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applied sciences Article Modeling of Temperature Time-Lag Eect for Concrete Box-Girder Bridges 1 1 , 2 1 3 Kang Yang , Youliang Ding *, Peng Sun , Hanwei Zhao and Fangfang Geng School of Civil Engineering, Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing 210096, China Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA School of Architecture Engineering, Nanjing Institute of Technology, Nanjing 211167, China * Correspondence: civilchina@hotmail.com Received: 1 July 2019; Accepted: 7 August 2019; Published: 9 August 2019 Featured Application: According to field research results, a time-lag between structural response and temperature load is commonly encountered in practice. Due to the fact that it cannot be neglected for accurate structure health monitoring, a phase-shifting method is proposed; with the method, the time-lag eect can be eectively reduced, leading to a sound understanding of temperature load and its eect. Abstract: It is common to assume the relationship between temperature and temperature response is instantaneous in bridge health monitoring systems. However, a time-lag eect between temperature and thermal strain response has been documented by the analysis of monitored field data of concrete box-girder s. This eect is clearly reflected by the ring feature in the temperature-strain correlation curve. Inevitably, the time-lag eect has an adverse impact on the accuracy and reliability of state assessment and real-time warning for structural health monitoring (SHM) systems. To mitigate the influence of the time-lag eect, a phase-shifting method is proposed based on the Fourier series expansion fitting method. The time-domain signal is firstly converted into the frequency domain signal to compute the phase dierence between temperature data and response strain data at each decomposed order. Subsequently, the total phase dierence can be obtained by weighted summation. The signal processing eectively reduces the hysteresis loop area and enhances the correlation between the structural response data and the temperature data. When processing the daily data in dierent seasons, it is found that after subtraction by the proposed method, the linear feature becomes dominant in the relationship between temperature and the strain during long-term observation. Keywords: structural health monitoring; temperature eects; time-lag eect; Fourier series expansion; box-girder bridges 1. Introduction For concrete box-girder bridges, integral parts of the structural health monitoring (SHM) system include the structural state assessment system and the real-time warning system. A necessary prerequisite to achieve a reliable state assessment and real-time warning is the clear understanding of environmental load eects on the structure, especially the temperature related eects. In fact, the temperature eects have been investigated by many researchers. It is considered that the stress generated by the nonlinear temperature distribution is usually equivalent to the live load, and the temperature-induced stress is significant in the concrete structure [1]. Taysi et al. [2] studied the thermal characteristics of concrete under the influence of temperature variation on box-girder utilizing experiments and finite element simulation. Their work highlighted the distribution of thermal dierence Appl. Sci. 2019, 9, 3255; doi:10.3390/app9163255 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3255 2 of 15 and its influencing factors. Besides, Catbas et al. [3] discovered that temperature eects possess an important impact on the reliability of full bridges by analyzing a vast amount of monitored data. In order to analyze the influence of temperature eects on structure assessment [4,5], Huang et al. [6] applied Kalman filtering and Kalman cointegration to identify the damage and recognized that external eects (e.g., temperature) may mask the changes induced by structural damage. Moreover, Liang et al. and Li et al. [7,8] carried a sensitivity study based on the structural fundamental frequencies in which it has been found that variations in ambient temperature might lead to the misjudgment of structural health conditions. Their work emphasized on temperature eects, but the observation and analysis of temperature distributions need to be further investigated. The temperature distribution is the state of temperature at various positions inside and outside a structure at a certain time [9]. Preliminary studies of the temperature eects only considered the general temperature eects. Xu et al. [10] examined more than 8 years’ temperature displacements using mean values of dierent temperatures sensors and obtained the statistical law of temperature change. Xu et al. [11] studied the change of structural dynamic response using uniform temperature rise and fall models. Nevertheless, they all ignored the dierence in temperature distribution. To further study the temperature stress, the nonlinear eect should be taken into consideration, when horizontal and vertical temperature gradients exist [12,13]. The external factors aecting the temperature distribution of concrete structures are solar radiation, nighttime cooling, cold air flow, wind, rain, snow, and other meteorological factors [1]. The internal factors that aect the temperature distribution of the concrete are mainly determined by the thermophysical properties of the concrete and the geometrical dimensions of the components. The spatial heterogeneity and time-dependent nature of temperature distribution make it dicult to determine the exact relationship between temperature and structural response [14]. Considering the monitored response data of a real bridge inevitably includes the influences of live load and environmental factors, making the problem more complicated. To explore the temperature influences, separation of the temperature-induced part in data should be performed beforehand. For instance, Chenet al. [15] used the linear fitting method to determine the relationship between ambient temperature and temperature-induced strain while Hedegaard et al. [16] separated the time-dependent deformations from the temperature-related deformations by means of linear regression. They all achieved the purpose of