# Evaluation of the Wind-Resistant Performance of Long-Span Cable-Stayed Bridge Using the Monitoring Correlation between the Static Cross Wind and Its Displacement Response

Authors: Gaoxin Wang; Youliang Ding; Dongming Feng; Jihong Ye
Journal: Shock and Vibration
 (2018)

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## Abstract

The wind-resistant performance of existing long-span cable-stayed bridge structures will inevitably deteriorate due to concrete carbonation, reinforcement corrosion, accumulated fatigue damage, etc., which can threaten the entire bridge safety. However, few studies currently focus on assessing the wind-resistant performance of existing long-span cable-stayed bridge structures, and at present, no studies have revealed how to identify the deterioration of the wind-resistant performance using a bridge health-monitoring technique. Therefore, based on the health-monitoring system installed on the Sutong Cable-stayed Bridge, the monitoring wind field and GPS displacement are captured to analyze the wind-resistant performance. First, the correlation between static cross wind and transversal displacement is analyzed, which is nearly linear but also contains discrete points caused by environmental noise. Second, considering that the discrete points can decrease the identification accuracy, one new method called the cross-correlation analysis of wavelet packet coefficients is put forward to effectively remove the discrete points. Third, considering that the traditional function cannot match the monitoring correlation very well, some new fitting functions are thoroughly studied to determine the best function for fitting the monitoring correlation. Fourth, the abnormal variation in the monitoring correlation caused by a deterioration in the wind-resistant performance is studied, and the root-mean-square (RMS) variable, which represents the difference between a good service state and a deteriorated service state, is used as a detection indicator to identify the deterioration of wind-resistant performance. Finally, the monitoring data from ten months are selected to evaluate the wind-resistant performance of the Sutong Bridge, and the result shows that its wind-resistant performance was still in a good service state during these ten months.


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## Body

## 1. Introduction

The wind-resistant performance of existing bridge structures will inevitably deteriorate due to the influence from synthetic factors, such as concrete carbonation, reinforcement corrosion, and accumulated fatigue damage, which can threaten the entire bridge safety [1–5]. For example, the boundary beam of the SR1014 Bridge in Pennsylvania laterally collapsed mainly due to serious performance deterioration [5]. Additionally, according to the relevant investigation results [5], the percentage of American bridges that have collapsed due to performance deterioration (mainly including steel brittle fracture, accumulated fatigue damage, and prestressed reinforcement corrosion) is 18.75%. The actual service life of bridges under real environment is usually lower than their designed life (i.e., 100 years) because of performance deterioration. Therefore, it is meaningful to research the deterioration behavior of bridges throughout their service life, which can help prevent bridge failure. Until now, studies on the theoretical exploration and numerical simulations of the deterioration mechanism have been extensively conducted for concrete bridge structures [1,6–12]. For example, Professor Shi chose a simply supported plate girder as a research object and studied the relationship between the material parameters and failure function through a sensitivity analysis [10]; Ye et al. researched the performance evolution of one continuous concrete box girder by changing the damage time and damage amount [11]; Tian et al. analyzed the performance variation of a continuous concrete girder under chloride penetration environment through finite-element simulation [12]; and Xue et al. explored the ultimate bearing capacity of one continuous rigid-frame bridge based on nonlinear simulation of the concrete material [1].However, the traditional methods based on theoretical deductions and numerical simulations cannot necessarily accurately represent the real deterioration process at the bridge site because of uncertain boundary conditions, imprecise assignments of initial parameters, and an improper ignorance of subordinate factors. In recent years, with the development of structural health monitoring, massive amounts of monitoring data (wind velocity, acceleration, strain, temperature, displacement, etc.) from bridge structures can be used to assess bridge performance [9,13–17]. For example, Liu et al. collected long-term monitoring strain data induced by heavy vehicle traffic on an existing bridge and then presented a series-parallel system model for a performance assessment [17]. However, few studies have specifically focused on assessing the wind-resistant performance of cable-stayed bridge structures using the bridge health-monitoring technique, and at present, no studies have revealed how to identify the deterioration of the wind-resistant performance of existing cable-stayed bridge structures.Therefore, this study focuses on evaluating the wind-resistant performance of existing cable-stayed bridge structures using the bridge health-monitoring technique. The basic idea of this research is that the transversal displacement response of the main girder under cross wind relates to the wind-resistant performance, so the wind-resistant performance can be estimated using the correlation between the transversal displacement response and the cross wind. In other words, the abnormal variation of the correlation indicates a deterioration in the wind-resistant performance. Using the monitoring wind field and the GPS displacement of the Sutong Cable-stayed Bridge, one new detailed method called the cross-correlation analysis of wavelet packet coefficients is invented in this research to accurately obtain the correlation between the transversal displacement and the cross wind. The root-mean-square (RMS) variable representing the difference between a good service state and a deteriorated service state of the wind-resistant performance is then used as a detection indicator to evaluate the wind-resistant performance, and finally, the wind-resistant performance of the Sutong Cable-stayed Bridge in the service period is evaluated using the monitoring RMS value.


