Assessing Static Performance of the Dashengguan Yangtze Bridge by Monitoring the Correlation between Temperature Field and Its Static Strains

1The Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Sipailou Road, Xuanwu District, Nanjing, Jiangsu 210096, China 2Department of Civil Engineering, Southeast University, Sipailou Road, Xuanwu District, Nanjing, Jiangsu 210096, China 3Department of Civil and Environmental Engineering, Rice University, Houston, TX 77005, USA 4China Railway Major Bridge (Nanjing) Bridge and Tunnel Inspect & Retrofit Co., Ltd., Nanjing 210032, China


Introduction
In recent years, with development of the structural health monitoring technology, it is feasible to install sensors for monitoring the performance of bridge structures [1][2][3][4].Static strain effect caused by combined load actions such as temperature field and traffic loading can present static performance of bridge structures [5][6][7].Relevant research showed that daily or seasonal temperature field caused critical strain levels, even higher than traffic-induced strains [8], so abnormal variation of correlation between temperature field and its static strain effect can indicate static performance degradation of bridge structures.For example, Duan et al. developed linear regression models for structural performance evaluation using temperature-strain correlation [8]; Li et al. obtained probability models for performance assessment application on cable stayed bridges through statistical analysis of temperature-strain linear relationships [9].However, current research efforts in this field are still insufficient in many specific details: (1) current research interests merely focus on correlation between temperature and static strain and have not considered the influence of temperature difference on static strain; (2) current research methods merely concentrate on single variable analysis, whereas static strain from one monitoring point is actually influenced by both temperature and temperature difference from many different monitoring points; (3) current research progress has not definitely pointed out degradation regulations of static performance when correlation between temperature field and its static strain effect abnormally changes.
Therefore, using health monitoring system installed on steel truss arch girder of the Dashengguan Yangtze Bridge, temperature field data and static strain data from top chord member, arch rib chord member, bottom chord member, and diagonal web member are continuously collected, and then variation regularities of their time history curves are analyzed by comparison.Wavelet packet decomposition method is introduced to extract static strains caused by temperature and temperature difference data and then principal component analysis is carried out to simplify the multivariate linear regression model.Finally, three kinds of degradation regulations of residual static strains are put forward to assess static performance of the Dashengguan Yangtze Bridge.

Bridge Health Monitoring and Data Collection
2.1.Bridge Health Monitoring.The bridge monitoring object for this research is the famous Dashengguan Yangtze Bridge, which is designed as the river-crossing channel for the Beijing-Shanghai high speed railway and Nanjing bidirectional subway.The structural type of its main girder is the continuous steel truss arch girder with the main span reaching 336 m as shown in Figure 1.According to its 1-1 profile as shown in Figure 2 In order to get the variation regularity of temperature field in steel truss arch girder, eight FBG temperature sensors are installed on the cross-sections of top chord member, diagonal web member, arch rib chord member, and bottom chord member, respectively, as shown in Figures 2(c)∼2(f) (where   denotes the th temperature sensor,  = 1, 2, . . ., 8) to continuously measure temperature field data.Besides, eight FBG strain sensors are installed on the same positions with temperature sensors as shown in Figures 2(c)∼2(f) respectively (where   denotes the th strain sensor,  = 1, 2, . . ., 8), to continuously obtain axial static strain data.Sampling frequency of data collection is set to 1 Hz, which is appropriate since the changes of temperature and strain are very slow and little.3 and 4, respectively.The history curves of global change and daily change in static strain data, taking  1 and  2 from training data, for example, are shown in Figures 5 and 6, respectively (negative values denote compressive static strain and positive values denote tension static strain).

Data of
From Figures 3 to 6, some findings can be obtained: (1) the history curves of global change and daily change in static strain data  1 are similar to that of temperature difference data  12 shown in Figures 4 and 5; that is, the global curves of  1 and  12 are both stationary and the daily curves of  1 and  12 both change little at night and change obviously at daytime in wave shape simultaneously, indicating that static strain data  1 mainly vary with temperature difference; (2) the history curves of global change and daily change in static strain data  2 are similar to that of temperature data  1 shown in Figures 3 and 6; that is, the global curves of  2 and  1 are both seasonable and the daily curves of  2 and  1 both change in the shape of one-cycle sine curve, indicating that static strain data  2 mainly vary with temperature; (3) daily history curves of  1 and  2 contain many sharp peaks caused by trains as shown in Figures 5(b) and 6(b), with each peak corresponding to the time when one train is across the bridge, whereas the proportion of its influence on static strain is obviously lower than that of temperature and temperature difference.

