The hinge joints of the assembled concrete multi-girder bridge (ACMGB) are the main components of transverse transmission of loads and the decline of their transmission capacity would lead to the degradation of the transverse co- operative working performance (TCWP) of the multi-beam system. As the key working performance of this kind of bridges, its severe degradation would result in the single-girder-bearing phenomenon inducing bridge collapse. In or- der to detect TCWP degradation timely, this paper proposes a vibration-based monitoring scheme of the ACMGB, where structural vibration information are collected by accelerometers and dynamic strain guages at the bottom of gird- ers in mid-span. These monitoring data are then used for identi ng transverse mode shape of the multi-beam system and its derivated index in order to eval- uate TCWP. This scheme has been implemented on three adjacent ACMGBs in Tongji Road Viaduct in shanghai and the preliminary analysis of monitoring data proves the e ectiveness and rationality of the monitoring scheme proposed in this paper, which has the potential of the TCWP evaluation. Corresponding author Email address: dandanhui@tongji.edu.cn (Danhui Dan) Preprint submitted to Engineering Structures February 21, 2020 Keywords: Assembled concrete multi-girder bridge, transverse mode shape, strain mode, transverse collaborative working performance, structure health monitoring 1. Introduction The assembled concrete multi-girder bridge (ACMGB), a common kind of medium and small span bridge, is widely used in China [1], South Korea[2], Japan[3] and USA[4]. For example, the concrete hollow core slab bridge[5], the adjacent box - beam bridge [6] and the concrete bulb-tee bridge[7] can be 5 classi ed as the ACMGB. Its universal construction method is to erect pre- cast concrete girders on site rstly, then construct hinge joints (shear keys) or diaphragm, and nally pour integrated deck pavement to ensure the whole struc- ture working monolithically. Therefore, hinge joints, girders and deck pavement are the main components of the superstructure of this type bridges, as shown 10 in Fig. 1 where reinforcement and accessory structures are ignored. Given its advantages over cast-in-place concrete bridges, ACMGB can be seen as an early but still popular practice of superstructure construction of accelerated bridge construction (ABC) in USA[4, 8]. From the viewpoint of design, the mechanical properties of ACMGB should 15 be similar to that of an orthotropic concrete slab or grillage system. In other words, when a vehicle acts on one of the girders, the adjacent girders should also bear part of the load through the load-transmitting function of transverse joints, which is de ned as transverse cooperative working performance (TCWP). TCWP prevents the single-girder-bearing phenomenon, as shown in Fig. 2. 20 However, after years of engineering practice, the ACMGB usually su ers from longitudinal re ective cracking, leakage as well as prestressed strand corrosion of the girders. It's believed that the cracking of hinge joints is the main cause of other diseases in the structural perspective, while vehicle load and environment conditions are the external cause of them. After damage of the hinge joints 25 develops, or to say, degradation of TCWP, the non-uniform load distribution 2 girderdeck pavement hinge joint Fig. 1. Schematic diagram of main compo- nents of ACMGB superstructure multi-girders work ing collaboratively (a) (b) single girder bearing balanced position loading positionFig. 2. Schematic diagram of working state of ACMGB in cross-sectional perspective problem between girders under vehicle load is intensi ed and the deck pavement has to play a bigger role on the transverse force transimission which may results in longitudinal cracks in the deck pavement. Also, the cracked hinge joint would be vulnerable to the moisture of rains leading to prestressed tendons corrosion, 30 and water stains at the bottom of girders. Obviously, after the occurrence of the above diseases, the TCWP of the ACMGB and resistance of girders would deteriorate. Therefore, the cracking of hinge joints can be regarded as a sign of deteriorated TCWP which may induce the single-girder-bearing phenomenon and even failures of the ACMGB[9{11]. 35 At present, studies on the cracking cause of hinge joints have developed from the early insucient strength of hinge joints under vehicle loads [12] to environmental conditions, like temperature and humidity[13], and cumulative fatigue damage[14]. Correspondingly, the design solution includes new hinge joint structures[6], transverse prestressing[15], the increase of pavement layer 40 thickness, and the recent use of UHPC material[16]. The application of UHPC to hinge joints seems to be a state-of-the-art solution[17], but it's still in the early stage of promotion and the improvement of the long-term performance of hinge joints has not been fully and widely veri ed in the industrial community. Consequently, in addition to improving hinge joint performance in terms of 45 design, it is more important for ACMGBs in service period to identify hinge joint transmission function in time, evaluate TCWP, and carry out appropriate hinge 3 joint maintenance. The hidden hinge joints assessment methods are divided into four ones: visual inspection, load test[18], nondestructive testing[19, 20] and structure health monitoring (SHM)[21{26]. The rst three methods above 50 can hardly indentify the early disease in hinge joints in time. Since the SHM technology takes full use of the sensors installed on the structure to continuously measure of the structure response, it can overcome the shortcomings of the above three methods, and has the potential of realizing early damage detection