## 2. Monitoring Data Analysis

### 2.1. Description of Bridge Health Monitoring

The monitoring bridge object for this research is the long-span Sutong Cable-stayed Bridge across the Yangtze River as shown in Figure1, which was the longest cable-stayed bridge in the world when it was open to traffic in May 2008. The main span of this bridge is 1088 m, and the structural type of the main girder is the flat steel box girder as shown in Figure 2. The complexity, scale, and design technology of this project rank first in the history of China. One obvious feature of this bridge is the low stiffness and long natural vibration period, making this bridge very sensitive to wind load [18, 19]. Therefore, the wind-resistant performance of this bridge is thoroughly studied using the monitoring wind load and its displacement response data.Figure 1
The longitudinal view of the Sutong Cable-stayed Bridge (unit: m).Figure 2
The locations of the 3DUA and GPSMS in the flat steel box girder (unit: mm).In detail, one 3D ultrasonic anemometer (3DUA) is installed at the upstream side of the main span as shown in Figures1 and 2 to record the wind load data, which include wind velocity, wind direction angle, and wind attack angle in the local bridge coordinate system (x, y, z), as shown in Figure 3. This ultrasonic anemometer can accurately record the wind speed ranging from 0 m/s to 65 m/s with a resolution of 0.01 m/s and the wind direction angle ranging from 0° to 359.9° with a resolution of 0.1°. Besides, the upper frequency limit can reach 32 Hz, and in this research, the sampling frequency is set as 1 Hz because only the static wind component is mainly concerned. With regard to the wind direction angle, the positive direction of x-axis is defined as 0° and the positive direction rotates clockwise; with regard to the wind attack angle, the z-axis upward is defined as 90° and the direction in the horizontal plane is defined as 0°.Figure 3
Illustration of the geographic coordinate system (X, Y, Z) and local bridge coordinate system (x, y, z).Moreover, the real-time kinematic global positioning system (RTK-GPS) device is employed to record the relative 3D positions of the rovers (i.e., the mobile stations) with respect to the base (i.e., the reference station) in the geographic coordinate system (X, Y, Z), as shown in Figure 3 [20–22]. The estimation accuracy of relative 3D positions can be improved by double-difference calculation of carrier phases from the base and rovers [20, 22]. In detail, the determination steps of relative 3D positions can be summarized as follows [23, 24]: (1) the reference station continuously observes all the GPS satellites and returns real-time monitoring data to the mobile station; (2) the mobile stations receive the observed data from the reference station through radio-receiving equipment while receiving GPS satellite signals as well; (3) the real-time calculations of the baseline vector between reference and mobile stations are conducted, and the coordinates of the mobile station are further obtained by summing the coordinates of the reference station and baseline vector together; and (4) the coordinates of the reference station and mobile station are calculated under the WGS84 coordinate system, so the coordinates of the reference station and mobile station are further projected onto the North-East-Depth geographic coordinate system (X, Y, Z); finally, the relative 3D positions of mobile stations with respect to the reference station in the geographic coordinate system are obtained according to the principle of relative positioning.In this bridge health-monitoring system, the RTK-GPS contains two GPS receivers (i.e., one base and one rover). The rover is installed at the upstream side of the main span as shown in Figures1 and 2, and the base is installed on the roof of the bridge management building near the bridge site. The base serves as a stationary checkpoint whose coordinates have been previously determined by conventional static GPS methods and constantly records the difference between its known position and the position calculated from the satellite data [24]. The sampling frequency of relative 3D positions is set as 1 Hz.


### 2.2. Monitoring Result

The monitoring data from one month are used for analysis. The monitoring wind data, which include wind velocity, wind direction angle, and wind attack angle, are decomposed to obtain the cross-wind velocity along they-axis using the skew wind decomposition method [25]:(1)Vc=Vcosβsinδ,where Vc denotes the cross-wind velocity along the y-axis, V denotes the monitoring wind velocity, β denotes the monitoring wind direction angle, and δ denotes the monitoring wind attack angle. Then, the 10-minute moving average process, which can be considered as one low-pass filter [26], is carried out to remove the dynamic displacement component and obtain the static cross-wind velocity Vs along the y-axis as shown in Figure 4(a). It can be seen that the maximum static cross-wind velocity is 23.92 m/s and the minimum static cross-wind velocity is −12.75 m/s, indicating that obvious static cross-wind velocity exists at the bridge site.Figure 4
Time histories ofVs and Ds, and the correlation between Vs and Ds: (a) time history of the static cross-wind velocity Vs; (b) time history of the static transversal displacement Ds; (c) the correlation between Vs and Ds.
(a)
(b)
(c)Moreover, it can be seen that the local bridge coordinate systemx-y-z and the geographic coordinate system X-Y-Z are not the same; i.e., there is one included angle about 10.6° between the X- and x-axes. Therefore, the relative 3D positions in the geographic coordinate system are decomposed to obtain the transversal bridge displacement along the y-axis through coordinate transformation as follows:(2)DT=DXsinα−DYcosα,where DT denotes the transversal displacement along the y-axis; DX and DY denote the relative displacements along the X- and Y-axes, respectively; and α denotes the included angle between the X- and x-axes. Then, the 10-minute moving average process is conducted to remove the dynamic displacement component and obtain the static transversal displacement data Ds along the y-axis, as shown in Figure 4(b). It can be seen that the maximum static transversal displacement is 0.259 m, while the minimum static transversal displacement is −0.082 m, indicating that an obvious transversal displacement exists in the Sutong Cable-stayed Bridge under static cross-wind velocity.By comparingVs and Ds, it is clear that they have similar trends. Accordingly, the correlation between Vs and Ds is plotted in Figure 4(c). It can be seen that Vs clearly increases as Ds increases, and the increasing trend appears to be nearly linear. Additionally, the correlation contains discrete points, which are caused by noise from the environment. Here, the increasing trend is treated as the principal component, and the discrete points are treated as the discrete component. The principal component plus the discrete component is the total trend of the correlation.