separating the temperature-induced parts in the raw structural response data. Previously, some scholars have noticed discrepancies in dierent temperature history data of steel bridges. For example, Zhou et al. [17] analyzed the lateral temperature distribution and temperature time history of the steel box-girder, and found a dierence existed in the lateral temperature distribution of the box-girder at the same time. Brownjohn et al. [18] revealed that temporal and spatial temperature variations dominate displacement in long-span bridges. Meanwhile, the dierence can reach 5 and 12 C in winter and summer seasons, respectively. Furthermore, Brownjohn et al. [19] noticed there exists a time-lag eect due to thermal inertia eects, giving us a clear insight into the temperature time-lag eect. To account for this temperature time-lag phenomenon, Zhao et al. [20] selected the first five principal components as the main components to determine the overall response of the -structure. Moreover, Guo et al. [21] noted that displacement data and temperature data for a steel box-girder cable-stayed bridge represented a lag time of approximately 45 min. After directly shifting the temperature data by 45 min, they found the correlation between the two was significantly improved. However, the temperature distribution and the temperature-induced eects of concrete small box-girder bridges still need to be further investigated. Based on the above analysis, this study establishes that there is a significant time-lag eect in concrete box-girder bridges. This eect can be handled by a phase shift method, which is illustrated in a case study using field monitoring (strain and temperature) data. As a result, the aims of this paper include: (1) Investigate the time-lag phenomenon and its basic characteristics of concrete box-girder bridges; (2) cope with the issue of insucient correlation between temperature data and strain data. Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 15 girder bridges; (2) cope with the issue of insufficient correlation between temperature data and strain data. Appl. Sci. 2019, 9, 3255 3 of 15 2. Time-Lag Effect 2. Time-Lag Eect In this section, the time-lag effect is demonstrated through the field measured data of a real bridge. Firstly, the general situation of the Lieshihe Bridge and its SHM systems will be introduced. In this section, the time-lag eect is demonstrated through the field measured data of a real bridge. Then the collected data is illustrated and the time-lag effect is put forward. Finally, the character of Firstly, the general situation of the Lieshihe Bridge and its SHM systems will be introduced. Then the the time-lag effect in different seasons is discussed. collected data is illustrated and the time-lag eect is put forward. Finally, the character of the time-lag eect in dierent seasons is discussed. 2.1. Introduction of Lieshihe Bridge 2.1. Introduction of Lieshihe Bridge Concrete small box-girder are commonly used in medium and small span bridges. Thanks to the stability and the structural characteristics suitable for both positive and negative bending moments, Concrete small box-girder are commonly used in medium and small span bridges. Thanks to the the cross-section form (concrete small box-girder) is becoming more and more popular in designing stability and the structural characteristics suitable for both positive and negative bending moments, the small and medium span bridges. This study focuses on the temperature effects on small concrete box- cross-section form (concrete small box-girder) is becoming more and more popular in designing small girders. and medium span bridges. This study focuses on the temperature eects on small concrete box-girders. Lieshihe Bridge is a typical small box-girder bridge, located in Jiangsu province, China. This case Lieshihe Bridge is a typical small box-girder bridge, located in Jiangsu province, China. This case study is used as an example to illustrate the thermal load on a bridge and the time-lag effect of study is used as an example to illustrate the thermal load on a bridge and the time-lag eect temperature-induced strains on the bridge. The superstructure of Lieshihe Bridge is a 5-span of temperature-induced strains on the bridge. The superstructure of Lieshihe Bridge is a 5-span continuous box-girder, as shown in Figure 1a. Each of the spans has a length of 25 m, with the total continuous box-girder, as shown in Figure 1a. Each of the spans has a length of 25 m, with the total length of the bridge reaching 125 m. The main beams are small box-girder with a height of 1.5 m, as length of the bridge reaching 125 m. The main beams are small box-girder with a height of 1.5 m, shown in Figure 1b. On the longitudinal axis, the locations for temperature and strain sensors are as shown in Figure 1b. On the longitudinal axis, the locations for temperature and strain sensors arranged in the middle section of the second span, while their positions on the lateral axis are are arranged in the middle section of the second span, while their positions on the lateral axis are illustrated in Figure 1b. illustrated in Figure 1b. (a) Photo of Lieshihe River Bridge WEST EAST T4 T2 S10 T3 S9 S8 S7 S6 S5 S4 S3 S2 T1 S1 : Temperature sensors : Strain sensors (b) Deployment locations of the sensors Figure Figure 1. 1. Photo Photo of of Lie Lieshihe shihe River River Bridge Bridge and and deployment deployment l locations ocations of of sens sensors. ors. 2.2. Time-Lag Eect Concrete is a non-homogeneous material that consists of multiple dierent phases with complex thermal properties; heat transfer through complex geometries common in structural applications often Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 15 2.2. Time-Lag Effect Concrete is a non-homogeneous material that consists of multiple different phases with complex Appl. Sci. 2019, 9, 3255 4 of 15 thermal properties; heat transfer through complex geometries common in structural applications often results in nonuniform temperature distributions [22]. Temporal, spatial, and structural results in nonuniform temperature distributions [22]. Temporal, spatial, and structural characteristics characteristics exist in temperature distributions on the bridge structure. Apart from the