of hinge joints and reliable evaluation of TCWP. 55 The transverse force transfer mechanism of ACMGB is not only re ected in the transverse distribution of quasi-static e ect between girders, but also in- cluded in the vibration response's transverse distribution. For the former one, random vehicle's self-weight is a interference factor for TCWP evaluation so it's dicult to directly apply quasi-static monitoring information for assessing 60 TCWP. To overcome this problem, the authors analyzed a large quantity of quasi-static information obtained from long-term monitoring to extract an con- vergent index in a statistical sense, and indirectly realized the monitoring of the TCWP of the ACMGB[22]. In order to achieve statistical stability, this method require calculation of a large amount of data, so the real-time performance is 65 poor. Since the modal components in the structural vibration response are al- most independent of the external excitation[27], it's convenient to realize the independent identi cation of the TCWP through vibration response monitor- ing of the ACMGB, without considering the random in uence of vehicle load. This technology route relies on the identi cation of modal parameters, so big 70 data seems not to be necessary. Besides, small computation and good real-time performance should be its advantages. Considering all this, this paper starts from the simpli ed dynamic model of the ACMGB of which modal analysis principle is studied and on this basis a vibration-based monitoring scheme for monitoring and evaluating the TCWP 75 of ACMGBs is proposed (section 2). Next, an implementation on an actual ACMGB is introduced (section 3). Then, Through the preliminary analysis of the monitoring data, the feasibility of the proposed scheme are revealed (section 4 4). Lastly, some problems and challenges in the further application are discussed (section 5). This paper can be seen as a preliminary study on the vibration- 80 based TCWP monitroing route of the ACMGB. 2. Vibration-based lateral cooperative performance monitoring scheme 2.1. Monitoring placement scheme In the previous work[23], it has been proved that, under certain assumptions, the modal vibration of ACMGBs can be analyzed by means of a multi-beam 85 model connected by distributed springs. In this section, rst of all, the modal analysis process of the multi-beam model is given. Secondly, the TCWP inver- sion route of ACMGB based on the modal parameter-derivated index is estab- lished and the corresponding sensor placement scheme of the TCWP monitoring system is put forward. 90 distributed springrigid arm distributed springrigid arm girder Fig. 3. Distributed springs-connected multi-beam model of ACMGB Considering the mechanical properties of the ACMGB structure, the follow- ing assumptions are made: Every beam is a uniform Euler-Bernoulli beam, ignoring the torsional e ect, and each beam has only vertical displacement degree of freedom. 5 The massless springs are longitudinally continuous distributed and only 95 transfer vertical shear force between adjacent beams. The TCWP between the single beams is characterized by the sti ness of the distributed spring. Each beam in longitudinal direction is simply supported. The cross-section characteristics and material properties of each beam are constant in longitudinal direction and the sti ness per unit of each 100 distributed spring is also constant in longitudinal direction. The schematic diagram of the simpli ed distributed spring-connected multi- beam model is shown in Fig. 3. The direction xis longitudinal and the direction yis transverse. The sti ness per unit of ith distributed spring is kiandviis the vertical displacement of the ith beam. Note that the virtual rigid arm in Fig. 3 105 is used to simulate the transverse transfer path of vertical shear force, which is an in nite, massless element of sti ness. The modal analysis process of distributed spring-connected multi-beam sys- tem is as followed: Firstly, execute time and space variables separation of dis- placement. Then assume global mode shape v(j) is a product of mode shape 110 for single beam system 'j(x) and transverse proportion coecient vector aj to separate longitudinal and transverse space variables. Thirdly, using the an- alytical characteristics of simply supported Euler-Bernoulli beam system, the solution of the jth modal parameters of the distributed spring-connected multi- beam system can be transformed into the following matrix eigenvalue problem 115 Kjaj=!2 jMaj (1) !jis natural frequency for mode jof the multi-beam system, matrix Kj andMare sti ness matrix and the mass matrix regarding multi-beam system 6 respectively, given by Kj=2 6666666666664~!2 k+11 0 0 0 1 2~!2 k+1+22 0 0 02 3~!2 k+2+33 0 .................. 0 0 0 0  n1 0 0 0 0  n~!2 k+n13 7777777777775 (2) M=2 66666666666641 0 0 0 0 020 0 0 0 030 0 .................. 0 0 0 0...0 0 0 0 0n3 7777777777775(3) ~!kis the corresponding modal frequency of the longitudinal base function or to say the longitudinal mode shape of a single beam, nis the sti ness of the nth 120 beam ,n1is the sti ness of the ( n1)th distributed spring, and nis the mass of the nth beam. The above parameters are all dimensionless. It can be further assumed that the sti ness and mass of each beam is iden- tical, which means i= 1,i= 1,i= 2;3;n, and Kj= ~!2 kI+K, hence the eigenvalue problem (1) can be transformed into Kaj=aj (4) K=2 666666666666411 0 0 0 11+22 0 0 022+33 0 .................. 0 0 0 0  n1 0 0 0 0  +n13 7777777777775;=!2 j~!2 k 7 ...longitudinal subsystem transverse subsystem global systemFig. 