## 2.1. Description of Bridge Health Monitoring

The monitoring bridge object for this research is the long-span Sutong Cable-stayed Bridge across the Yangtze River as shown in Figure1, which was the longest cable-stayed bridge in the world when it was open to traffic in May 2008. The main span of this bridge is 1088 m, and the structural type of the main girder is the flat steel box girder as shown in Figure 2. The complexity, scale, and design technology of this project rank first in the history of China. One obvious feature of this bridge is the low stiffness and long natural vibration period, making this bridge very sensitive to wind load [18, 19]. Therefore, the wind-resistant performance of this bridge is thoroughly studied using the monitoring wind load and its displacement response data.Figure 1
The longitudinal view of the Sutong Cable-stayed Bridge (unit: m).Figure 2
The locations of the 3DUA and GPSMS in the flat steel box girder (unit: mm).In detail, one 3D ultrasonic anemometer (3DUA) is installed at the upstream side of the main span as shown in Figures1 and 2 to record the wind load data, which include wind velocity, wind direction angle, and wind attack angle in the local bridge coordinate system (x, y, z), as shown in Figure 3. This ultrasonic anemometer can accurately record the wind speed ranging from 0 m/s to 65 m/s with a resolution of 0.01 m/s and the wind direction angle ranging from 0° to 359.9° with a resolution of 0.1°. Besides, the upper frequency limit can reach 32 Hz, and in this research, the sampling frequency is set as 1 Hz because only the static wind component is mainly concerned. With regard to the wind direction angle, the positive direction of x-axis is defined as 0° and the positive direction rotates clockwise; with regard to the wind attack angle, the z-axis upward is defined as 90° and the direction in the horizontal plane is defined as 0°.Figure 3
Illustration of the geographic coordinate system (X, Y, Z) and local bridge coordinate system (x, y, z).Moreover, the real-time kinematic global positioning system (RTK-GPS) device is employed to record the relative 3D positions of the rovers (i.e., the mobile stations) with respect to the base (i.e., the reference station) in the geographic coordinate system (X, Y, Z), as shown in Figure 3 [20–22]. The estimation accuracy of relative 3D positions can be improved by double-difference calculation of carrier phases from the base and rovers [20, 22]. In detail, the determination steps of relative 3D positions can be summarized as follows [23, 24]: (1) the reference station continuously observes all the GPS satellites and returns real-time monitoring data to the mobile station; (2) the mobile stations receive the observed data from the reference station through radio-receiving equipment while receiving GPS satellite signals as well; (3) the real-time calculations of the baseline vector between reference and mobile stations are conducted, and the coordinates of the mobile station are further obtained by summing the coordinates of the reference station and baseline vector together; and (4) the coordinates of the reference station and mobile station are calculated under the WGS84 coordinate system, so the coordinates of the reference station and mobile station are further projected onto the North-East-Depth geographic coordinate system (X, Y, Z); finally, the relative 3D positions of mobile stations with respect to the reference station in the geographic coordinate system are obtained according to the principle of relative positioning.In this bridge health-monitoring system, the RTK-GPS contains two GPS receivers (i.e., one base and one rover). The rover is installed at the upstream side of the main span as shown in Figures1 and 2, and the base is installed on the roof of the bridge management building near the bridge site. The base serves as a stationary checkpoint whose coordinates have been previously determined by conventional static GPS methods and constantly records the difference between its known position and the position calculated from the satellite data [24]. The sampling frequency of relative 3D positions is set as 1 Hz.


## 2.2. Monitoring Result

The monitoring data from one month are used for analysis. The monitoring wind data, which include wind velocity, wind direction angle, and wind attack angle, are decomposed to obtain the cross-wind velocity along they-axis using the skew wind decomposition method [25]:(1)Vc=Vcosβsinδ,where Vc denotes the cross-wind velocity along the y-axis, V denotes the monitoring wind velocity, β denotes the monitoring wind direction angle, and δ denotes the monitoring wind attack angle. Then, the 10-minute moving average process, which can be considered as one low-pass filter [26], is carried out to remove the dynamic displacement component and obtain the static cross-wind velocity Vs along the y-axis as shown in Figure 4(a). It can be seen that the maximum static cross-wind velocity is 23.92 m/s and the minimum static cross-wind velocity is −12.75 m/s, indicating that obvious static cross-wind velocity exists at the bridge site.Figure 4
Time histories ofVs and Ds, and the correlation between Vs and Ds: (a) time history of the static cross-wind velocity Vs; (b) time history of the static transversal displacement Ds; (c) the correlation between Vs and Ds.
(a)
(b)
(c)Moreover, it can be seen that the local bridge coordinate systemx-y-z and the geographic coordinate system X-Y-Z are not the same; i.e., there is one included angle about 10.6° between the X- and x-axes. Therefore, the relative 3D positions in the geographic coordinate system are decomposed to obtain the transversal bridge displacement along the y-axis through coordinate transformation as follows:(2)DT=DXsinα−DYcosα,where DT denotes the transversal displacement along the y-axis; DX and DY denote the relative displacements along the X- and Y-axes, respectively; and α denotes the included angle between the X- and x-axes. Then, the 10-minute moving average process is conducted to remove the dynamic displacement component and obtain the static transversal displacement data Ds along the y-axis, as shown in Figure 4(b). It can be seen that the maximum static transversal displacement is 0.259 m, while the minimum static transversal displacement is −0.082 m, indicating that an obvious transversal displacement exists in the Sutong Cable-stayed Bridge under static cross-wind velocity.By comparingVs and Ds, it is clear that they have similar trends. Accordingly, the correlation between Vs and Ds is plotted in Figure 4(c). It can be seen that Vs clearly increases as Ds increases, and the increasing trend appears to be nearly linear. Additionally, the correlation contains discrete points, which are caused by noise from the environment. Here, the increasing trend is treated as the principal component, and the discrete points are treated as the discrete component. The principal component plus the discrete component is the total trend of the correlation.


## 3. Principal Component Analysis

The discrete points have a negative influence on the estimation accuracy of the wind-resistant performance, so it is necessary to decrease the amount of discrete points. One new method referred to as a cross-correlation analysis of wavelet packet coefficients (CAWPC) is introduced to effectively remove the discrete component while extracting the principal component from the correlation.


