influence from measured feature data (Chandola et al., 2009). The reason for this is that algorithms used for warning system to detect anomalies should not raise alarms if the component of interest changes because of benign operational or environmental variations, whereas the abnormal changes may be critical for the structure’s safety. As for a cable-stayed bridge, the cable force monitoring results are influenced by several factors including dead weight, vehicle loadings temperature, wind, and so on (Ren et al., 2019). The variation of cable force caused by environmental temperature sometimes masks the anomaly of stay cable itself (Xia et al., 2013). Zhou et al. (2019) studied the thermal response of a long-span cable-stayed bridge from monitoring phenomena to underlying mechanisms. The correlation of cable force and environmental temperature can be approximately fitted by the linear regression method, according to Ren et al. (2019). Based on the monitoring data and numerical method, Wang et al. (2019) separated cable forces induced by temperature variations. In addition, with the development of monitoring techniques, a type of fiber Bragg grating (FBG) force-testing ring that achieves effectively a higher accuracy of cable force measurements as well as temperature compensation is proposed for the long-term cable force monitoring (Li et al., 2015). The existing cable anomaly warning systems usually determine the threshold directly based on statistical characteristics of cable force values. The phenomena of false alarm or omission always happen for warning index’s sensitivity to the environmental temperature. In consideration of the limitations of cable anomaly warning, more effective methods need to be explored.

A core link of structural warning systems is to seek indices which are representative of anomaly, a subject extensively investigated in past decades (Hsieh et al., 2006). As a statistical concept, cointegration is now extensively used to model the long-term common trends among economic variables in the field of econometrics. Recently, cointegration has been successfully implemented in the context of SHM, where it has been used to remove the confounding influences of operational variations (especially the environmental temperature) that can often mask the signature of structural anomaly (Cross and Worden, 2011). Peeters and De Roeck (2001) studied SHM data of the Z24-Bridge in Switzerland, and it is concluded that temperature may have an impact on the boundary conditions and the material properties and raise an alarm in the monitoring system (i.e. a false positive). Then, Shi et al. (2018) established a nonlinear cointegration method and examined with real SHM data from the Z24-Bridge; analysis results demonstrated that the environmental and operational variations can be neatly eliminated. Dao and Staszewski (2018) demonstrated the method based on the cointegration technique and fractal signal processing to remove the undesired temperature effect from Lamb wave data in the purpose of structural damage detection. For cable-stayed bridges, Liang et al. (2018) proposed a frequency-based cointegration technique to eliminate the influence of the changing environmental temperature and identify the structural damage, which is verified by a practical application on Tianjin Yonghe cable-stayed bridge.

In theory then, the cointegration process is ideally suited to remove environmental temperature from monitoring cable force data (Cross et al., 2011). Considering the requirements of engineering applications, this article aims to find patterns in cable force data that do not confirm to the expected behaviors by developing a cointegration-based framework of cable anomaly warning system. Subsequently, the complex mathematics behind the cointegration process will be introduced in a comprehensive manner with the purpose of providing a self-contained paper tailored. Meanwhile, the procedure of cable anomaly warning under the influence of environmental temperature is illustrated in detail. Finally, a case study is presented to illustrate the proposed cointegration-based cable anomaly warning method, includes summary and discussions.

Methodology for cable anomaly warning

In this article, the proposed methodology is raising alarms on the calculated warning index residual series, with a pre-processing of cointegration on input SHM measurements. The general framework in Figure 1 gives a visualized procedure of cable anomaly warning system, whose details are given in the following.

Figure 1.

Framework of the cable anomaly warning methodology.

First, the preliminary analysis is conducted on the monitoring cable force (indirect measurements such as vibrational frequency should be transferred to force data) and environmental temperature data for the following utilization. For each collected cable force series, the correlation with nearby measurements as well as available environmental temperature is respectively studied as basic checking. Then, the cable anomaly warning system is operated with cointegration approach based on input data. Basically, cointegration processing can be divided into three main parts: stationarity test, cointegration test, and cointegrated residual series calculation. The threshold is simultaneously determined by statistical means. Finally, the warning system achieves to raise alarm when cable anomaly is detected.