4. The global system is decomposed along longitudinal and transverse direction into two types of subsystems. The green quadrilateral in the global system represents a spec c cross-section plane. The purple point is the intersection of the plane and the balance position of girders. The yellow point is the intersection of the plane and a certain mode shape of the global system. It can be seen that the value of the tranverse participation coecient vector ajis only determined by Kand its order jdepends on the magnitude ordering of!j. The sti ness matrix Kis only related to the sti ness of the distributed 125 spring and can be interpreted as a nidentical mass block system connected by n1 springs. In other words, the spring sti ness of the mass block system decide aj. At this point, the distributed spring-connected multi-beam global system can be equivalent to the product of the transverse subsystem and the longitudinal subsystem, as shown in Fig. 4. The longitudinal subsystem is a 130 simply supported single beam describing the longitudinal characteristics of each girder, while the transverse subsystem is a spring-connected mass block system which reprsent the characteristics of hinge joints. ajis the mode shape of the transverse subsystem and is de ned as transverse mode shape for its novelty. The acorrding modes are named as tranverse modes. The detailed qualitative 135 analysis and parameter analysis of the transverse mode shape can be found in the previous work[23]. As mentioned above, the sti ness of the distributed spring is used to char- 8 Spring stiffness 结构：装配式梁桥 ACMGBTheoretical world Key performance: TCWP Real worldHinge joint conditioningVibration response Transverse mode shapeTransverse mode shape- derivated indexTransverse mode shape Transverse mode shape- derivated indexdistributed springrigid arm girder Multi-beam system Transverse subsystem Forward analysis Theory-reality counterpart Inversion routeabc dFig. 5. Technical route diagram: a represents modal identi cation, b stands for index ex- traction, c means index characterizing, and d indicates diagnosis and evaluation. acterize the conditioning of hinge joints and is further linked with the TCWP. The transverse mode shape contains the information of the transverse subsys- 140 tem or to say the spring sti ness's information. Therefore, the index derived from the transverse mode shape can be used to represent TCWP. The forward analysis logic and TCWP inversion route are shown in Fig. 5. Inside it the theory-reality counterpart presents the connection between the theorical word and the real world which justify the rightness of inversion route in concept. 145 To identify transverse mode shape, arranging sensors on any cross-section to measure the vertical vibration information is reasonable. However, considering that the ACMGB is generally of relatively large sti ness, the problem vibration signal-noise-ratio (SNR) may be low and the amplitude of midspan section is often higher than other sections, so arrangement of sensors at the bottom of the 150 girders in midspan section is a more concrete and suitable scheme. 9 2.2. Choice of sensors for vibration information collecting For vibration monitoring of civil engineering structures, the most commonly used sensors is the accelerometer. In addition, the dynamic strain gauges is also an option. In general, with appropriate sampling frequency, strain data of 155 bridge structures contains four components[28{33]. According to di erent time scales, they are static component induced by deal load, temperature-induced component, quasi-static component resulted from static e ect of vehicle load and vibration component caused by vehicle load dynamic e ect. Since the longitudinal normal strain at the bottom of girders is mainly con- 160 trolled by the bending of the girders while the longitudinal subsystem shows the bending vibration of the girders when the multi-beam system vibrates, the longitudinal normal strain of at the bottom of girders must contain abundant vibration information. Furthermore, the lateral arrangement should re ect the vibration characteristics of the tranverse subsystem. Therefore in this study, 165 vibration information of ACMGB comes from both accelerometers and strain gauges. The vibration component of strain is further used to identify the trans- verse strain mode shape and then utilized for structure performance evaluation. 3. Real bridge implementation of vibration monitoring system Built in 2001, the Shanghai Tongji Road Viaduct has been in safe service for 170 19 years. Because of its heavy or even overloading trac load which is mostly composed of container trucks, its TCWP has been greatly degraded. Apparent diseases, longitudinal cracks in the deck pavement (to see Fig. 6a) and water leakage in hinge joints is easy to be found. Exactly, in its service period, the pavement and hinge joints have undergone repairing and strengthening of many 175 times, but due to no suitable solution to the heavy load problem , these diseases recurred again and again. In the summer of 2016, three adjcent spans, K92, K93 and K94, are chosen to be monitoring span where vibration-based TCWP monitoring system proposed above implemented, as is shown in Fig. 7a. 10 (a)Location (b)Longitudinal cracks in K92's pavement Fig. 6. Shanghai Tongji Road Viaduct MS1 MS2 MS3 continuous slab-deck expansion joint22m monitoring section11m 11m Yixian Road22.3m 11.15m 11.15m girderK92 K93 K94 BAOSTEEL22m 11m 11m pier 93# pier 94# pier 92# pier 94# MS1 MS2 MS3 continuous slab-deck expansion joint22m monitoring section11m 11m Yixian Road22.3m 11.15m 11.15m girderK92 K93 K94 BAOSTEEL22m 11m 11m pier 93# pier 94# pier 92# pier 94# (a)Planar graph (b)Cross-section graph Fig. 7. Schematic diagram of sensor placement 11 (a)Sensors installation (b)Sensors details (c)Monitoring spans after installation (d)Aerial view of monitoring points in K92 Fig. 8. Monitoring system implemnetation