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Damage Identification Method Based On Wavelet Energy Band In Shield Tunnel Structure Li Hongqiao 1, Xie Xiongyao 2 Department of Geotechnical Engineering, Tongji University, Shanghai, 1410260@tongji.edu.cn Department of Geotechnical Engineering, Tongji University, Shanghai, xiexiongyao@tongji.edu.cn ABSTRACT Rail Transit System is a traffic lifeline of a city. It s crucial for the health service of the shield tunnel structure on the safe operation of the city. The damage identification of tunnel structure (lining cracking, bolt loosening, concrete deterioration and soil erosion after lining of tunnel) is a key technology in structural health monitoring system. However, it s very difficult to identify damages on the tunnel structure due to significant non-linear characteristics of tunnel structure and hardly measured excitation and unpredictable interaction between soil and structure. In this study, we propose a method to identify the tunnel structure damages based on the vehiclemounted accelerometer of the service train, in which the distribution of structure stiffness of the tunnel can be continuously measured. Firstly, the vehicle-tracktunnel coupling model is established to analyze the vehicle response by change of stiffness of the track and tunnel. Then, the wavelet energy band damage index is constructed based on the principle of wavelet transform. Finally, the validity of wavelet energy band damage index is verified in numerical simulation. The results show that the index can accurately locate the damage and has good robustness. Key Words: Damage Identification, Wavelet Energy Band, Shield Tunnel 1. INTRODUCTION The construction of urban rail transit in our country is in a stage of rapid development. According to the data of China Urban Rail Transit Association, by the end of 2016, the number of cities that have opened rail transit operation in our country reached 30 and the total length of operation lines reached 3,133 kilometres. China s urban metro mileage has already ranked first in the world. As the lifeblood of the city, rail transportation is vital to the normal operation of the city. As the lifeblood of the city, rail transportation is vital to the normal operation of the city. Once any node of urban rail transit goes wrong, it will affect the entire subway network and hinder the travel of millions of people, causing paralysis of the urban traffic system and bad social influence and even serious loss of people and property. For example, December 22, 2009, Shanghai Metro Line 1 due to the top of the tunnel structure of carbon fibres shedding led to the South Shaanxi Road to the People s Square intermittent power supply network trip fault, Which caused by the suspension of trains in the area, the rare paralysis on the lines and a large number of passengers were Forced by the ground traffic, affecting over millions of residents travel. For another example, December 6, 2017, Shenzhen Metro Line 11 due to illegal piling led to the train outage. It is understood that at that time, the high-speed train hit the pile head directly and the train was severely damaged. The related subway equipmentwas damaged to varying degrees. No passengers were injured in the incident. However, if the pilehead can be further re-imprinted, we are afraid it will cause a catastrophic accident. Among the above accidents, there is not no health monitoring system. Instead, the traditional monitoring system for tunnel settlement, deformation and convergence does not provide early warning in time 2

and damages and abnormalities inside and outside the tunnel structure cannot be detected by traditional monitoring system. As mentioned above, for shield tunnels, monitoring systems such as settlement, deformation and convergence are not enough to provide a real-time and reliable warning. In recent years, the method of structural damage identification by structural vibration has become a hot research topic of non-destructive monitoring. Damage identification methods based on structural vibration can be divided into modelbased methods and model-free methods based on whether need to structure the physical model. Model-based approaches can locate structural damage and generally also distinguish the type and extent of damages.model-based recognition relies heavily on the accuracy of established physical models (Fu, Lu, & Liu (2013);Huang, Gardoni, & Hurlebaus (2012);Link & Weiland (2009);Zapico-Valle, Alonso-Camblor, Gonzalez-Martinez, & Garcia-Dieguez (2010). The shield tunnel structure usually extends underground through different types of soil layers, with the existence of soil-structure interaction, complex service conditions, boundary conditions, material parameters and structural mechanics mechanism in the modeling process (Hussein & Hunt (2007)). Therefore, the shield tunnel structure is more suitable for model-free methods before the advent of new modeling techniques. Model-free method Based on the measured data to construct damage indicators, currently can be divided into two types of methods, namely direct type and indirect type. The direct method is to arrange the sensor on the structure under test, and to identify the structural damage through the change of the sensor signal (Lam, Wong, & Yang (2012)). Direct method is effective for small structure and simple structure. However, the large number of sensors needed to be installed for large and complicated structures is not only economical but also increases the difficulty of installation. Obviously, the direct method is not suitable for such long linear structures such as tunnels. Our remaining option is indirect method. Indirect methods have yielded many beneficial results. Rolling stiffness measurement vehicles (RSMV) developed by Swedish researchers are used to measure the dynamic stiffness of railway tracks.deutsche Bahn has integrated two different measurement methods into a high-speed measurement train, measuring the orbital geometry and the vehicle s dynamic response(berggren EG (2009). Correlation analysis of the measured defects found that for the 18% of the vehicle dynamic response overrun, no abnormalities were found in the corresponding geometric measurement data. This indicates that the degradation of the underlying orbital property may lead to unacceptable vertical acceleration of the vehicle, whereas the traditional geometric measurement of the orbit cannot be detected (Kratochwille R(2015)). In this study, In order to quickly and economically locate the tunnel structure damage, this paper preliminary studies indirect method by Train-Track-Tunnel-Model (TTTM) and discusses its feasibility. Through the TTTM, the response signals of the wheel axles can be obtained. The response signals of the wheel axles from the TTTM under different working conditions are analysed and the damage index based on the wavelet energy band is extracted and finally used to locate the damage of the tunnel structure. 3