### 3.1. Analysis Method

The basic idea of the CAWPC method contains three parts:Part 1: to more easily conduct the analysis, the correlation in Figure4 is divided into several segments, as shown in Figure 5. The correlations in each segment are considered to be linear, so the correlations are treated as combinations of linear segments.Part 2: the correlation in each segment contains a principal component as well as a discrete component. One distinct difference between the principal component and the discrete component is that the value ofDs caused by Vs in the principal component changes gently, but the value of Ds caused by Vs in the discrete component clearly fluctuates; so the principal component and discrete component have different frequency bands. Therefore, the wavelet packet method is introduced to decompose Ds and Vs in each segment into different frequency bands. Based on a pair of low-pass and high-pass conjugate quadrature filters hj and gj of the wavelet packet [27, 28], Ds or Vs in each segment can be decomposed scale by scale into different frequency bands. In each frequency band, one corresponding decomposition coefficient is obtained, which is calculated as follows [29]:(3)Cpi,2l=∑q∈ZCqi+1,lhq−2j,(4)Cpi,2l+1=∑q∈ZCqi+1,lgq−2j,where Cpi,2l denotes the decomposition coefficient in the kth frequency band under the 2lth scale and Cpi,2l+1 denotes the decomposition coefficient in the kth frequency band under the 2l+1th scale.Part 3: after the wavelet packet decomposition,Ds and Vs in each segment have many decomposition coefficients, and each decomposition coefficient from Ds corresponds to one decomposition coefficient from Vs in the same frequency band, i.e., a pair of decomposition coefficients. Some pairs of decomposition coefficients mainly contain the principal component, and the other pairs of decomposition coefficients mostly contain the discrete component; so the pairs of decomposition coefficients containing the principal component should be selected from all the decomposition coefficients. Each decomposition coefficient can be reconstructed into one time-domain signal. If one pair of decomposition coefficients contains a principal component, their reconstructed time-domain signals should have good linear cross-correlation characteristics to each other, which can be judged by the following formula:(5)c=∑m=1nVRm−V¯RDRm−D¯R∑m=1nVRm−V¯R2∑m=1nDRm−D¯R20.5,where c denotes the linear cross-correlation coefficient of the reconstructed time-domain signals of Ds and Vs; VRm and DRm denote the mth values in the reconstructed time-domain signals of Vs and Ds, respectively; V¯R and D¯R denote the average values of VRm and DRm, respectively; and n denotes the total amount of VRm or DRm. Therefore, if the calculation result of c is close to 1, it can be concluded that the reconstructed time-domain signals of Ds and Vs have good linear cross-correlation characteristics to each other, so the corresponding pair of decomposition coefficients is retained. Otherwise, if the calculation result of c is close to 0, the corresponding pair of decomposition coefficients is removed. Finally, after cross-correlation analysis, only the pairs of decomposition coefficients containing the principal component are retained. Therefore, the reconstructed time-domain signals of Ds and Vs mainly contain the principal component.Figure 5
Eleven segments of the correlation.


### 3.2. Analysis Steps

In detail, this CAWPC method is carried out by the following four steps:(1)
The correlation betweenVs and Ds is divided into w segments, as shown in Figure 5. The correlation in each segment contains a principal component as well as a discrete component. The static wind velocity and static displacement in the kth segment are denoted by Vs,k and Ds,k, respectively, where k=1,2,…,w.(2)
Vs,k and Ds,k are each decomposed into 2s frequency bands using the wavelet packet method. In each frequency band, the corresponding decomposition coefficients are obtained. The jth decomposition coefficient of Vs,k is reconstructed into the time-domain static wind velocity Vkj, and the jth decomposition coefficient of Ds,k is reconstructed into the time-domain static displacement Dkj, where j=1,2,…,2s. The jth decomposition coefficients of Vs,k and Ds,k are the jth pair of decomposition coefficients in the jth frequency band.(3)
Based on (4), the linear cross-correlation coefficient of Vkj and Dkj, which is denoted by ckj, is calculated to judge whether the correlation between Vkj and Dkj contains good linear characteristics:(6)ckj=∑m=1nVkj,m−V¯kjDkj,m−D¯kj∑m=1nVkj,m−V¯kj2∑m=1nDkj,m−D¯kj20.5,where Vkj,m and Dkj,m denote the mth values of Vkj and Dkj, respectively, and V¯kj and Dkj denote the average values of Vkj and Dkj, respectively. If the calculation result of ckj is close to 1, it can be concluded that Vkj and Dkj have good linear cross-correlation characteristics, so the jth pair of decomposition coefficients is retained; otherwise, the jth pair of decomposition coefficients is removed.(4)
After cross-correlation analysis, only the pairs of decomposition coefficients containing good linear cross-correlation characteristics are retained. Then, all the reconstructed time-domain static wind velocity dataVkj from the remaining decomposition coefficients are superposed to obtain the static wind velocity V˜s,k in the kth segment, and all the reconstructed time-domain static displacement data Dkj from the remaining decomposition coefficients are superposed to obtain the static displacement data D˜s,k in the kth segment. The correlation between V˜s,k and D˜s,k mainly contains the principal component in the kth segment. Furthermore, the principal components from all the segments constitute the principal component of the whole correlation between Vs and Ds; that is, V˜s=V˜s,1,V˜s,2,…,V˜s,w and D˜s=D˜s,1,D˜s,2,…,D˜s,w.


### 3.3. Results

According to the four steps mentioned above, the parameter values ofw and s in the CAWPC method during analysis are set to 11 and 8, respectively. The principal component of the correlation between Vs and Ds from Figure 4 is further extracted and shown in Figure 6. It can be seen that, after the CAWPC analysis, the principal component obviously presents a nearly linearly increasing trend with few discrete points compared to that in Figure 4(c), verifying that the CAWPC method is effective and can be further used for evaluating the wind-resistant performance.Figure 6
The principal component of the correlation.