Diagonal web member Shanghai Beijing
Top chord member

Bottom chord member
Vertical web member 6 6 Arch rib chord member The hierarchical tree of decomposition coefficients in three scales.

Extracting Static Strain Caused by Temperature Field.
According to the analysis above, static strain data mainly contains three parts: static strain I caused by temperature, static strain II caused by temperature difference, and static strain III caused by train.In order to research the correlation between temperature field and its static strain effect, static strain I and static strain II (denoted by  I,II ) must be extracted from the static strain data.Considering that primary period of  I,II is one day, which is far longer than that of static strain III, wavelet packet decomposition method is introduced to extract  I,II .
In detail, based on a pair of low-pass and high-pass conjugate quadrature filters ℎ() and () of wavelet packet, static strain data can be decomposed scale by scale into different frequency bands, and each decomposition coefficient corresponding to its frequency band and its scale is calculated as follows [10,11]: ,2  or  ,2+1  denotes the decomposition coefficient within the th frequency band and the 2th or (2 + 1)th scale, respectively, which is visually described by a hierarchical tree shown in Figure 7 (only three scales are listed).Each decomposition coefficient can be reconstructed into time-domain signals with constraints of its own frequency band; therefore decomposition coefficients within primary frequency band of  I,II can be specially selected and then reconstructed as  I,II .
Taking the daily history curve of static strain data  1 in Figure 5(b) as an example, it is decomposed within 8 scales and the decomposition coefficient  0,8  is specially selected and then reconstructed as  I,II shown in Figure 8(a), which effectively retains main component caused by temperature field and removes sharp peaks caused by trains as well.Using the method above,  I,II of each static strain data is extracted, such as the extraction results of  1 from training data shown in Figure 8(b).

Modeling Method.
In order to better present the correlation between  I,II and temperature field from training data, two steps are specifically carried out as follows: (1) temperature and temperature difference data from sufficient solar radiation days, whose daily variation trends are similar to one smooth single-period sine curve as shown in Figure 3(b) [12], are specially selected and then  I,II are selected from the same corresponding days; (2) daily maximum and minimum values are furthermore chosen from the selected temperature field and  I,II data.The scatter plots between  I,II of  1 and temperature difference  12 , between  I,II of  2 and temperature  1 , are shown in Figures 9(a) and 9(b), respectively, both presenting apparent linear correlation.Moreover, considering that  I,II of each static strain data may be influenced by many temperature and temperature difference data,  I,II are expressed as follows: where   is the th temperature data from set { 1 ,  Considering that total 24 linear expansion coefficients and one constant term are very complicated for calculation, principal component analysis is introduced to simplify (2).Generally speaking, principal components can be derived from either an algebraic or a geometric viewpoint.Given 8 random variables { 1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ,  8 } with a known covariance matrix ∑, the th ( = 1, 2, . . ., 8) principal component   is a linear expression of the 8 random variables defined as follows [13,14]:    according to the Pareto diagram of explained variances of   shown in Figure 10(a), it can be seen that the explained variance of  1 is 96.7%, very higher than the other principal components, so  1 is specially selected with its calculation equation as follows: where  = [ 1 ,   [13,14], with the Pareto diagram of their explained variances shown in Figure 10(b).It can be seen that the sum of explained variances of  1 ,  2 , and  3 is 98.8%, so  1 ,  2 , and  3 are specially selected with their calculation equations as follows: where ) is an eigenvector of covariance matrix and their values are = [ 12 ,  34 ,  56 ,  78 ,  13 ,  15 ,  17 ,  35 ,  37 ,  57 ,  24 ,  26 ,  28 ,  46 ,  48 ,  68 ]  .Therefore, the calculation of  I,II can be simplified as follows: where  1 is performance parameter of  1 and  ⇀  1×3 is one vector containing three performance parameters corresponding to and  can be estimated by multivariate linear regression method [15], and then the correlation models between  I,II and temperature field can be expressed as follows: where   denotes simulative  I,II ,   denotes monitoring  I,II , and ,  are fitting parameters, with the fitting results shown in Figures 11(a)∼11(f) as well.The fitting results show that all the fitting curves can well describe the linear correlation of scattered points with  close to 1 and  close to 0, verifying good modeling effectiveness of the correlation models.
(a), the continuous steel truss arch girder is composed of steel truss arch and steel bridge deck.Moreover, the steel truss arch comprises chord members with box-shaped cross-sections (top chord, arch rib chord, and bottom chord, resp.), diagonal web members with I-shaped cross-sections, vertical web members, and horizontal and vertical bracings as shown in Figures2(a) and 2(b); the steel bridge deck comprises top plate and transverse stiffening girder as shown in Figure 2(a).