graph The selected spans' transverse layout is 0.5m barrier+8.5m lane+0.5m me- 180 dian+8.5m lane+0.5m barrier, so the total width is 18.5m. 17 pieces of slab girder of 0.9 m height is connected by pavement and hinge joints to form multi- beam system. Except that the K92 and K93 are entirely isolated by the ex- pansion joint in the decks, the structure in the rest of the supporting location are simply supported with continous deck details, shown in Fig. 7a. In order 185 to facilitate the follow-up evaluation, from southwest to northeast, girders is numbered as girder 1 to girder 17 and hinge joint number rule is similar (to see Fig. 7b). The Fiber Bragg Grating (FBG) accelerometers and longitudinal normal strain gauges are arranged at the bottom of each girders in the midspan section 190 of three spans, as shown in Fig. 7. For studying the e ect of environmental 12 FBG sensor FBG demodulatorMain optical cable Telecom networkRouter MySQL databaseFig. 9. Monitoring system topology graph temperature on the monitoring index, FBG thermometers are also installed at the side girders. All the sensors in the midspan section share a main cable guranteeing good synchronization of multi-channel measurement. The optical signals of di erent spans' sensors are also connected by the main cable and are 195 further converted into electrical signals through FBG demodulator with a sam- pling frequency of 50Hz. After that, the electrical signals are transmitted into the telecom network by the router, and are nally transmitted to the server for storage. The monitoring system implementation diagram and topology diagram are shown in Fig. 8 and Fig. 9, respectively. 200 4. Preliminary analysis of vibration monitoring data of K92 The preliminary analysis of monitoring data starts with the time-frequency feature analysis of measured vibration response (i.e., data quality assessment) and identi cation of modal parameters follows. At the end there would be a ten- tative performance assessment using transverse mode shape-derivated indexes. 205 4.1. Time and frequency domain analysis of measured acceleration and strain response data In order to truly re ect the time-frequency characteristics of the two type of measured response data, raw data are analyzed without preprocessing like tering. Unless noted specially, the data analyzed are taken from the FBG 210 acceleration and strain monitoring data of K92 on July 1, 2018. In the case of single-channel data analysis or to say time domain analysis, the corresponding monitoring data of girder 15 is used whilie in the case of multi-channel data analysis or to say frequency domain analysis, the monitoring data of all girders are taken. 215 13 4.1.1. Time domain features To better analyze the time domain features of the two types of data at di erent scales, scales of hours, minutes and seconds are selected to study the time domain features. As is known to all, the acceleration measurement in time domain should 220 present symmetry oscillations form around a certain baseline[27], which is also a typical time domain feature of vibration. It can be seen from Fig. 10 that the three -scale time domain diagrams of the acceleration at the bottom of K92 show obvious symmetric oscillations. Note that due to the measurement setup problem, the acceleration data has a baseline close to 105mg. 225 Compared with acceleration data, strain data is more complex, or in other words, contains more abundant information. It can be seen from Fig. 11 a that the most evident component of strain data in hour-scale is the chaning baseline or trend term under the action of dead load and temperature. The biggest di erence between it and the above acceleration data's is that, in hour-scale, 230 the trend term of strain is presented as nonlinear o set, while the acceleration's is linear drift. Take the strain data of 5min around 12:00 as an example to observe the time domain feature in minute-scale, as shown in Fig. 11 b. Due to high trac ow density at noon, the strain data in minute-scale includes many peaks with burring. The shape of each peak is similar to the shape of the bending 235 moment in uence line of the simply supported beam bridge, which is mainly the quasi-static response generated by moving vehicles. Furthermore, a section of data of 16s is selected to observe the features in second-scale, as shown in Fig. 11 c. It is evident that in second-scale, more high-frequency oscillating components can be observed riding above the slowly changing baseline in the strain data. 240 The oscillating component is related to the vibration response of the structure while the slowly changing baseline corresponds to the quasi-static response. As for such oscillatory components, some existing studies regard them as noise, and ignore them by ltering or signal reconstruction. Of course, this depends on the research purposes. However, it is undoubtedly a waste of valuable monitoring 245 14 0 6 12 18 24 Time(h)0100200acceleration(mg) 012345 Time(min)50100150acceleration(mg) 02468 1 0 1 2 1 4 1 6 Time(s)80100120acceleration(mg) D B B C Fig. 10. Time domain diagrams of acceleration at di erent scales: (a)hour (b)minute (c)second 0 6 12 18 24 Time(h)-200-150-100strain( ) 012345 Time(min)-150-125-100strain( ) 02468 1 0 1 2 1 4 1 6 Time(s)-160-155-150strain( ) D E F Fig. 11. Time domain diagrams of strain at di erent scales: (a)hour (b)minute (c)second 15 0 5 10 15 20 2 5 Frequency(Hz)-20020406080PSD(dB)4.14Hz 4.64Hz 5.46Hz 7.05Hz 8.4Hz 10.8Hz 12.65Hz 14.55Hz(a)Auto-power spectrum of acceleration 0 5 10 15 20 2 5 Frequency(Hz)-20020406080PSD(dB)4.14Hz 4.64Hz 5.46Hz 7.03Hz 8.26Hz (b)Auto-power spectrum of strain Fig. 12. Frequency domain feature gure of monitoring data: Each solid line of di erent color represents the auto-power