Theory of cointegration

The basic theory of cointegration is originally proposed by Engle and Granger (1987) as a typical method for dealing with non-stationary time series in the field of econometrics. Cointegration is now widely used in statistical arbitrage, macroeconomic analysis, and fiscal policy research. The central idea is that even if two or more variables are non-stationary, the combination of them will create a new stationary series. This stationary linear combination is defined as cointegration, which means that the variables can be represented to have a stable relationship known as a long-term equilibrium.

The link between the econometric method and cable anomaly warning based on SHM data is the existence of common stochastic trend. These characteristics can be understood as that economic time series are simultaneously affected by markets, policies, and so on, while the recorded force of each cable is significantly influenced by environmental temperature. Non-stationary series are cointegrated if there exists a linear combination of them that is stationary. Recently, cointegration has been adopted successfully to address the challenge of environmental variation in structural monitoring (Worden et al., 2016).

In the definition of cointegration, it is assumed that the input variables series are ${X_{1}}$ , ${X_{2}}, \dots, {X_{k}}$ , the corresponding response variable series is ${Y_{t}}$ , and the regression model is expressed as

Y_{t} = β_{0} + \sum_{i = 1}^{k} β_{i} X_{i} + ε_{t}

where $β = (β_{0}, β_{1}, \dots, β_{k})$ are cointegrating vectors and $ε_{t}$ is the regression residual. If the regression residual series ${ε_{t}}$ is stationary, there exists cointegration between ${Y_{t}}$ and ${X_{1}}$ , ${X_{2}}, \dots, {X_{k}}$ . Therefore, the eventual aim of cointegration process is to identify the cointegration vector to ensure the residual series stationary.

For a stationary variable series ${Y}$ , it is called integrated of order 0 if the variable itself is stationary, denoted as ${Y} \sim I (0)$ . As for a non-stationary variable series, if it becomes stationary after d times difference operation, it is integrated of order d, which indicates d unit root is existed in series ${Y}$ , denoted as ${Y} \sim I (d)$ . To obtain the cointegration vectors β, the order of integration should be determined at the beginning, since the cointegration procedure can only operate for the non-stationary variables with the same order of integration.

Subsequently, the cointegration vectors β can be computed from the cointegration test. The integration tests can be classified to regression parameter-based method (such as Johansen test and so on) and regression residual based method (such as the Engle–Granger (E-G) test and so on). In this article, the E-G test is introduced since only two types of variables series are related, namely environmental temperature and cable forces data (Johansen, 1988; Shin, 1994).

In addition, four key points should be considered in the cointegration procedure (Liang et al., 2018; Pena and Box, 1987; Tiao and Box, 1981): First, the cointegration vectors β to represent the relationship between non-stationary variables are not unique. Second, the variables series should have the same integrated order for the operation of a cointegration procedure. Third, there are at most n – 1 cointegration vectors with linear independence. Fourth, the variables series obey a similar series trend.

Non-stationarity and unit root test

With regard to non-stationarity, researchers have developed different methods for non-stationary time-series testing while the unit root test is the most proven one. It has been demonstrated that the time series will be non-stationary if a unit root existed. The unit root test used in this article is the augmented Dickey–Fuller (ADF) test (Cross et al., 2011). To illustrate how the ADF test works, the argument will start from the form of a first-order autoregressive model AR(1), which can be formulated as

y_{i} = a_{1} y_{i - 1} + ε_{i}

where $ε_{i}$ is the Gaussian white noise series of independent normal random series with mean value of 0 and variance of $σ^{2}$ ; a ₁ is a real number and its value defines the root of the characteristic equation of the process in the above case. The root of the characteristic equation of any process determines its stationarity. The process y_i is non-stationary when a ₁ is higher or equal to 1 in magnitude and it is stationary when a ₁ is lower than 1 in magnitude. When a ₁ = 1, there exist a unit root in the process and equation (2) changes to

y_{i} = y_{i - 1} + ε_{i} \to Δ y_{i} = ε_{i}

The non-stationary process becomes stationary after 1 time difference operation, namely $y_{i} \sim I (1)$ . The information about stationarity of the process can be received from the defining characteristic root parameters when a process is fitted to the model AR(1). This is commonly conducted by testing a null hypothesis of $a_{1} = 1$ . Then, the t-test statistic is carried out and be compared with critical values that constructed by Dickey and Fuller (1979).