## 3.1. Analysis Method

The basic idea of the CAWPC method contains three parts:Part 1: to more easily conduct the analysis, the correlation in Figure4 is divided into several segments, as shown in Figure 5. The correlations in each segment are considered to be linear, so the correlations are treated as combinations of linear segments.Part 2: the correlation in each segment contains a principal component as well as a discrete component. One distinct difference between the principal component and the discrete component is that the value ofDs caused by Vs in the principal component changes gently, but the value of Ds caused by Vs in the discrete component clearly fluctuates; so the principal component and discrete component have different frequency bands. Therefore, the wavelet packet method is introduced to decompose Ds and Vs in each segment into different frequency bands. Based on a pair of low-pass and high-pass conjugate quadrature filters hj and gj of the wavelet packet [27, 28], Ds or Vs in each segment can be decomposed scale by scale into different frequency bands. In each frequency band, one corresponding decomposition coefficient is obtained, which is calculated as follows [29]:(3)Cpi,2l=∑q∈ZCqi+1,lhq−2j,(4)Cpi,2l+1=∑q∈ZCqi+1,lgq−2j,where Cpi,2l denotes the decomposition coefficient in the kth frequency band under the 2lth scale and Cpi,2l+1 denotes the decomposition coefficient in the kth frequency band under the 2l+1th scale.Part 3: after the wavelet packet decomposition,Ds and Vs in each segment have many decomposition coefficients, and each decomposition coefficient from Ds corresponds to one decomposition coefficient from Vs in the same frequency band, i.e., a pair of decomposition coefficients. Some pairs of decomposition coefficients mainly contain the principal component, and the other pairs of decomposition coefficients mostly contain the discrete component; so the pairs of decomposition coefficients containing the principal component should be selected from all the decomposition coefficients. Each decomposition coefficient can be reconstructed into one time-domain signal. If one pair of decomposition coefficients contains a principal component, their reconstructed time-domain signals should have good linear cross-correlation characteristics to each other, which can be judged by the following formula:(5)c=∑m=1nVRm−V¯RDRm−D¯R∑m=1nVRm−V¯R2∑m=1nDRm−D¯R20.5,where c denotes the linear cross-correlation coefficient of the reconstructed time-domain signals of Ds and Vs; VRm and DRm denote the mth values in the reconstructed time-domain signals of Vs and Ds, respectively; V¯R and D¯R denote the average values of VRm and DRm, respectively; and n denotes the total amount of VRm or DRm. Therefore, if the calculation result of c is close to 1, it can be concluded that the reconstructed time-domain signals of Ds and Vs have good linear cross-correlation characteristics to each other, so the corresponding pair of decomposition coefficients is retained. Otherwise, if the calculation result of c is close to 0, the corresponding pair of decomposition coefficients is removed. Finally, after cross-correlation analysis, only the pairs of decomposition coefficients containing the principal component are retained. Therefore, the reconstructed time-domain signals of Ds and Vs mainly contain the principal component.Figure 5
Eleven segments of the correlation.


## 3.2. Analysis Steps

In detail, this CAWPC method is carried out by the following four steps:(1)
The correlation betweenVs and Ds is divided into w segments, as shown in Figure 5. The correlation in each segment contains a principal component as well as a discrete component. The static wind velocity and static displacement in the kth segment are denoted by Vs,k and Ds,k, respectively, where k=1,2,…,w.(2)
Vs,k and Ds,k are each decomposed into 2s frequency bands using the wavelet packet method. In each frequency band, the corresponding decomposition coefficients are obtained. The jth decomposition coefficient of Vs,k is reconstructed into the time-domain static wind velocity Vkj, and the jth decomposition coefficient of Ds,k is reconstructed into the time-domain static displacement Dkj, where j=1,2,…,2s. The jth decomposition coefficients of Vs,k and Ds,k are the jth pair of decomposition coefficients in the jth frequency band.(3)
Based on (4), the linear cross-correlation coefficient of Vkj and Dkj, which is denoted by ckj, is calculated to judge whether the correlation between Vkj and Dkj contains good linear characteristics:(6)ckj=∑m=1nVkj,m−V¯kjDkj,m−D¯kj∑m=1nVkj,m−V¯kj2∑m=1nDkj,m−D¯kj20.5,where Vkj,m and Dkj,m denote the mth values of Vkj and Dkj, respectively, and V¯kj and Dkj denote the average values of Vkj and Dkj, respectively. If the calculation result of ckj is close to 1, it can be concluded that Vkj and Dkj have good linear cross-correlation characteristics, so the jth pair of decomposition coefficients is retained; otherwise, the jth pair of decomposition coefficients is removed.(4)
After cross-correlation analysis, only the pairs of decomposition coefficients containing good linear cross-correlation characteristics are retained. Then, all the reconstructed time-domain static wind velocity dataVkj from the remaining decomposition coefficients are superposed to obtain the static wind velocity V˜s,k in the kth segment, and all the reconstructed time-domain static displacement data Dkj from the remaining decomposition coefficients are superposed to obtain the static displacement data D˜s,k in the kth segment. The correlation between V˜s,k and D˜s,k mainly contains the principal component in the kth segment. Furthermore, the principal components from all the segments constitute the principal component of the whole correlation between Vs and Ds; that is, V˜s=V˜s,1,V˜s,2,…,V˜s,w and D˜s=D˜s,1,D˜s,2,…,D˜s,w.