spectrum of a channel's data while the vertical dotted lines and dash-dotted lines suggest the location of peaks identi ed manually. information to overlook the details of high-frequency oscillatory signals. To sum up, the acceleration data and strain data should include vibration information. When the system is excited, the acceleration and strain would be synchronously excited, so if the acceleration oscillation is apparent, the strain should also contain vibration component. This can be found by comparing 250 Fig. 10 c and Fig. 11 c. 4.1.2. Frequency domain features In addition to oscillations in time domain, the vibration behavior of mechan- ical systems is also featured by the power distribution in frequency domain. But ACMGBs belong to the category of medium and small bridges, and their global 255 sti ness and mass is often larger than that of composite bridges of the same span. Therefore, weak-excitement[34] is indeed a common problem. Fig. 12 shows the auto-power spectrum of acceleration and strain data, which is esti- mated by Welch's method. The adopted window function is a hanning window with a length of 4096 points and its segment overlap rate is 50 %. Besides, the 260 FFT calculation points are 4096. As can be seen from Fig. 12, due to the limited exciting level of the vehicles for ACMGBs, the spectrum peaks from measured data are not as abundant as 16 those obtained in the laboratory vibration test[35], regardless of acceleration or strain. Although the excitation e ect is not very good, a number of peaks are 265 still apparent and the modal frequencies can still be identi ed by peak picking manually. Comparing the strain and acceleration auto-power spectrum, the lower fre- quency bounds of the vibration components of both are about 4Hz and the peak locations of power spectrum have good consistency. Of course, the accel- 270 eration data have a richer high-frequency vibration component in the range of 9-15 Hz. This phenomenon can be explained as follows: under the Euler beam hypothesis, the longitudinal strain is proportional to the second order partial derivative of the de ection curve with respect to space, so the frequency domain characteristic is consistent with the de ection. Moreover, acceleration can be 275 tranformed into de ection by a low-pass system ltering with an amplitude of 1=!2, therefore the quasi-static and trend term of the strain are particularly obvious in the frequency domain, while the high-frequency components would be masked by noise due to small power spectral density. Three components can be clearly seen in the Fig. 12b. The static component 280 with the maximum amplitude stays at 0Hz, the quasi-static component is be- tween 0 and 4Hz with a second-largest amplitude and vibration components is in the range of 4-9hz. Therefore, according to the features of di erent scales (or di erent frequency bands) of each component of the strain, a rough component separation method is FFT bandpass ltering. 285 4.2. Operational modal identi cation based on frequency domain decomposition method As mentioned above, the main technical route of this paper is output-only health monitoring and its corresponding modal identi cation technology is often referred to as operational modal analysis (OMA) technology. OMA usually as- 290 sumes that the random environment excitation (mainly vehicle load) over long time is presented as white noise[36], it mainly depends on the second order moments of the structure under white noise as 'theoretical proxy'[37]. The 'the- 17 oretical proxy' estimating and identifying modal parameters from 'theoretical proxy' by parametric or nonparametric method are two main steps for OMA. 295 Depending on taking the correlation function or power spectrum as 'theoretical proxy', OMA can be divided into time domain method and frequency domain method and the representatives of the two kinds of methods are stochastic sub- space identi cation (SSI)[38] and frequency domain decomposition (FDD)[39], respectively. SSI can realize more automated modal identi cation, but its pa- 300 rameter selection is always a dicult problem and choosing large model order and identifying the real mode by stable diagram is a widely-used solution. In the case of low SNR, it tends to identify many false modes. Besides, large calculation time is its another shortcoming. Though FDD needs manual peak picking, its robustness is easy to guarantee. Considering that only preliminary 305 data analysis is conducted and big data analysis is not necessary in this paper, FDD with less in uence from noise is adopted. 4.2.1. Principle of frequency domain decomposition method Under white noise excitation, the power spectrum density matrix of the civil enginneriing structures, Gyy(i!), can be expressed by Eq. (5), where Nis the number of modes, dris a scalar constant, risrth pole, is the mode shape matrix of which each column is rth mode shape r(r= 1N). The super- script * and Trepresents the complex conjugate and transpose, respectively. Besides,diag() means the diagnal matrix operator while Re() denotes the real-part operator. Also note that ihere stands for the imaginary unit. Gyy(i!)NX r=1 dr  r T r i!r+dr  r T r i!r! = diag 2Redr i!r T (5) It can be observed that Eq. (5) have a similar form as matrix singular value decomposition (SVD)[40]. Therefore it's possible to pick peaks from the mea- 310 sured vibration signals' singular value spectrum using manual and automated algorithms[41] to identify modal frequency while the corresponding singular vec- tors is taken as mode shape. Generally, the rst singular walue is considered in 18 vibration signal PSD matrixWelch’s method singular value spectrumSVD peak location of first singular value spectrumpeak picking modal frequency modal shapeFig. 