The ADF test follows the same principle as expounded above. The data are fitted into a more complex time-series model, which is given as

Δ y_{i} = ρ y_{i - 1} + \sum_{j = 1}^{p - 1} b_{j} Δ y_{i - j} + ε_{i}

where Δ is the difference operator with the definition of $Δ y_{i - j} = y_{i - j} - y_{i - j - 1}$ , ρ is a real number, b_j is a coefficient of the model, and p is the number of lags, which is expected to insure that $ε_{i}$ becomes a white noise series. The following substitutions should be made to transform AR(p) model to the form of equation (4), which are

{\begin{cases} a_{1} = 1 + ρ + b_{1} \\ a_{n} = - b_{n - 1} + b_{n} \end{cases}, for n = 2, \dots, p - 1, and a_{p} = - b_{p - 1}

where a_j are coefficients of the AR(p) model. The characteristic equation (4) can be gathered from the characteristic equation of an AR(p) process with the above substitutions, which is

1 - λ^{- 1} - ρ λ^{- 1} = \sum_{j = 1}^{p - 1} b_{j} (1 - λ^{- 1}) λ^{- j}

where λ are roots of equation (6). Independent roots with a number of p are exited in time-series model as equation (4). If at least one root of the characteristic equation is unity, it can be drawn from equation (6) that $ρ = 0$ . When there is only one unit root and consider remaining (p - 1) roots of the characteristic equation. For $ρ = 0$ , equation (6) is then demonstrated as

1 - \sum_{j = 1}^{p - 1} b_{j} λ^{- j} = 0

When y_i has one unit root, the values of all the remaining roots are smaller than 1. Meanwhile, equation (7) is the characteristic equation of differenced time-series AR model, which is

Δ y_{i} = \sum_{j = 1}^{p - 1} b_{j} Δ y_{i - j} + ε_{i}

In view of that all roots in equation (7) must smaller than 1, the first difference $Δ y_{i}$ is stationary. For y_i to be integrated of order one, the requirement that $ρ = 0$ in equation (4) is necessary. Additional hypotheses and test statistics are needed if the form of time-series regression model includes constant or trends term. The ADF procedure depends on estimating the following model

Δ y_{i} = c + a t + ρ y_{i - 1} + \sum_{j = 1}^{p - 1} b_{j} Δ y_{i - j} + ε_{i}

where c is the constant term and a is the trend coefficient. To sum up, the aim of the ADF procedure is to estimate the parameters in equation (4) and test the null hypothesis of $ρ = 0$ . The test statistic $t_{ρ}$ is conducted as follows

t_{ρ} = \frac{\hat{ρ}}{σ_{ρ}}

where $\hat{ρ}$ is the least squares estimate of ρ and $σ_{ρ}$ is the variance estimate. The result of the hypothesis test is obtained by comparing $t_{ρ}$ with the critical value in the ADF statistic table. If the hypothesis is accepted, time series $y_{i} \sim I (1)$ . If the hypothesis is rejected, ADF procedure is continued for the each difference of y_i , until the order for integration of y_i is ascertained.

Cointegration test

When the non-stationarity order of variable series is determined, an attempt to create a stationary residual series through integrating variable series to the same order is made. For this purpose, the so-called the E-G procedure based on the regression residual is then outlined with 2 steps.

First step: estimating the cointegration regression model

It starts by estimating the cointegration regression equation, as given in the following

x_{1, t} = β_{1} + β_{2} x_{2, t} + \dots + β_{p} x_{p, t} + u_{t}

where p is the number of variables in regression equation, $β_{i}$ is the regression coefficient of variables, and u_t is the cointegration error. Assuming that all the variables series are non-stationary and accord with $I (1)$ ; hence, a stationary residual series is given as follows

{\hat{u}}_{t} = x_{1, t} - β_{1} - β_{2} x_{2, t} - \dots - β_{p} x_{p, t}

When variables series are cointegrated, they will share a common trend and form a long-run stationary relationship.