difference exist in temperature distributions on the bridge structure. Apart from the dierence in temperature in temperature distributions (between different seasons and different days), a notable feature of the distributions (between dierent seasons and dierent days), a notable feature of the time delay time delay between temperature and temperature-induced response indeed exists. The phenomenon between temperature and temperature-induced response indeed exists. The phenomenon that the that the temperature-induced structural response lags following the temperature is referred to as the temperature-induced structural response lags following the temperature is referred to as the time-lag time-lag phenomenon of the temperature-induced response. A considerable time-lag effect between phenomenon of the temperature-induced response. A considerable time-lag eect between temperature temperature and temperature-induced strain can be found in concrete box-girder bridges. and temperature-induced strain can be found in concrete box-girder bridges. The measured data of temperature and strain history at location S10 on 4 April 2017 is shown in The measured data of temperature and strain history at location S10 on 4 April 2017 is shown Figure 2a,b, respectively. Furthermore, the correspondence of temperature and strain is plotted in in Figure 2a,b, respectively. Furthermore, the correspondence of temperature and strain is plotted in Figure 2c. Figure 2c. (a) Temperature at T4 location (b) Strain at S10 location (c) Temperature-strain correlation curve Figure 2. Time history plots of (a) temperature, (b) strain, and (c) temperature-strain correlation curve Figure 2. Time history plots of (a) temperature, (b) strain, and (c) temperature-strain correlation curve on 4 April 2017. on 4 April 2017. In Figure 2a, from the 24 h time history data, starting from 00:00 AM to 23:59 PM, the temperature In Figure 2a, from the 24 h time history data, starting from 00:00 AM to 23:59 PM, the temperature data shows a trend of decreasing between 00:00 AM and 04:15 AM, then an increasing between 04:15 data shows a trend of decreasing between 00:00 AM and 04:15 AM, then an increasing between 04:15 AM and 15:00 PM, and a downward trend between 15:00 PM and 23:59 PM at last. By comparing AM and 15:00 PM, and a downward trend between 15:00 PM and 23:59 PM at last. By comparing Figure Figure 2a,b, the strain is found to follow the same overall trend as the temperature. The curve of 2a,b, the strain is found to follow the same overall trend as the temperature. The curve of temperature temperature vs. corresponding strain at the same time are plotted in Figure 2c. The relationship vs. corresponding strain at the same time are plotted in Figure 2c. The relationship between the two between the two represents a fusiform annular shape, showing a significant nonlinear correlation. represents a fusiform annular shape, showing a significant nonlinear correlation. As a structural As a structural response, strain changes cyclically due to the process of heating and cooling from response, strain changes cyclically due to the process of heating and cooling from sunshine, which lags sunshine, which lags the temperature change, as a result showing the annular feature. The strain the temperature change, as a result showing the annular feature. The strain data is inevitably data is inevitably contaminated by some live load, such as trac load from moving vehicles, which contaminated by some live load, such as traffic load from moving vehicles, which determine a high determine a high number of local small-period fluctuations, as is shown in Figure 2b. The strain number of local small-period fluctuations, as is shown in Figure 2b. The strain obtained after the obtained after the removal of the live load is the temperature-induced strain. The separation methods will be introduced in Section 4.1. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 15 removal of the live load is the temperature-induced strain. The separation methods will be introduced Appl. in SSci. ect2019 ion ,4.1 9, 3255 . 5 of 15 The primary cause of the temperature time-lag effect may be the hysteresis of temperature transfer, as transfer of heat throughout the concrete cross section takes time. This hysteresis manifests The primary cause of the temperature time-lag eect may be the hysteresis of temperature transfer, as uneven temperature distributions, called thermal gradients or thermal inertia effects [19]. As strain as transfer of heat throughout the concrete cross section takes time. This hysteresis manifests as uneven represents an overall response of the structure measured at a single point, considering only one temperature distributions, called thermal gradients or thermal inertia eects [19]. As strain represents temperature measurement point may cause some inconsistencies. This paper investigates the an overall response of the structure measured at a single point, considering only one temperature temperature time-lag effect in concrete box-girder bridges, but further research is required to reveal measurement point may cause some inconsistencies. This paper investigates the temperature time-lag the ultimate cause of this phenomenon. It`s worth noting that the time-lag effects mentioned in this eect in concrete box-girder bridges, but further research is required to reveal the ultimate cause of this paper all refer to the time scale of a single day. Considering that the daily trend of temperature is phenomenon. It‘s worth noting that the time-lag eects mentioned in this paper all refer to the time most distinct and has a direct influence on the structure, this paper mainly deals with the daily time- scale of a single day. Considering that the daily trend of temperature is most distinct and has a direct lag effect. influence on the structure, this paper mainly deals with the daily time-lag eect. 2.3. Seasonal Characteristics 2.3. Seasonal Characteristics Firstly, the data of a typical day in spring and summer was selected, respectively. Then, the time-lag Firstly, the data of a typical day in spring and summer was selected, respectively. Then, the curve of temperature and temperature-induced strain was drawn, see Figure 3. It is obvious that the time-lag curve of temperature