13. FDD owchart 0 5 10 15 20 2 5 Frequency(Hz)-40-2002040Singular value of PSD matrix(dB ) 1st singular value 2nd singular value 3rd singular value (a)Auto-power spectrum of acceleration 0 5 10 15 20 2 5 Frequency(Hz)-40-2002040Singular value of PSD matrix(dB ) 1st singular value 2nd singular value 3rd singular value (b)Auto-power spectrum of strain Fig. 14. Singular value spectrum of monitoring data: black points are peaks picked manualy this process. In this paper, FDD owchat is shown in Fig. 13. 4.3. Modal frequency and mode shape results 315 The singular value spectrum of K92's acceleration and strain data collected on July 1st in 2018 is shown in Fig. 14. By picking peaks in the rst singular value spectrum, modal frequencies of the displacement mode (DM) and strain mode (SM) are identi ed and listed in Table 1. The number of strain modes identi ed is obviously less than that of displacement modes, but the consistency 320 of the displacement mode's and the strain mode's modal frequencies indicates the possibility of mining vibration information in the strain data. 19 Table 1. Modal frequencies of K92's identi ed displacement modes and strain modes Mode 1st 2nd 3rd 4th 5th 6th 7th 8th !(Hz)DM 4.15 4.65 5.47 7.01 8.41 10.80 12.65 14.55 SM 4.15 4.65 5.46 7.01 8.36 | | | It should be pointed out that the above singular value spectra of acceleration and strain are based on Welch power spectrum density matrix by multi-segment averaging of one day's data. If the vehicle-passing event is taken as the sign of 325 data collection, the size of collected data would be much less and the estimated power spectra also contain more burring, which is adverse to the later modal identi cation. Therefore, the authors believes that although the excitation level of the ACMGB is not high under the ambient actions, as long as enough op- erational state data are collected, even the strain's vibration information with 330 low SNR can be recognized and further used for the characterization of key performance. The corresponding transverse displacement mode shapes (TDMSs) and trans- verse strain mode shapes (TSMSs) in K92 are shown in Fig. 15 and Fig. 16. It can be seen that, not taking mode 1 into account, the 2nd-7th TDMS of 335 K92 is very similar to the theoretical qualitative mode shape of the simpli ed transverse subsystem model: The second TDMS of K92 presents an approximate horizontal line, which has a good consistency with rigid body mode shape (a horizontal line) of the simpli ed transverse subsystem. The di erence originated from the 340 not exactly same the sti nes and mass of each girder of K92, which is not completely complied with the assumption of the simpli ed model. The other six mode's TDMS is similar to harmonic shapes and the har- monic order rises with the increase of the mode order, which is consistent with the non-rigid body mode shape of the simpli ed transverse subsys- 345 tem. 20 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shapeMode 1 Mode 4Mode 2 Mode 3 Mode 3 Mode 5(a)TDMS 13579 1 1 1 3 1 5 1 7 Girder-0.800.8Strain mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.800.8Strain mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.800.8Strain mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.800.8Strain mode shape 13579 1 1 1 3 1 5 1 7 Girder-0.800.8Strain mode shapeMode 1 Mode 4Mode 2 Mode 3 Mode 3 Mode 5 (b)TSMS Fig. 15. Comparison TDMS and TSMS of the rst ve modes in K92 21 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shapeMode 6 13579 1 1 1 3 1 5 1 7 Girder-0.600.6Displacement mode shapeMode 7Fig. 16. 6th and 7th TDMS in K92 In fact, the transverse rigid body mode corresponds to the vertical bending mode of the ACMGB, while transverse non-rigid body modes correspond to torsion modes. Given that the vertical bending mode is mainly controlled by the behavior of a single beam, the transverse rigid body modal frequency and 350 mode shape are not sensitive to hinge joint damage, that is, the spring sti ness decrease in the transverse subsystem model. By the above analysis, the concept of transverse subsystem for the ACMGB proposed in this paper is to some extent proved to be in line with the actual situation. However, the 1st TDMS at 4.1hz appears to be similar to that of the 355 3rd TDMS, which is inconsistent with the transverse subsystem theory of the simpli ed model. This issue would be discussed in section 5.1. In the authors' previous work[23], it's deduced that there is one-to-one cor- respondence in shape between TSMS and TDMS of the multi-beam system ac- cording to the transformation relationship between euler beam displacement and 360 strain. However, as obsevrd between Fig. 15a and Fig. 15b, the rst ve mode's TDMSs are seemingly di erent from the corresponding TSMSs. If comparing identi ed TSMS with the ideal transverse mode shapes (harmonic shapes) of the simpli ed transverse subsystem , the discrepancy is more serious. Some similar situations exist in strain or curvature modal tesing [42] and 365 the reason for them is that the strain measurement is sensitive to local sti ness. If the local sti ness of di erent strain measuring points are of great di erence, it's easy to nd some outliers relative to the baseline in strain mode shape, which have a great in uence on the understanding of strain mode shapes and 22 their further use. Actually, for strain measuring points arranged as we proposed 370 above, the local sti ness may be distinct, which not only can be attributed to low construction quality, but also be due to random size e ect of concrete material. Therefore, it is an objective fact that the identi ed TSMSs shown in Fig. 15b present abnormal shapes due to the existence of outliers. Considering these, if removing outliers and comparing the identi ed TSMSs