Second step: test for a unit root in the residual process of the cointegration regression

The stationary test for residual series ${\hat{u}}_{t}$ will indicate the result of cointegration test for variables series; therefore

Δ {\hat{u}}_{t} = α + π {\hat{u}}_{t - 1} + \sum_{i = 1}^{k} γ_{i} Δ {\hat{u}}_{t - i} + υ_{t}

where α, π, and γ are coefficients of the model corresponding to the cointegration equation and $υ_{t}$ is the residual series of cointegration regression. Once the residual series ${\hat{u}}_{t}$ passes the ADF test and is confirmed as stationary, a cointegration relationship is existed between variables. Under the null of no cointegration, the estimated residual is $I (1)$ since that $x_{1, t} \sim I (1)$ and all parameters are 0. Thus, the coefficient π implies cointegration (Osarumwense and Mbegbu, 2017). The test statistics is

\begin{array}{l} H_{0} : π = 0 No cointegration \\ H_{a} : π < 0 Cointegration \end{array}

Warning index of cable anomaly

Various warning indexes regarding the actual condition and historical monitoring data about cable anomaly for cable-stayed bridges have been studied in recent years. However, the most widely adopted method by now is still directly using the cable force measurement. For instance, according to China’s evaluation standards of highway bridge (JTG/T H21-2011), cable anomaly can be controlled by monitoring the variation rate of the cable forces for a single cable (Liu et al., 2018). Anomaly for a stay cable may happen in the case of over 10% variation compare to initial data is detected. In AASHTO standard of America, similarly, cable force is regarded as anomaly warning index and the 75% limit value of normal condition is set as warning threshold. However, it is difficult and faces challenges to conduct the cable anomaly warning system effectively in engineering application, which is largely due to the phenomena of false alarm or omission. Figure 2 shows the application of the cable anomaly warning methodology that commonly used nowadays. Minutely monitoring data within 1 day from a long-span cable-stayed bridge are plotted in Figure 2(a), including cable force of a typical cable on the bridge and environmental temperature. Suspect omission happens at midnight and early morning (about 0:00 to 6:00 a.m.) while false alarm happens in the afternoon (about 3:00 to 6:00 p.m.). To investigate the reason of failure warning and find a solution of effectively cable anomaly warning, the relationship of cable force and environmental temperature is also analyzed, as shown in Figure 2(b). The Pearson correlation coefficient (PCC) between two variables is 0.643. Figure 2 indicates that influence of environmental temperature variation debases the effectiveness of cable anomaly warning systems.

Figure 2.

Application of the common cable anomaly warning methodology: (a) warning result for a typical cable-stayed bridge and (b) correlation of environmental temperature and cable force.

One of the most fundamental problems in anomaly warning is that of filtering out environmental variations from measured feature data. In view of the cable anomaly warning system should not raise alarms if cable force changes because of benign environmental temperature variations, whereas the abnormal changes are critical for the bridge’s safety. A core aim of cable anomaly warning is to seek index which is representative of cable anomaly. A novel cointegration approach for obtaining warning index is developed in this article. Based on the cointegration theory, the stationary residual series ${\hat{u}}_{t}$ (see equation (12)) can be calculated through cable force and environmental temperature series, see Figure 3.

Figure 3.

Obtaining of warning index.

The essential principle is that the acquired cable force data influenced by the variation of environmental temperature can be transformed to stationary series by combining two non-stationary monitoring time series using the cointegration algorithm. In this article, the stationary residual series ${\hat{u}}_{t}$ created from the cointegration procedure is extracted as a cable-anomaly-sensitive feature that is independent of the environmental conditions. Several successful applications involve cointegration-based index have been adopted in the field of SHM, mechanics, and industrial (Chen et al., 2009; Zolna et al., 2016); the results show the advantages of steady and sensitiveness to environmental changes.

Threshold determination

According to China’s evaluation standards for cable-stayed bridge, the thresholds for warning index are usually set directly based on the monitoring cable force data. In view of the novel warning index developed in this article, the statistics-based methodology is introduced to determine the warning threshold.

The cable anomaly warning technology is based on the actuality that the warning index will obviously exceed the normal variance range. Therefore, cable anomaly warning for a complex structure can be simplified to outlier identification in digital signal processing. Usually, the anomalous data instances are defined as those signals outside the threshold. In the field of SHM, the Pauta criterion (also known as 3σ criterion) is a well-known statistical method to calculate abnormal boundaries of warning index based on the distribution of parameters under different operational conditions (Li et al., 2019; Liang et al., 2018; Shi et al., 2019; Zhu et al., 2019). The Pauta criterion is applicable to large sample data such as SHM measurements, mean value, and standard deviation of the index are first calculated. Then an interval is determined according to a certain confidence probability (generally 99.7%, corresponding to μ ± 3σ), see Figure 4. It is considered that the warning indexes that exceed this interval are abnormal data, namely the cable anomalies.

Figure 4.

A diagram of the Pauta criterion.