and temperature-induced strain was drawn, see Figure 3. It is obvious temperature and response strain curve possess typical seasonal characteristics. By comparing Figure 3a,b, that the temperature and response strain curve possess typical seasonal characteristics. By comparing it is observed that the hysteresis loop in spring, as shown in Figure 3a, is relatively more compact Figure 3a,b, it is observed that the hysteresis loop in spring, as shown in Figure 3a, is relatively more than that in summer, as shown in Figure 3b. It reflects that in spring the change in temperature is compact than that in summer, as shown in Figure 3b. It reflects that in spring the change in temperature comparatively slower. Hence, the degree of strain response lagging temperature is also less than that is comparatively slower. Hence, the degree of strain response lagging temperature is also less than that in summer. It also gives us a new way to measure the extent of time-lag effect through the hysteresis in summer. It also gives us a new way to measure the extent of time-lag eect through the hysteresis loop area. loop area. (a) 2017/04/20 (b) 2017/07/22 Figure 3. Plots of strain vs. temperature. Figure 3. Plots of strain vs. temperature. Furthermor Furthermore e, ,to to illustrate illustrate the the time-lag time-lage efect fect as as a a common common phenomenon phenomenon of of concr concr ete ete box-gir box-girde der r bridges, bridges, this thispaper paper investigated investigated thr thee reemonths months of of data. data. Additionally Additionally, , the the hyster hysteresi esiss loop loop ar area ea of of temperatur temperature e and andstrain straindata data of ofevery everysingle singleday dayar are e plotted plotted in in Figur Figure e 4 .4.Fr From om Figur Figure e 4 ,4,a a time-lag time-lag phenomenon phenomenon of of di d ifferent erent extent extenin t in almost almost every every day day indeed indeed exists. exists. Mor Moreov eover er, , the the hyster hyster esis esis loop loop ar area ea in in sum summer mer (July) (July) is is lar lar ger ger than than that that i inn winter winter (Nov (November). ember). In summary, this part gives us a clear understanding of the time-lag phenomenon of temperature eects, a very common phenomenon in concrete box-girder bridges. Moreover, this phenomenon in the form curve annular feature can be directly reflected by the temperature vs. strain graph. Based on the analysis of data in dierent seasons, a seasonal characteristic was found. Appl. Sci. 2019, 9, 3255 6 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 15 Figure 4. Hysteresis loop area in three months of 2017. Figure 4. Hysteresis loop area in three months of 2017. 3. Methods In summary, this part gives us a clear understanding of the time-lag phenomenon of For eliminating the time-lag eect, the premise is obtaining the phase dierence between the temperature effects, a very common phenomenon in concrete box-girder bridges. Moreover, this temperature and the temperature-induced strain. To scientifically and accurately acquire the phase phenomenon in the form curve annular feature can be directly reflected by the temperature vs. strain dierence, a method based on Fourier fitting [23] is proposed. graph. Based on the analysis of data in different seasons, a seasonal characteristic was found. Since any continuous periodic signal can be composed of a set of appropriate sinusoids, the Fourier transform was firstly performed on the temperature and the temperature-induced strain [24]. 3. Methods Specifically, the original data was fitted to obtain the phase dierence of various orders [25]. Then, the For eliminating the time-lag effect, the premise is obtaining the phase difference between the phase dierence between the two signals was weighted and summed, so the total phase dierence temperature and the temperature-induced strain. To scientifically and accurately acquire the phase was obtained. Furthermore, the flow chart of the Fourier series expansion fitting method is shown in difference, a method based on Fourier fitting [23] is proposed. Figure 4. Since any continuous periodic signal can be composed of a set of appropriate sinusoids, the According to Figure 5, the specific processes of the method are as follows. Fourier transform was firstly performed on the temperature and the temperature-induced strain [24]. Specifically, the original data was fitted to obtain the phase difference of various orders [25]. Then, the phase difference between the two signals was weighted and summed, so the total phase difference was obtained. Furthermore, the flow chart of the Fourier series expansion fitting method is shown in Figure 4. According to Figure 5, the specific processes of the method are as follows. Appl. Sci. 2019, 9, 3255 7 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 15 Raw data (Temperature&strain) Wavelet decomposition & reorganization Smoothed data Fourier frequency decomposition RMSE <0.2 P order decomposition P=max(m,n) P=max(m,n) Minimum expansion orderorder Minimum expansion order order m of temeperature n of strain Phase difference of each same frequenciy Weighted summation Total phase difference Data conversion Lag time Figure 5. The flow chart of the Fourier series expansion. Figure 5. The flow chart of the Fourier series expansion. 3.1. Temperature-Induced Response Separation 3.1. Temperature-Induced Response Separation The temperature history data and structural strain data of the same day are represented as f temp temp The temperature history data and structural strain data of the same day are represented as and f , respectively. The wavelet decomposition and reconstruction method is used to separate the sr measured strain data. Consequently, the temperature-induced strain f and live-load strain f f sr,tem sr,load sr and , respectively. The wavelet decomposition and reconstruction method is used to separate the are extracted. The principle and process of the method are described by Zhao et al. [20]. sr,tem measured strain data. Consequently, the temperature-induced strain and live-load strain 3.2. Fourier Frequency Decomposition sr,load are extracted. The principle and process of the method are described by Zhao et al. [20]. The Fourier frequency decomposition in the time domain of f and f is used to obtain the sr,tem temp frequency components of the data signal. The specific steps are as follows. 