## 3.3. Results

According to the four steps mentioned above, the parameter values ofw and s in the CAWPC method during analysis are set to 11 and 8, respectively. The principal component of the correlation between Vs and Ds from Figure 4 is further extracted and shown in Figure 6. It can be seen that, after the CAWPC analysis, the principal component obviously presents a nearly linearly increasing trend with few discrete points compared to that in Figure 4(c), verifying that the CAWPC method is effective and can be further used for evaluating the wind-resistant performance.Figure 6
The principal component of the correlation.


## 4. Evaluation Method Analysis

At the beginning of the service period, the Sutong Bridge had a good wind-resistant performance. However, as the service time increases, the wind-resistant performance will inevitably deteriorate due to the influence of synthetic factors, such as concrete carbonation, reinforcement corrosion, and accumulated fatigue damage. The transversal displacement response of the main girder under cross winds relates to the wind-resistant performance, so the wind-resistant performance can be estimated using the correlation between the transversal displacement response and the cross wind; in other words, the abnormal variation of the correlation indicates a deterioration in the wind-resistant performance.


### 4.1. Deterioration Process of Wind-Resistant Performance

The static cross-wind loadF relates to the static wind velocity V˜s, which can be expressed as follows [30]:(7)F=0.5ρCHHV˜s2,where ρ is the air density, CH is the resistance coefficient, and H is the projection height of the flat steel box girder. Furthermore, the static displacement response D˜s is induced by the static cross-wind load F, which can be modeled using one second-order polynomial function (abbreviated as TPF) as follows [31]:(8a)D˜sTPF=Fδ=0.5ρCHHδV˜s2=0.5ρCHHδV˜s2,(8b)orD˜sTPF=0.5ρCHHδV˜s2,where D˜sTPF denotes the static displacement D˜s in the function TPF, D˜sTPF denotes the absolute static displacement D˜s in the function TPF, and δ denotes the wind-resistant flexibility of the Sutong Bridge. According to (8b), the static displacement response D˜s should be 0 if the static wind velocity V˜s is 0, so the monitoring correlation between D˜s and V˜s is shifted along the horizontal axis to meet this condition. Then, (8b) is fitted using the monitoring correlation between V˜s and D˜s in the least-squares sense. Before fitting (8b), the monitoring correlations between V˜s and D˜s are divided into two parts: the monitoring correlation within the wind velocity intervals [0 m/s, 15 m/s] is used as training data to fit (8b), and the monitoring correlation within the wind velocity intervals (15 m/s, 24 m/s] is used as test data to verify the fitting results of (8a). After the least-squares analysis, the fitted result of (8b) is D˜s = 0.00071V˜s2, as shown in Figure 7. It can be seen that the fitted result of (8b) does not match the monitoring correlation very well in the test phase. Therefore, more possible functions, including the power function (PF), Fourier function (FF), fourth-order polynomial function (FPF), sine function (SF), and Gauss function (GF), are concretely studied to fit the monitoring correlation. Using the mathematical functions PF, FF, FPF, SF, and GF to fit the in situ data belongs to the data-driven method, which has been commonly used in current research [32, 33]. According to the definition of this method, although this method does not contain the basic physical meaning regarding the relation between displacement and wind velocity, the calculated results are equivalent to the real physical model if the mathematical functions are well fitted. These mathematical equations are expressed as follows.(1)
Power function (PF):(9a)D˜sPF=aV˜sb,where D˜sPF denotes the static displacement D˜s in the function PF, and a and b denote the parameters to be estimated in the function PF.
(2)
Fourier functionFF:(9b)D˜sFF=a0+∑i=14aicoswiV˜s+bisinwiV˜s,where D˜sFF denotes the static displacement D˜s in the function FF, and a0,a1,…,a4 and b1,b2,…,b4 denote the parameters to be estimated in the function FF.
(3)
Fourth-order polynomial function (FPF):(9c)D˜sFPF=∑i=04piV˜si,where D˜sFPF denotes the static displacement D˜s in the function FPF, and p0,p1,…,p4 denote the parameters to be estimated in the function FPF.
(4)
Sine functionSF:(9d)D˜sSF=∑i=14aisinbiV˜s+ci,where D˜sSF denotes the static displacement D˜s in the function SF, and a1,a2,…,a4, b1,b2,…,b4, and c1,c2,…,c4 denote the parameters to be estimated in the function SF.
(5)
Gauss functionGF:(9e)D˜sGF=∑i=14aieV˜s−bi/ci2,where D˜sGF denotes the static displacement D˜s in the function GF, and a1,a2,…,a4, b1,b2,…,b4, and c1,c2,…,c4 denote the parameters to be estimated in the function GF.Figure 7
The fitted results of the monitoring correlation.After the least-squares analysis, the fitted results of all the functions are shown in Figure7. It can be seen that although all the fitted functions match the monitoring correlation well in the training phase, only the power function (PF) matches the monitoring correlation well in the test phase, so the power function (PF) is selected to represent the monitoring correlation, with the estimated values of a and b in the PF at 0.002374 and 1.476, respectively.Furthermore, the static displacementD˜sPF in (8a) is induced by the static cross-wind load F, which can be expressed as follows:(10)D˜sPF=Fδ.Suppose that the wind-resistant performance of the Sutong Bridge gradually deteriorates in the service period, and the deterioration amount isΔδ. Then, the static displacement D^s after deterioration can be reexpressed as follows:(11)D^s=Fδ+Δδ=1+ΔδδaV˜sb.Taking the principal component in Figure6 as an example, if Δδ = 0, 0.3δ, and 0.6δ, respectively, the deterioration results of the principal component corresponding to different Δδ values are shown in Figure 8(a). It can be seen that the principal component obviously deviates from its good service state during deterioration, and the principal component gradually becomes thinner as the deterioration amount Δδ increases. What should be emphasized here is that the CAWPC method plays one important role in obtaining the distinct deterioration of the wind-resistant performance in Figure 8(a). If the correlation between Vs and Ds in Figure 4(c), which does not undergo CAWPC analysis, is used to analyze the deterioration of the wind-resistant performance as shown in Figure 8(b), it can be seen that the deterioration of wind-resistant performance is fuzzy compared with that in Figure 8(a), which can obviously decrease the identification accuracy of deterioration identification.Figure 8
The deterioration of wind-resistant performance: (a) after the cross-correlation analysis; (b) before the cross-correlation analysis.
(a)
(b)