In addition, cable force in an actual cable-stayed bridge will change gradually for the variance of operational condition and cable performance degradations. For instance, Liu et al. (2018) revealed the degradation of cable force from 2006 to 2012 for two adjacent cables on the Third Nanjing Yangtze River Bridge, as shown in Figure 5. The effectiveness of anomaly warning will be sharply weakened using lagged warning thresholds under current service status. It is important to update the threshold periodically using the latest monitoring data. In this article, it is suggested to update the warning threshold annually.

Figure 5.

Change of force data for two adjacent cables from 2006 to 2012.

Case study

This study investigates the application of cointegration-based cable anomaly warning from a typical cable-stayed bridge, namely, the Third Nanjing Yangtze River Bridge. In total, 240-day valid environmental temperature and cable force measurements in 2007 are utilized for the statistical studying (including 11 months except June), since part of SHM data during 2007 unavailable. More data are supplied for case studying including monitoring data in disaster conditions and analog signals. Meanwhile, this study also verifies that the essential requirements for management are met when emergencies occur.

Descriptions of the bridge its monitoring system

The Third Nanjing Yangtze River Bridge, a long-span cable-stayed bridge with a main span of 648 m across the Yangtze River in Nanjing city, Jiangsu Province, China. The bridge located at the latitude of 32.0°N and the longitude of 118.6°E. It was open to traffic in 2005, a sophisticated SHM system was then established in 2006 involving several monitoring items related to bridge operational safety. The geographical location of the Third Nanjing Yangtze River Bridge is shown in Figure 6(a). The superstructure has a 3.2 m deep and 37.5 m wide steel box girder that accommodates three traffic lanes for each direction. The deck is supported by a total of 168 stay cables and each cable consists of 109 to 241 wires having a diameter of 7 mm.

Figure 6.

Location and sensor layouts of the Third Nanjing Yangtze River Bridge: (a) geographical location and (b) sensors layout on the bridge.

Sensors installed on the bridge include anchor load cell (ALC), which for receiving cable force measurements. The air temperature and relative humidity sensors (RHS) are designed and mounted on the bridge for measuring the environmental temperature. Meanwhile, other sensors were also designed to provide comprehensive data. The overall layout of sensors on the bridge is shown in Figure 6(b). The ALC sensors are installed in the anchors of stay cables that achieve measuring cable force directly. The total of 168 stay cables are all equipped with ALCs with an acquisition frequency of 10 Hz and a relative error of ±1%. According to Ren et al. (2019), the ALC sensors have a higher precision compared to the other cable force measurement approaches (e.g. vibration frequency measurement). The measurements of ALC sensors will not be significantly influenced by pretension forces if the supports (below and above the load cell) provide enough rigidity. However, due to the degradation of the material and uneven tensions, the measurement precision will reduce over time. In addition, ALC sensors that are difficult to be replaced for the sensors are embedded in the anchor system. A total of six R. M. YOUNG-41372 thermometers are installed around two towers with the measure range of –50°C to 50°C and the precision of ±0.3%. The environmental temperature, temperature inside the girder, and towers are respectively monitored with an acquisition frequency of 10 Hz.

Characteristic analysis of the monitoring data

In order to comprehend the SHM data of cable force and environmental temperature from the Third Nanjing Yangtze River Bridge, 240 daily measurements in 2007 are collected and plotted with respect to time history (note that missing data are all removed), as shown in Figure 7. In consideration of the symmetry for the studied cable-stayed bridge, quarter cable planes at the upstream of the south tower are investigated (total 42 cables). The cables are marked the same as actual situation, which are from SA21 to SA1 and SJ1 to SJ21, as illustrated in Figure 7. The cable force time series of all cables share a similar variation tendency. As for environmental temperature, it reaches a higher value of over 30°C in summer and reduces in winter.

Figure 7.

Parts of monitoring cable force series and environmental temperature series plotted against time.