and TDMSs, there are 375 some similarities between them at the baseline level. For example, for mode 2 and 3, ignore the ouliers at girder 2, 3, 5, 12, 14 and 15 and the obtained baseline will show an approximate horizontal line and an anti-symmetric harmonic shape. Of course, there is another possible reason for the abnormal shape of the identi ed TSMSs in Fig. 15b, that is, they are results of spurious modes. In 380 order to falsify this possibility, more strain modes were identi ed from the strain data of the following six consecutive days (July 2-7, 2018). The variation of them relative to the rst day's is studied and the mode shape di erence was measured by modal assurance criterion (MAC). See Eq. (6) for MAC calculation, where i;Arepresents the ith element of mode A,nindicates the total element 385 number of A's mode shape, and the mode of Banother mode shape that is compared with the mode shape of A. MAC value varies from 0 to 1, with 0 for no correlation (huge di erence) and 1 for full correlation (no di erence). MAC A;B=jPn i=1 i;A i;Bj2 Pn i=1( i;A)2Pn i=1( i;B)2(6) Fig. 17 shows the MAC results of TSMSs identi ed from the next six days' data and the rst day's. Obviously, these 6 days of TSMSs are very similar to 390 the rst day's, as most MAC values are greater than 0.9, so it can be considered that the TSMS identi ed on July 1, 2018 is relatively stable. Even though they are far from the ideal TSMSs, they do represent the tranverse characteristics of operational strain to some extent and are not spurious. 4.4. Preliminary evaluation of K92 based on transverse mode shape 395 For preliminary evaluation, only short-term (1 day) monitoring data are used. According to the previous study[23], modal absolute di erence (MAD) 23 2018/7/3 2018/7/5 2018/7/7 Date0.850.90.951MAC Mode 1 Mode 2 Mode 3 Mode 4 Mode 5Fig. 17. MAC between the rst day's TSMSs and the next six days' index is an appropriate choice, which is de ned by Eq. (7). In essence, MAD captures the di erence of vibration amplitude and phase between adjacent gird- ers, on which the degradation of TCWP caused by hinge joint damage is inferred 400 based. Therefore, locating outliers on MAD is the main step of evaluation. The previous modal parameter identi cation results have shown that the mode 1 is not consistent with the theory of this paper and the previous work while the mode 2 is the 'rigid body mode', which is not sensitive to hinge joint damage. Besides, TSMSs have ouliers on baselines. Considering all this, in this section, 405 the 3rd-7th TDMS are utilized for MAD calculation and preliminary TCWP evaluation. MAD j(i;i+ 1) =j i;j i+1;jj (7) The MAD results are shown in Fig. 18 and outliers are listed in Table 2. Because the transverse modes have di erent sensitivity to hinge joint damage, the selected oueliers are also di erent. It should be noted that, since there are 410 additional masses on the girder 1, 9 and 17 from the accessory structures (the barriers and median), the related MAD may present outliers which are excluded as outliers in this paper. In order to indirectly verify the feasibility of MAD method, a eld survey was conducted in K92 on November 23, 2018. and the girder bottom picture 415 is shown in Fig. 19. It can observed that there were evident marks of water 24 13579 1 1 1 3 1 5 Hinge joint00.050.10.15MAD 13579 1 1 1 3 1 5 Hinge joint0.050.10.150.2MAD 1 3 5 7 9 11 13 15 Hinge joint00.10.20.30.4MAD 13579 1 1 1 3 1 5 Hinge joint00.10.20.30.4MAD 13579 1 1 1 3 1 5 Hinge joint00.10.20.30.40.5MADFig. 18. MAD of 3th-7th TDMS. The black points indicate identi ed outliers and show the TCWP degradation locations of preliminary evaluation. leakage in di erent length under hinge joint 3, 4, 5, 6, 9, 11, 12, 13, 14 and 15, which are considered of obvious damage, and the corresponding TCWP had deteriorated to di erent degrees. Therefore, most of the selected MAD outliers can correspond to the damage of the actual hinge joints, indicating that 420 the proposed MAD method has the potential to evaluate the TCWP of the ACMGB. The preliminary application also shows that the MAD method has the fol- lowing problems in practice which are common in damage detection based on vibration index : 425 The inconsistency of the results of multi-mode MAD outliers makes it more dicult to accurately locate the damage of hinge joints. Although most hinge joint damage can be located, the damage degree is dicult to assess. The purpose of this paper mainly lies in studying the TCWP evaluation of 430 ACMGB, i.e., the damage detection of hinge joints. In addition to the transverse cooperative working system, the evaluation of the performance and damage 25 Table 2. Outliers identi ed from MAD Mode Hinge joint number of outliers 3 3,4,11,13 4 3,7,11,14 5 3,12,14 6 3,5,7,11 7 5,13 Fig. 19. Field picture of K92 on November 23, 2018. The blue line denotes locations of damaged hinge joints casuing evident water leakage. 26 of the girder is also of great importance to the ACMGB management. As mentioned above, the ideal transverse strain 'rigid body mode shape' should be close to the horizontal line. If there are evident ouliers from the horizontal line, 435 they can be regarded as results of the corresponding girder sti ness changing. The 2nd TSMS does contains many outliers, mainly at girder 2, 3, 5, 12, 14, 15 and it's reasonable to infer that the sti ness of these girders have decreased signi cantly. 