3.2. Fourier Frequency Decomposition 3.2.1. The Fourier Expansion The Fourier frequency decomposition in the time domain of and is used to sr ,tem temp According to the Fourier expansion [26], the phase corresponding to each frequency component obtain the frequency components of the data signal. The specific steps are as follows. can be obtained as 3.2.1. The Fourier Expansion 1 1 1 X X X a a 0 0 f (x) = + a cos(kx) + b sin(kx) = + c sin(kx + ) (1) f ourier k k k k According to the Fourier expansion [26], the phase corresponding to each frequency component 2 2 k=1 k=1 k=1 can be obtained as where a , a , a ,::: , a are the cosine coecients of the Fourier factor and b , b ,::: , b are the sine 0 1 2 k 1 2 k aa f (x)= + a cos k x + b sin k x = + c sin k x+ ( ) ( ) ( ) (1) fourier k k k k coecients of the Fourier factor. And x is a discrete time variable, as to the temperature and strain data k=1 k=1 k=1 in Lieshihe bridge, x = 1/1440 [1, 2, 3,. . . ,1440], owing to the sampling frequency is 1/1 min. is the where a ,a ,a ,...,a are the cosine coefficients of the Fourier factor and b ,b ,,b are the sine reference 0frequency 1 2 k of the raw data, and is the phase of k-th order. Moreover12 , the amplitude k of each coefficients of the Fourier factor. And x is a discrete time variable, as to the temperature and strain order data signal is 2 2 data in Lieshihe bridge, x = 1/1440 × [1, 2, 3...1440], owing to the sampling frequency is 1/1 min. c = a + b (2) k k is the reference frequency of the raw data, and is the phase of k-th order. Moreover, the amplitude of each order data signal is Appl. Sci. 2019, 9, 3255 8 of 15 The phase of each order of the signal is = arctan(a /b ) , k = 1, 2, 3,::: (3) k k k 3.2.2. Determining the Minimum Order of the Fourier Series Expansion P The expansion order is determined by trial-by-level trials until the residuals root mean square error (RMSE) meets the requirements. The S order RMSE(S) between the Fourier expansion value f f ourier f f ( temp) f ourier i=1 and the temperature data f can be calculated by RMSE(S) = , where S is the temp expansion order. The minimum expansion order M of temperature is determined by RMSE(m) < 0.2. Then, the expansion order N of strain data is also obtained in the same way. At last, the larger value of M and N is taken as P as the final expanded order. The value of RMSE is related to the absolute value of the data value, the degree of dispersion, and so on. As a result, RMSE has no certain criterion for dierent kinds of data. The criterion of RMSE is determined by the correlation coecient of fitted data and raw data in this paper. 3.3. Calculating the Phase Dierence The phase dierence D between the separated structural response data f and the temperature i sr,tem data f are solved at the same frequency. The specific steps are as follows. temp 3.3.1. Obtain the Phase of Temperature and Strain Respectively Calculate the phase of the structural response data f and the phase of the sr,tem sr,k temp,k temperature data f according to the Fourier series approximation expression [27]. The Fourier series temp approximation expression for the structural response data f and temperature data f are sr, f ourier temp, f ourier sr,0 f (x) = + c sin k x + (4) sr sr, f ourier sr,k sr,k k=1 temp,0 f (x) = + c sin k x + (5) temp temp, f ourier temp,k temp,k k=1 where and represent the reference frequency of strain data and temperature data, respectively. sr temp They can be computed by 2/L. L is the length of the normalized cycle, which is closely related to the baseline period of the raw data and can be calculated automatically by the MATLAB Fourier series fitting program. Lambda varies with data of dierent days. It is mainly dependent on the shape feature of the data. As the shape of data in dierent single days is roughly similar, so the value of Lambda is approximate 5/2. 3.3.2. Calculate the Phase Dierence The phase dierence D can be calculated according to the following formula: D = , i = 1, 2, 3,::: , P (6) i temp,i sr,i where is the i-th order temperature data phase and is the i-th order structure response temp,i sr,i data phase. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 15 where is the i-th order temperature data phase and is the i-th order structure response temp,i si r, data phase. Appl. Sci. 2019, 9, 3255 9 of 15 3.4. Determining the Total Phase Difference and Lag Time 3.4. Determining the Total Phase Dierence and Lag Time Through a mass data research, the total phase difference can be obtained by the weighted summation of phase differences in each order. Moreover, the weight is proportional to the square of Through a mass data research, the total phase dierence can be obtained by the weighted the frequency amplitude of each order. summation of phase dierences in each order. Moreover, the weight is proportional to the square of the frequency amplitude of each order. P w x , x , x =cc / ( ) (7) jP 12 temp,, j temp j j=1 2 2 w (x , x , x ) = c / c (7) j 1 2 P temp,j temp,j j=1 where is the phase weight of the -th order. where w is the phase weight of the j-th order. The delay effect on the correlation is eliminated or reduced by translating phase difference , The delay eect on the correlation is eliminated or reduced by translating phase dierence ', which can be calculated by the following equation: which can be calculated by the following equation: = w (8) ii i=1 ' = w D (8) i i where is the total delay phase to be eliminated. i=1 Since the temperature data changes in cycles of days and the overall trend is a half-sine function, where ' is the total delay phase to be eliminated. the lag time can be determined from the ratio of the lag phase difference to the half cycle of the sine Since the temperature data changes in cycles of days and the overall trend is a half-sine function, function: the lag time can be determined from the ratio of the lag phase dierence to the half cycle of the sine T = 1440 / (9) lag sr function: T = ' 1440/ (9) lag sr where is the lag time in minutes. lag where T is the lag time in minutes. lag 4. Case Study 4. Case Study This section includes two parts. First, the processing of separating temperature-induced strain This section includes two parts. First, the processing of separating temperature-induced strain from the field measured data is given. Then, according to the above data, the phase subtraction from the field measured data is given. Then, according to the above data, the phase subtraction method method based on the Fourier fitting method is described in detail and its effectiveness is verified. based on the Fourier fitting method is described in detail and its eectiveness is verified. 