On further examining the mutual relationship between cable force and environmental temperature, correlation of two variables is analyzed for different placements. In this article, several typical cables are selected as examples to expound the proposed cointegration-based cable anomaly warning methodology. In view of the optimal ALC sensors placement for cable force monitoring (Li et al., 2018) and the critical control positions of bridge according to management department, six typical cables with marks of SA21, SA13, SA1, SJ1, SJ10, and SJ21 are taken out for studying and further anomaly warning procedure. The correlations of environmental temperature and cable forces of the selected cables are shown in Figure 8, and PCCs are also calculated for the selected samples. The cable location serves as a factor that obviously influences the PCCs. The PCCs are Cable SA21 = 0.901, Cable SA13 = 0.973, Cable SA1 = 0.745, Cable SJ1 = 0.568, Cable SJ10 = 0.722, and Cable SJ21 = 0.983, respectively. In general, the force of long cable close to 1/2 mid-span (such as Cable SJ21) shows the most significant correlation with environmental temperature. That characteristic about correlation may be explained as follows: force measurements of the short cable (such as Cable SJ1) are substantially influenced by its boundary condition. According to Lepidi and Gattulli (2012), stronger correlation between two variables holds on only for strongly prestressed cables. Based on the results of the E-G test, cointegration procedure is still workable for these short cables. Linearity is not a necessary postulate for the effectiveness of the proposed cointegration-based method.

Figure 8.

Correlation of environmental temperature and cable forces of the selected typical cables from the Third Nanjing Yangtze River Bridge.

Cable anomaly warning procedure

In order to demonstrate the availability of the proposed methodology, recorded measurements in the practical engineering are used for cable anomaly warning. The aim is to build damage indexes based on the healthy state of the Third Nanjing Yangtze River Bridge. Following the steps in the warning system and based on software MATLAB, the cointegration is conducted including the E-G procedure and ADF test. Then cointegrated residual series are obtained to use estimated coefficients produced in cointegration processing, as given in the following form

I_{w} = [\begin{matrix} y_{C i} & y_{T i} \end{matrix}] \times [\begin{matrix} 1 \\ - b \end{matrix}] - a

where I_w is the warning index, y_C is the cable force series, y_T is the environmental temperature series, and a and b are estimated coefficients produced in cointegration processing. The cable anomaly warning results of six cables selected above are shown in Figure 9. Two red horizontal dashed lines indicate three standard deviation intervals, which represent warning thresholds. As demonstrated in Figure 9, the cointegration residual that acts as warning index becomes more stationary than the recorded original cable force measurements. The influence of environmental temperature is eliminated; therefore, the anomaly information is revealed. According to the proposed methodology in this article, the potential cointegration relationship changes; therefore, the warning index (cointegrated residual series) will show a significant indication of anomaly, as the warning index becomes nonstationary again.

Figure 9.

Cable anomaly warning results of six selected cables: (a) Cable SA 21, (b) Cable SA 13, (c) Cable SA 1, (d) Cable SJ 1, (e) Cable SJ 10, and (f) Cable SJ 21.

In actual cable-stayed bridges, these nonconforming data patterns may result from structural damages or accidents during operational stages. For instance, the anomalous scenario may be overloading vehicles across the bridge or other sudden disasters, not limited to structural damage. In consideration of verifying the effect of proposed cable anomaly warning method for bridge in improper loading, disaster and damage conditions, three scenarios are simulated or observed to validate the effectiveness of the anomaly detection method, which are

Case 1: Adding a 6-ton overloaded vehicle in the normal traffic flow on 1 February, which should be raised alarm as the anomalous operational event.

Case 2: Supplying additional SHM measurements from January to March in 2008, which including the famous snow disaster happened in China.

Case 3: Damage of cable wire with a fracture area of 1% is introduced on 3 July (see Figure 10), the change of cable force is achieved by finite element (FE) method.

Figure 10.

The view of Case 3: damage of the cable with an area of 1%.

Aiming to build anomalies, a FE model as shown in Figure 11 is established by software MIDAS/CIVIL for analog signal implementation. Beam and truss elements are used to simulate the girder and stay cables. There are a total of 962 nodes and 1128 elements (960 beam elements and 168 truss elements) in the model. The model is also calibrated to ensure it is reliable, refer to Xu et al. (2019b). With the FE method, Case 1 is achieved by adding vehicle weight (nodal load) in relevant position while Case 3 is achieved by reducing the cross-sectional area of cable element. To be brief, only Cable SJ21 is selected to do further investigation in the above three cases, since this cable is close to the location of 1/2 mid-span and usually be regarded as the critical cable for bridge management department. When a 60-ton overload goods vehicle crossed past the position of Cable SJ21, the additional cable force induced by it is 56.71 kN, based on the FE model as shown in Figure 12.

Figure 11.

The FE models of the Third Nanjing Yangtze River Bridge.