### 4.2. Method of Deterioration Identification

As mentioned above, the principal component will gradually deviate from its good service state during deterioration as shown in Figure8(a), and the deviation amount can be used to identify the deterioration of the wind-resistant performance. The deviation amount can be concretely described using the root-mean-square (RMS) value, which represents the difference between a good service state and a deteriorated service state. Specifically, the steps of deterioration identification of the wind-resistant performance are shown as follows:(1)
The monitoring data of the static cross-wind velocity and its transversal displacement response are divided by months. The principal component of the correlation between the static cross-wind velocity and the transversal displacement in each month is extracted through CAWPC analysis. Suppose that the principal component in the first month corresponds to the good service state of wind-resistant performance, which is mathematically modeled by (8a).
(2)
The principal components of the monitoring correlation in themth month are denoted by V˜sm and D˜sm, respectively. V˜sm is substituted into (8a) to obtain D^sm as follows:(12)D^sm=aV˜smb.The deviation amount between the deteriorated service state and the good service state is calculated by D^sm−D˜sm, and then, the RMS value of D^sm−D˜sm is calculated as follows:(13)Rm=∑k=1NmD^smk−D˜smk2Nm,where Rm denotes the RMS value in the mth month; Nm denotes the total amount of D^sm or D˜sm; and D^smk and D˜smk denote the kth values of D^sm and D˜sm, respectively.
(3)
As mentioned earlier, the principal component will gradually become thinner as the deterioration amountΔδ increases (Figure 8(a)), indicating that the values of Rm will gradually increase with the deterioration process of wind-resistant performance. Therefore, a gradual increase in Rm can indicate the deterioration of the wind-resistant performance.


### 4.3. Results

According to the three steps mentioned above, the monitoring data of the static cross-wind velocity and its transversal displacement in ten months are used to identify whether or not the wind-resistant performance of the Sutong Bridge is in a good service state. Finally, the calculation results ofR1,R1,…,R10 are shown in Figure 9, which does not present an increasing trend, indicating that the wind-resistant performance of the Sutong Bridge was still in a good service state during these ten months.Figure 9
The calculation results ofR1,R1,…,R10.


## 4.1. Deterioration Process of Wind-Resistant Performance

The static cross-wind loadF relates to the static wind velocity V˜s, which can be expressed as follows [30]:(7)F=0.5ρCHHV˜s2,where ρ is the air density, CH is the resistance coefficient, and H is the projection height of the flat steel box girder. Furthermore, the static displacement response D˜s is induced by the static cross-wind load F, which can be modeled using one second-order polynomial function (abbreviated as TPF) as follows [31]:(8a)D˜sTPF=Fδ=0.5ρCHHδV˜s2=0.5ρCHHδV˜s2,(8b)orD˜sTPF=0.5ρCHHδV˜s2,where D˜sTPF denotes the static displacement D˜s in the function TPF, D˜sTPF denotes the absolute static displacement D˜s in the function TPF, and δ denotes the wind-resistant flexibility of the Sutong Bridge. According to (8b), the static displacement response D˜s should be 0 if the static wind velocity V˜s is 0, so the monitoring correlation between D˜s and V˜s is shifted along the horizontal axis to meet this condition. Then, (8b) is fitted using the monitoring correlation between V˜s and D˜s in the least-squares sense. Before fitting (8b), the monitoring correlations between V˜s and D˜s are divided into two parts: the monitoring correlation within the wind velocity intervals [0 m/s, 15 m/s] is used as training data to fit (8b), and the monitoring correlation within the wind velocity intervals (15 m/s, 24 m/s] is used as test data to verify the fitting results of (8a). After the least-squares analysis, the fitted result of (8b) is D˜s = 0.00071V˜s2, as shown in Figure 7. It can be seen that the fitted result of (8b) does not match the monitoring correlation very well in the test phase. Therefore, more possible functions, including the power function (PF), Fourier function (FF), fourth-order polynomial function (FPF), sine function (SF), and Gauss function (GF), are concretely studied to fit the monitoring correlation. Using the mathematical functions PF, FF, FPF, SF, and GF to fit the in situ data belongs to the data-driven method, which has been commonly used in current research [32, 33]. According to the definition of this method, although this method does not contain the basic physical meaning regarding the relation between displacement and wind velocity, the calculated results are equivalent to the real physical model if the mathematical functions are well fitted. These mathematical equations are expressed as follows.(1)
Power function (PF):(9a)D˜sPF=aV˜sb,where D˜sPF denotes the static displacement D˜s in the function PF, and a and b denote the parameters to be estimated in the function PF.
(2)
Fourier functionFF:(9b)D˜sFF=a0+∑i=14aicoswiV˜s+bisinwiV˜s,where D˜sFF denotes the static displacement D˜s in the function FF, and a0,a1,…,a4 and b1,b2,…,b4 denote the parameters to be estimated in the function FF.
(3)
Fourth-order polynomial function (FPF):(9c)D˜sFPF=∑i=04piV˜si,where D˜sFPF denotes the static displacement D˜s in the function FPF, and p0,p1,…,p4 denote the parameters to be estimated in the function FPF.
(4)
Sine functionSF:(9d)D˜sSF=∑i=14aisinbiV˜s+ci,where D˜sSF denotes the static displacement D˜s in the function SF, and a1,a2,…,a4, b1,b2,…,b4, and c1,c2,…,c4 denote the parameters to be estimated in the function SF.
(5)
Gauss functionGF:(9e)D˜sGF=∑i=14aieV˜s−bi/ci2,where D˜sGF denotes the static displacement D˜s in the function GF, and a1,a2,…,a4, b1,b2,…,b4, and c1,c2,…,c4 denote the parameters to be estimated in the function GF.Figure 7
The fitted results of the monitoring correlation.After the least-squares analysis, the fitted results of all the functions are shown in Figure7. It can be seen that although all the fitted functions match the monitoring correlation well in the training phase, only the power function (PF) matches the monitoring correlation well in the test phase, so the power function (PF) is selected to represent the monitoring correlation, with the estimated values of a and b in the PF at 0.002374 and 1.476, respectively.Furthermore, the static displacementD˜sPF in (8a) is induced by the static cross-wind load F, which can be expressed as follows:(10)D˜sPF=Fδ.Suppose that the wind-resistant performance of the Sutong Bridge gradually deteriorates in the service period, and the deterioration amount isΔδ. Then, the static displacement D^s after deterioration can be reexpressed as follows:(11)D^s=Fδ+Δδ=1+ΔδδaV˜sb.Taking the principal component in Figure6 as an example, if Δδ = 0, 0.3δ, and 0.6δ, respectively, the deterioration results of the principal component corresponding to different Δδ values are shown in Figure 8(a). It can be seen that the principal component obviously deviates from its good service state during deterioration, and the principal component gradually becomes thinner as the deterioration amount Δδ increases. What should be emphasized here is that the CAWPC method plays one important role in obtaining the distinct deterioration of the wind-resistant performance in Figure 8(a). If the correlation between Vs and Ds in Figure 4(c), which does not undergo CAWPC analysis, is used to analyze the deterioration of the wind-resistant performance as shown in Figure 8(b), it can be seen that the deterioration of wind-resistant performance is fuzzy compared with that in Figure 8(a), which can obviously decrease the identification accuracy of deterioration identification.Figure 8
The deterioration of wind-resistant performance: (a) after the cross-correlation analysis; (b) before the cross-correlation analysis.
(a)
(b)