4.1. Separation of Temperature-Induced Strain 4.1. Separation of Temperature-Induced Strain The measured strain data includes the interaction of live load and temperature effects. Therefore, The measured strain data includes the interaction of live load and temperature eects. Therefore, the separation of temperature strain is a prerequisite for thermal strain response studies. The the separation of temperature strain is a prerequisite for thermal strain response studies. The measured measured data of the S6 strain measuring point of the bridge on 9 June 2017 is shown in Figure 6. data of the S6 strain measuring point of the bridge on 9 June 2017 is shown in Figure 6. (a) Raw strain and temperature-induced strain. Figure 6. Cont. Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 15 Appl. Sci. 2019, 9, 3255 10 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 15 (b) Live load-induced strain. (b) Live load-induced strain. Figure 6. Strain time history of the raw data and the processed data. Figure 6. Strain time history of the raw data and the processed data. Figure 6. Strain time history of the raw data and the processed data. In Figure 6, the separation of temperature effects is achieved by the wavelet decomposition and reconst In Fi In gu Figur ructio re 6, e tn he 6,met separ the hod separation ation [20 ] of . As tem of shown peratu temperatur re ineffect Figure e s e is ects 6 ac , th hieved is is achieved met by hod th ecan by wavelet the separ wavelet deco ate tem mp decomposition osition perature an d strain reconst and effect ructio reconstr ively. n met Th uction e hod raw method [str 20ai ]. n As data [20 shown ]. is As finally shown in F decom igure in Figur po 6, sed th eis 6, into met thistem hod method perature can can separ -ind separate atuced e tem stemperatur peratu train (Fi reg ure str eai strain 6n a, the effect ered ively. ectively lineTh ) .and e The ra li w raw ve str lo ai strain ad n data stra data i is n (Fig finally is finally ure decom 6b). decomposed It po is wort sed into h no into tem ting temperatur perature that the -ind live e-induced uced loads -in train du strain ced (Fig (Figur str ure ain 6a e in , 6 th Fi a, e g the ure red rli ed 6ne b line) i) s and larger and live th live lo an ad load th stra at strain in in F(Fig igure (Fi ure gur 2b 6 e .b Th 6 ). b). It e is re It as wort is on worth h isno th noting ting e location ththat at th o the e f lsen ive live sloa or load-induced S d6 -in is du in ced the str cent strain aier n in of in Fi Figur th ge ure bridge e 6b 6b iis s where t llar arger gerth he g than an ir th that der wi at in inF ll Figur igure with e2b stand more 2b. . Th The e reras eason v on ehicle is is ththe l eo lo ads. cation location of of sen sensor sor S6 S6 is in is th ine the cent center er of th ofe the bridge bridge where t wher he g e the irder wi girderll will with withstand stand more mor vehicle e vehicle loads. loads. 4.2. Phase Subtraction 4.2. Phase Subtraction 4.2. Phase Subtraction The separated temperature-induced strain and temperature data was fitted subject to the Fourier The separated temperature-induced strain and temperature data was fitted subject to the Fourier de Th com e separ posi ate tiod n. tem Furtherm peratur o ere, -ind thuced e tem str per aiature n and ftem itting pera curves ture d oata f th e wa 1st s fi order, tted s ub 4th ject order, to thand e Fou 8th rier order decomposition. Furthermore, the temperature fitting curves of the 1st order, 4th order, and 8th order decom decom positio pon. sitio Furtherm n on 20 o Apr re, th il e 2017 temar per e ature shown fitting in Figure curves 7ao . f W th ith e 1st th e order, increas 4th e of order, the Four and 8th ier exp order ans ion decomposition on 20 April 2017 are shown in Figure 7a. With the increase of the Fourier expansion decom order, positio the n co on nsis 20 ten Apr cy il wa 2017 s bet are ter sho , as wn show in Fn igure in Fi 7g au . re With 7b. th When e increas the e data of th wa e Four s exp ier anded expans to ion the 6th order, the consistency was better, as shown in Figure 7b. When the data was expanded to the 6th order, order, order, the the consis acc ten ura cy cy re waq s ui bet reter men , as t wa show s me n t. in Figure 7b. When the data was expanded to the 6th the accuracy requirement was met. order, the accuracy requirement was met. (a) Fitting process. (a) Fitting process. Figure 7. Cont. Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 15 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 15 Appl. Sci. 2019, 9, 3255 11 of 15 (b) RMSE of different orders. Figure 7. The Fourier curve fitting results (20 April 2017). (b) RMSE of different orders. Figure 7. The Fourier curve fitting results (20 April 2017). Figure 7. The Fourier curve fitting results (20 April 2017). The direct consequence of the time-lag phenomenon is the decline in correlation between the temperature and the temperature response. The hysteresis loop area directly reflects the degree of The direct consequence of the time-lag phenomenon is the decline in correlation between the The direct consequence of the time-lag phenomenon is the decline in correlation between the the time-lag phenomenon. Specifically, the larger the area, the more significant the lag effect. temperatur temperature e and and the th temperatur e temperature e response. response. The Thyster he hysteresi esis loop s loop areaarea directly directly reflects reflthe ects degr the ee deof gree theof Therefore, the hysteresis loop area and the correlation coefficient can be used as indicators to verify time-lag phenomenon. Specifically, the larger the area, the more significant the lag eect. Therefore, the the time-lag phenomenon. Specifically, the larger the area, the