Figure 12.

The diagram of Case 1: an overload goods vehicle passes through the bridge.

Meanwhile, it is important to note that only the data before introducing simulative scenarios are used for estimation, namely the same time series set from above are used for statistical calculation purpose.

The anomaly warning results of Cases 1 and 2 are shown in Figure 13, which eliminate the influence of temperature. In Case 1, a 60-ton overload vehicle goes through the bridge on 1 February (in the winter with low environmental temperature). The response of the warning index for Cable SJ21 reaches 61.78 that exceeds the upper threshold of 14.85, and the anomalous operating state is successfully warned. However, corresponding original cable force measurement increases from 3577.06 to 3633.77 kN that even lower than many normal measurements in summer (maximum daily value is 3654.75 kN on 3 August 2007). The conventional cable anomaly warning method based on original measurements may lead to omission. In the same vein, false alarm may happen in summer corresponding to comparative higher environmental temperature and cable force. In Case 2, the effectiveness of the proposed cable anomaly warning methodology is tested by the famous great snow disaster in early 2008, south China. The great snow begins on 26 January and continues for 5 days; a countermeasure of bridge closure is taken for the first 2 days. It is clear from Figure 13 that warning indexes significantly increase during disaster with a maximum value of 128.47 (upper threshold is 14.85) while only about 4.3% increase for cable force measurement (from 3555.83 kN before disaster to maximum 3710.21 kN). Once disaster occurs, the anomaly information for Cable SJ21 is revealed.

Figure 13.

Cable anomaly warning result in Cases 1 and 2.

In Case 3, the cable anomaly warning result under 1% damage area is shown in Figure 14. The influence of cable wire damage is artificially introduced by reducing the cross-sectional area was reduced. As shown in Figure 14, a significant jump is observed for the warning index amplitude under the cable damage condition. It demonstrates that this anomaly can be detected accurately by employing the proposed cointegration-based cable anomaly warning methodology, such that after the cable damage introduction, the value of warning indexes goes beyond the threshold (three standard deviation intervals). As for conventional warning method, when cable damage area is not high enough, anomalous information may be overwhelmingly masked by the variations caused by environmental temperature.

Figure 14.

Cable anomaly warning result in Case 3, the black dashed line indicates where damage is simulated: (a) cable force data and (b) residual series calculated using the cointegration approach.

To sum up, all the investigation results in case study of the Third Nanjing Yangtze River Bridge suggest that the proposed methodology for cable anomaly warning clearly retains the power to eliminate the effect of the environmental temperature variation and receives greater accuracy than conventional warning method. The result is encouraging, in that all three anomalous scenarios are warning successfully. However, there are still some restrictions on the current approach, which will be future directions for the authors.

Conclusion

In this article, a cointegration approach for cable anomaly warning based on SHM data is proposed and then be applied the methodology to a typical cable-stayed bridge, namely the Third Nanjing Yangtze River Bridge. The following conclusions can be drawn from this study:

The cointegration approach method is applied to eliminate environmental temperature effects from the original monitoring measurements. Based on statistical theories, residual series is calculated with the cointegration procedure including ADF test and the E-G procedure, which is introduced as warning index series.

Framework of cointegration-based cable anomaly warning is proposed with the input–output mechanism. The warning index is calculated by three main parts of cointegration procedure while threshold is simultaneously determined by the Pauta criterion based on monitoring data. Moreover, the warning threshold is required to be updated periodically to consider the variance of operational condition and cable performance degradations.

The proposed cable force anomaly warning method is applied to the Third Nanjing Yangtze River Bridge. With monitoring data, anomalous scenarios of the stay cable are successfully identified. The proposed cointegration-based anomaly warning method has the advantages to effectively eliminate environmental temperature effects in operational conditions and has potential in this field, as demonstrated in the Third Nanjing Yangtze River Bridge in this contribution.

Footnotes

Acknowledgements

The authors thank the anonymous reviewers for their constructive comments and advice, which greatly improved the quality of this article.

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 study was supported by the Natural Science Foundation of Jiangsu Province under grant no. BK20181278, Transportation Science Research Project in Jiangsu under grant no. 2019Z02, the Postgraduate Research & Practice Innovation Program of Jiangsu Province under grant no. KYCX19_0099, and the Fundamental Research Funds for the Central Universities.

ORCID iDs

Zi-Yuan Fan

Xiang Xu

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