## 4.2. Method of Deterioration Identification

As mentioned above, the principal component will gradually deviate from its good service state during deterioration as shown in Figure8(a), and the deviation amount can be used to identify the deterioration of the wind-resistant performance. The deviation amount can be concretely described using the root-mean-square (RMS) value, which represents the difference between a good service state and a deteriorated service state. Specifically, the steps of deterioration identification of the wind-resistant performance are shown as follows:(1)
The monitoring data of the static cross-wind velocity and its transversal displacement response are divided by months. The principal component of the correlation between the static cross-wind velocity and the transversal displacement in each month is extracted through CAWPC analysis. Suppose that the principal component in the first month corresponds to the good service state of wind-resistant performance, which is mathematically modeled by (8a).
(2)
The principal components of the monitoring correlation in themth month are denoted by V˜sm and D˜sm, respectively. V˜sm is substituted into (8a) to obtain D^sm as follows:(12)D^sm=aV˜smb.The deviation amount between the deteriorated service state and the good service state is calculated by D^sm−D˜sm, and then, the RMS value of D^sm−D˜sm is calculated as follows:(13)Rm=∑k=1NmD^smk−D˜smk2Nm,where Rm denotes the RMS value in the mth month; Nm denotes the total amount of D^sm or D˜sm; and D^smk and D˜smk denote the kth values of D^sm and D˜sm, respectively.
(3)
As mentioned earlier, the principal component will gradually become thinner as the deterioration amountΔδ increases (Figure 8(a)), indicating that the values of Rm will gradually increase with the deterioration process of wind-resistant performance. Therefore, a gradual increase in Rm can indicate the deterioration of the wind-resistant performance.


## 4.3. Results

According to the three steps mentioned above, the monitoring data of the static cross-wind velocity and its transversal displacement in ten months are used to identify whether or not the wind-resistant performance of the Sutong Bridge is in a good service state. Finally, the calculation results ofR1,R1,…,R10 are shown in Figure 9, which does not present an increasing trend, indicating that the wind-resistant performance of the Sutong Bridge was still in a good service state during these ten months.Figure 9
The calculation results ofR1,R1,…,R10.


## 5. Conclusions

Based on the bridge health-monitoring system, this research conducted a thorough evaluation on the wind-resistant behavior of the Sutong Cable-stayed Bridge using the correlation between the static cross wind and transversal displacement. The following main conclusions were drawn:(1)
The monitoring correlation between the static cross wind and transversal displacement response shows a nearly linearly increasing trend and contains discrete points influenced by environmental noise.(2)
A new method called the cross-correlation analysis of wavelet packet coefficients (CAWPC) can effectively remove the discrete points from the monitoring correlation.(3)
After a comparison of many fitted functions, the fitted power function (PF) can match the monitoring correlation best in both the training and test phases, so the PF is selected to represent the monitoring correlation.(4)
The deterioration process of wind-resistant performance related to the abnormal variation of the correlation is revealed, and additionally, the root-mean-square variable, which represents the difference between a good service state and a deteriorated service state, is put forward to identify the deterioration of wind-resistant performance;(5)
The monitoring data from ten months are selected to evaluate the wind-resistant performance of the Sutong Bridge, and finally, the result shows that the Sutong Cable-stayed Bridge maintained a good service state during these ten months.What should be mentioned is that the deterioration process of wind-resistant performance is long-term and slow, so ten-month monitoring data may not contain obvious deterioration information. Hence, our bridge health-monitoring system will continuously monitor the Sutong Bridge, and more monitoring data in the future will be used to judge the wind-resistant performance of the Sutong Bridge.



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*Source: 5369281-2018-08-27.xml*