more significant the lag effect. the effectiveness of subtracting the time-lag phenomena by the Fourier series expansion method. hyster Therefore esis loop , the ar hystere ea andsi the s loop corr elation area and coe th e cient correl can atio be n coeff usedicien as indicators t can be used to verify as indi thece ators ectiveness to verify With the Fourier fitting method, the phase difference of the measured temperature and strain of subtracting the time-lag phenomena by the Fourier series expansion method. the effectiveness of subtracting the time-lag phenomena by the Fourier series expansion method. data on 22 July 2017 (summer) was calculated. The lag time of the S8 strain measuring point and T4 WWith ith the thFourier e Fourier fitting fitting method, method, the the phase phase did ifferenc erence of e of the thmeasur e measured ed temperatur temperature e and and strain strain temperature measuring point can be calculated through Equation 9. As a result, the lag time was data on 22 July 2017 (summer) was calculated. The lag time of the S8 strain measuring point and data on 22 July 2017 (summer) was calculated. The lag time of the S8 strain measuring point and T4 approximately 176 min in the summer season. Then, a translation of the temperature data by the lag T4 tem temperatur perature e meas measuring uring point point can can be be cal calculated culated th thr rough ough Eq Equation uation 9. (9). As As a res a ul result, t, the the lag lag time time was time achieves the goal of subtracting the time-lag phenomenon. The temperature and strain was approximately 176 min in the summer season. Then, a translation of the temperature data by approximately 176 min in the summer season. Then, a translation of the temperature data by the lag correlation before and after the subtraction is shown in Figure 8. Using the same method to subtract the time lag time achieve achieves s the th goe al goal of of subtract subtracting ing ththe e time time-lag -lag ph phenomenon. enomenon. Th The e temperatur temperature e and andstrain strain the time-lag phenomenon in winter, Figure 9 is obtained. Meanwhile, through Equation 9, the lag correlation before and after the subtraction is shown in Figure 8. Using the same method to subtract correlation before and after the subtraction is shown in Figure 8. Using the same method to subtract time was calculated to be around 129 min in the winter season. As mentioned before, the the thtime-lag e time-lag phenomenon phenomenon in in winter winter, , Figur Figure e 9 is 9 obtained. is obtaineMeanwhile, d. Meanwhile thr , th ough roug Equation h Equation (9), 9, the the lag lag displacement and temperature data for steel bridges possess a lag time of approximately 45 min in time was calculated to be around 129 min in the winter season. As mentioned before, the displacement time was calculated to be around 129 min in the winter season. As mentioned before, the the research by Guo et al. [21]. As a result, the lag time in concrete structures is longer than that in and distemperatur placement e and data tem for per steel ature bridges data fo possess r steel a bri lag dges time poof sses appr s a oximately lag time of 45approx min inimatel the resear y 45 ch min by in steel structures. Guo et al. [21]. As a result, the lag time in concrete structures is longer than that in steel structures. the research by Guo et al. [21]. As a result, the lag time in concrete structures is longer than that in steel structures. (a) Before the phase difference is translated (b) After the phase difference is translated Figure Figure 8.8.T Temper emperatur atur e e vs. vsstrain . strain plots plots i inn summer summer ( (22 22 July July 2017). 2017). (a) Before the phase difference is translated (b) After the phase difference is translated Figure 8. Temperature vs. strain plots in summer (22 July 2017). Appl. Appl. Sci. Sci. 2019 2019 , 9 , ,93255 , x FOR PEER REVIEW 12 12of of 15 15 (a) Before the phase difference is translated (b) After the phase difference is translated Figure 9. Temperature vs. strain plots in winter (27 November 2017). Figure 9. Temperature vs. strain plots in winter (27 November 2017). By calculating the hysteresis areas and lag time of raw data and phase dierence eliminated data, By calculating the hysteresis areas and lag time of raw data and phase difference eliminated Table 1 was obtained. From Table 1, the hysteresis curve and lag times in winter with those in summer, data, Table 1 was obtained. From Table 1, the hysteresis curve and lag times in winter with those in respectively, were compared. Consequently, after subtracting, the hysteresis loop was notably reduced summer, respectively, were compared. Consequently, after subtracting, the hysteresis loop was and the correlation coecient was significantly improved. Therefore, the Fourier fitting method was notably reduced and the correlation coefficient was significantly improved. Therefore, the Fourier verified to be eective. fitting method was verified to be effective. Table 1. Comparison of indicators before and after phase dierence elimination. Table 1. Comparison of indicators before and after phase difference elimination. Date Correlation Coecient Hysteresis Loop Area Date Correlation Coefficient Hysteresis Loop Area Before elimination 0.698 29.67 Before elimination 0.698 29.67 22 July 2017 2017/07/22 After elimination 0.957 4.41 After elimination 0.957 4.41 Before elimination 0.885 13.65 Before elimination 0.885 13.65 27 November 2017 2017/11/27 After elimination 0.974 3.27 After elimination 0.974 3.27 With the proposed method, the time-lag phenomenon can be reduced. Furthermore, the long- With the proposed method, the time-lag phenomenon can be reduced. Furthermore, the long-term stable term r stable elationship relation between ship bet temperatur ween temper e load ature and load the an str d uctural the struct corr uresponding al correspornd esponse ing respon can be se mor can be e more clearly reflected. Specifically, the same temperature and strain measurement points of four days clearly reflected. Specifically, the same temperature and strain measurement points of four days in the in the four four s seasons easons of 2017 of 2017 wer were e selected selected to draw to draw the the