5. Discussion 440 5.1. Rationality of the simpli ed model for qualitative analysis As previously stated, most of the identi ed transverse mode shapes are ba- sically the same as those of the transverse subsystem of the simpli ed model. However, it seems that the transverse mode shapes of mode 1 are dicult to be explained by it, and may be identi ed as the spurious mode. Here we provide a 445 possible explanation from the view point of the simpli ed model's rationality: In fact, the transverse subsystem model proposed in this paper is simpli ed. Only unidirectional linear displacement spring is used to connect the multi-beam sys- tem while the actual hinge joints should also have the e ect of transmitting bending moment, i.e., there should also be torsional springs. Therefore, the 450 proposed transverse subsystem would have a mode of integral rotation (also a 'rigid body mode') in which the torsional springs between girders would not work at all and the according transverse mode shape would appear an oblique line, to which the identi ed 1st TDMS is highly close. 5.2. Comprehensive utilization of various information in strain 455 As analyzed in section 4, it is clear that there is more than one compo- nent in the strain of the ACMGB, and its quasi-static information can also be used to evaluate the TCWP[22]. By utilizing various information in strain comprehensively, the quality of evaluation should be enhanced. However, it seems that how to realize strain multi-component coseparation in an accurate 460 27 and automated way is still a unsolved problem. Although there have been a lot of attempts to separate strain components[28, 32, 43], but according to the author's preliminary testing, some of these methods are still far from the pareto optimal solution. It's still dicult to achieve high quality of coseparation of the quasi-static and vibration components. 465 5.3. Necessity of long-term analysis The data analysis in this paper is still of the category of short-term or small data analysis and only the index in no need of the baseline can be used. Al- though this method has't reached a satisfactory level due to some factors, it still shows some applicability and provides useful enlightenment for future research, 470 i.e., long-term monitoring and analysis of indexes. Long-term continuous mon- itoring makes it possible to attain massive indexes of di erent time, and the corresponding analysis method is called long-term analysis method[44{48]. The analysis of long-term monitoring data mainly evaluates the structural perfor- mance by identifying the trend of changeing indexes in the background of big 475 monitoring data, which is considered to be a more robust data utilization tech- nology route. The above long-term analysis ideas would provide better tools for this paper's goal realization and demonstrate necessity for research. 5.4. Physical concept orientated middle and small span bridge monitoring sys- tem design 480 Consndiering limited management funds, the monitoring system of medium and small bridges is often not equipped with a large number of sensors of var- ious types like the one of long-span bridges. However, the main structures of them are relatively simple and the disease patterns are also comparatively clear. Therefore, the corresponding monitroing system design should begin with dis- 485 ease pattern mining to nd out the speci c key performance. Then, the key performance is modeled in mechanical system which should not only be sim- pli ed enough to model key performance but also be of enlightment to indexes 28 construction. Next, depending on how to identify the indexes, the sensor place- ment, selection and other hardware details are determined. Last but not the 490 least, the index anlysis based on monitoring data are designed. The ACMGB monitoring system proposed in this paper can be served as an solid example and the above four steps are concise summary for section 2 and 3. 6. Conclusions This paper rst introduced the distributed spring-connected simply-supported 495 multi-beam model for the modal analysis of the ACMGB, nding that if assum- ing each beam to be identical, the global vibration system can be divided into simply-supported single beam subsystem and spring-connected mass block sys- tem along the longitudinal and transverse direction. Besides, the mode shapes of transverse subsystem were de ned as the transverse mode shape, which is 500 completely determined by the spring sti ness. Inspired by this, this paper pro- posed an TCWP monitroing and evaluation scheme based on the transverse mode shape-derivated index which is identi ed from vibration information col- lected from sensors arranged at the bottom of girders in midspan. Moreover, the vibration information can be sensed by accelerometer and dynamic strain 505 gauge. This vibration-based TCWP monitoring system is implemented in Tongji Road Viaduct and through the preliminary analysis of the short-term vibration monitoring data of it, the purpose of evaluating the TCWP of the ACMGB is realized. The concrete conclusions are as follows: The measured longitudinal normal strain of girder bottom contains abun- 510 dant information and its vibration information can be used to identify the transverse strain mode. The transverse displacement mode identi ed by acceleration data are qual- itatively consistent with those of the simpli ed transverse subsystem, prov- ing the rationality of the simpli ed model to some extent. 515 29 The MAD index based on the transverse displacement mode shape has the potential to evaluate the TCWP of the actual ACMGB, while the transverse strain mode shape can provide additional information about the girder degradation. Acknowledgements 520 This work is supported by the Opening Foundation of National Key Lab- oratory of Bridge Structural Health and Safety; the National Nature Science Foundation of China (Grant No. 5187849); the National key R&D Program of China (2017YFF0205605); and Shanghai Urban Construction Design Research Institute Project `Bridge Safe Operation Big Data Acquisition Technology and 525 Structure Monitoring System Research'.