8 (Figure 2 :
Figure 2: Layout of temperature and strain sensors on the steel truss arch girder (unit: mm).

Figure 3 :
Figure 3: History curves of global change and daily change in temperature data  1 .

Figure 4 :Figure 5 :
Figure 4: History curves of global change and daily change in temperature difference data  12 .

Figure 6 :
Figure 6: History curves of global change and daily change in static strain data  2 .

1 Figure 8 :
Figure 8: Extraction results  I,II of static strain data  1 .

Temperature T 1 (
I,II of  1 and temperature difference  12 ∘ C) (b)  I,II of  2 and temperature  1

Figure 9 :
Figure 9: Correlation scatter plots between  I,II and temperature and temperature difference.

Figure 10 :
Figure 10: Pareto diagram of explained variances of temperature and temperature difference data.
)∼13(f) respectively.Through analysis of their change trends, all of the residual static strains do not obviously present any

Figure 11 :
Figure 11: Correlation between simulative  I,II and monitoring  I,II .
Temperature data from temperature sensor   is denoted by   , temperature difference data acquired by   minus   is denoted by   , and static strain data from strain sensor   is denoted by   (,  = 1, 2, . . ., 8).The history curves of global change and daily change in temperature data and temperature difference data, taking  1 and  12 from training data, for example, are shown in Figures 2 ,  3 ,  4 ,  5 ,  6 ,  7 ,  8 },   is the th temperature difference data from set { 12 ,  34 ,  56 ,  78 ,  13 ,  15 ,  17 ,  35 ,  37 ,  57 ,  24 ,  26 ,  28 ,  46 ,  48 ,  68 },   is the th linear expansion coefficient of   ,   is the th linear expansion coefficient of   , and  is the constant term.

Table 1 :
Performance parameter values  1 ,  1 ,  2 ,  3 , and  of correlation models.,  1 ,  2 ,  3 , and  shown in Table1.In order to test the modeling effectiveness of the correlation models, temperature and temperature difference data from test data are substituted into (9) to calculate simulative  I,II after two-step analysis (detailed in Section 3.2), and then the scattered points between simulative  I,II and monitoring  I,II are plotted as shown in Figures11(a)∼11(f), all of which present good linear correlation.Therefore, the scattered points are furthermore least-square fitted by linear function: strains are obtained by simulative  I,II minus monitoring  I,II .Augmented Dickey-Fuller test at nominal significance level of 0.05 indicates that time history of residual static strains is one stationary stochastic process around the central line of 0  and within the scope of upper and lower limit [−50, 50], such as that of  1 shown in Figure12(a), which are considered relative initial state of static performance of Dashengguan Yangtze Bridge. 1 from the first combination gradually deviate from the central line with time and then surpass the upper limit; (2) residual static strains of  1 from the second combination fluctuate more fiercely with time around the central line and then surpass the upper and lower limit; (3) residual static strains of  1 from the third combination can be considered the summation of the first and second situations.Therefore, the degradation of static performance can be confirmed if one of the degradation regulations exits in time history of residual static strains.4.2.Assessment Results.Using the analytical method above, daily maximum and minimum values of temperature and temperature difference from assessment data are substituted into (9) to calculate simulative  I,II and then residual static strains of each static strain data are acquired by simulative  I,II minus monitoring  I,II as shown in Figures13(a 4.1.Assessment Method.Daily maximum and minimum values of temperature and temperature difference from training data are substituted into (9) to calculate simulative  I,II and then residual static