temperatur temperature e–strain –strain curve. curveThen, . Then, the the subtracted subtracted data curve was plotted in Figure 10. It is clear to see, although the specific distribution difference of data curve was plotted in Figure 10. It is clear to see, although the specific distribution dierence of strai strain n var varying ying with with tem temperatur peraturee ex exists ists in in d di ifferent erent seasons, seasons, th the e hyster hysteres esis is loops loops in in four four seasons seasons possess similar slopes overall. In particular, by eliminating the phase difference, the reduced curve possess similar slopes overall. In particular, by eliminating the phase dierence, the reduced curve can can more clearly reflect the corresponding linear relationship characteristics of temperature-induced more clearly reflect the corresponding linear relationship characteristics of temperature-induced eects effects and temperature in different seasons. This feature reflects the specific mapping relationship and temperature in dierent seasons. This feature reflects the specific mapping relationship between between temperature and structure response and it is related to structure physical properties, such temperature and structure response and it is related to structure physical properties, such as structural as structural geometry and material properties. geometry and material properties. In summary, the wavelet decomposition and reconstruction method is useful and handy to extract temperature-induced strain. The lag time in a concrete structure is longer than that in a steel structure. Moreover, according to the comparison of hysteresis loop areas and correlation coecient before and after subtracting the time-lag eect, the eectiveness was verified. Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 15 Appl. Sci. 2019, 9, 3255 13 of 15 Figure 10. Temperature–strain curve and the subtracted data curve in dierent seasons. Figure 10. Temperature–strain curve and the subtracted data curve in different seasons. 5. Conclusions In summary, the wavelet decomposition and reconstruction method is useful and handy to Temperature is the most common and critical environmental load acting on bridge structures. extract temperature-induced strain. The lag time in a concrete structure is longer than that in a steel The non-uniformity distribution and time-dependent character of the temperature leads to the structure. Moreover, according to the comparison of hysteresis loop areas and correlation coefficient complexity of temperature response. The existence of a structural response lag behind a reference before and after subtracting the time-lag effect, the effectiveness was verified. temperature was investigated by analyzing the monitored data of a small concrete box-girder bridge. This time-lag eect causes interference between temperature load and structural response, leading to 5. Conclusions diculties in the real-time warning of the SHM system. To address this problem, this study proposed Temperature is the most common and critical environmental load acting on bridge structures. a phase dierence subtraction method based on the Fourier series fitting algorithm. Then, the method The non-uniformity distribution and time-dependent character of the temperature leads to the was verified to be eective through a case study. The main conclusions are as follows: complexity of temperature response. The existence of a structural response lag behind a reference temperature was investigated by analyzing the monitored data of a small concrete box-girder bridge. 1. The temperature-induced eect of concrete small box-girder determine a time-lag phenomenon This time-lag effect causes interference between temperature load and structural response, leading to under the nonuniformed temperature load. Specifically, the time-lag curve in summer is fuller difficulties in the real-time warning of the SHM system. To address this problem, this study proposed than that in winter, indicating that the time-lag phenomenon in summer is more notable. a phase difference subtraction method based on the Fourier series fitting algorithm. Then, the method 2. The hysteresis loop areas and correlation coecients can be used as two indicators to describe the was verified to be effective through a case study. The main conclusions are as follows: extent of the time-lag eect. 3. The eectiveness of the Fourier series expansion and least-squares fitting method to subtract the 1 The temperature-induced effect of concrete small box-girder determine a time-lag phenomenon time-lag eect was confirmed through a case study. Furthermore, using this method, the time-lag under the nonuniformed temperature load. Specifically, the time-lag curve in summer is fuller eect is reduced and the corresponding linear relationship characteristics between temperature than that in winter, indicating that the time-lag phenomenon in summer is more notable. load and temperature-induced strain in dierent seasons can be more clearly reflected. 2 The hysteresis loop areas and correlation coefficients can be used as two indicators to describe 4. The lag time in a concrete structure is longer than that in a steel structure. the extent of the time-lag effect. 3 The effectiveness of the Fourier series expansion and least-squares fitting method to subtract the time-lag effect was confirmed through a case study. Furthermore, using this method, the time- Author Contributions: Original ideas K.Y. and Y.D.; data analysis, K.Y.; funding acquisition, Y.D.; methodology, K.Y. and H.Z.; writing—original draft preparation P.S.; writing—review and editing, P.S. and F.G. lag effect is reduced and the corresponding linear relationship characteristics between temperature load and temperature-induced strain in different seasons can be more clearly Funding: This research was funded by the National Natural Science Foundation of China (51578138 and 51608258) and the Fundamental Research Fund for the Central Universities. reflected. 4 The lag time in a concrete structure is longer than that in a steel structure. 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Applied Sciences – Multidisciplinary Digital Publishing Institute
Published: Aug 9, 2019
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