2. TRAIN-TRACK-TUNNEL-MODEL A dynamic computational model for the vehicle and track and tunnel coupling system is developed by means of finite element method in this paper. Due to its simplicity and high computational efficiency, the moving element method (MEM) is adopted (Koh, Ong, Chua, & Feng (2003)). Defining a whole vehicle with 26 degrees of freedom (DOF) as a computational moving element, the nodal displacement vector r for this element, as shown in Figure 1, can be expressed as (1) Where Vi, Oi ( i = 1, 2, 3...8) are the vertical rail displacement and the rail slope for node i, vci, (i = 1, 2, 3,4) is the vertical rail displacement at the ith wheel/rail contact point, v9, 09 are the vertical displacement and pitch motion of the car body, vi, 01 (i = 10,11) are the vertical displacement and pitch motion of two bogies, and vi (i = 12,13,14,15) is the vertical displacement for the ith wheel. Figure 1. Vehicle element model with 26 DOF 4 In order to formulate finite element equations for the vehicle element, Lagrange equation need to be used. The specific derivation process can refer to the XiaoyanLei s essay (Lei & Zhang (2011)). From that, Me U KeU can be obtained. To construct a TTTM system, some simplified assumption of the tunnel structure is also required. Subway shield tunnel diameter is generally 6.2 meters, while the length between the two stations is about 1 000 meters. Slenderness ratio is greater than 160, so in theory the shield tunnel can be approximated as beam model. It is also assumed that the contact between the global track bed and the shield tunnel is not smooth, the layers have strong adhesion and friction, and the shear force can be sufficiently transmitted. The bearing mechanism of this combination is similar to that of the composite beam. Accordingly, this paper considers the whole tunnel bed and shield tunnel as a finite continuous elastic Euler beam model with discrete point support. As shown in Figure 2, the top floor is the rail, which is also treated as a finite length Euler beam, regardless of the seam between the rails is the bending stiffness of the rail and is the bending stiffness of the composite beam. The irregularity of the track vertical profile is expressed by the function w Composite beam of tunnel and track bed connects with rail upward through springs and dampers, down directly to the formation through springs and damperis corresponding damping and spring coefficient. In this way, a double-beam model of the tunnel structure and rail and track bed is formed. When using finite element method to discrete, is the length of each unit, which is the spacing between the two

sleepers. It is also easy to obtain the element stiffness matrix and the damping matrix mass matrix for the double-beam modelnamely Figure 2. Double-beam model of tunnel structure Finally, the governing equation for the coupling system can be obtained as M&a&+Ca&+Ka =Q (2) Where, a is a global displacement vector for the vehicle and track coupling system. Equation (2) can be solved by numerical method such as Newmark integration scheme. In numerical calculation, it is necessary only one time to integrate the total stiffness matrix, total mass matrix, total damping matrix and total load vector of the double-beam structure, and then assemble the vehicle unit matrix with them in each time step calculation. Therefore, the TTTM model calculated by moving element method has high efficiency and good accuracy. 3. ANALYSIS METHOD BASED ON TIME - WAVELET ENERGY SPECTRAL 3.1. Continuous Wavelet Transform and Time - Wavelet Energy Spectrum Assumed be a finite energy function,, if the Fourier transform φ satisfies the allowable conditions: is called the mother wavelet, and the can scale and shift. Supposing that the scale parameter is a and the shift parameter is b, a set of functions is called wavelet basis function. The continuous wavelet transform of signal x(t) is defined as: In the formula,is the conjugate of. According to the energy conservation principle of wavelet transform, namely Equation (7) can be rewritten as According to equation (9), is defined as the time-wavelet energy spectrum. 3.2. Headings Wavelet Energy Band The above mentioned wavelet time energy spectrum is ideal, specific to this research topic, need tobe transformed into practical indicators available. First, in the actual programming calculation, wavelet scale and translation values are discrete values. turns to be, And then This integral becomes summation. Second, when calculating no longer divided by. Although this does not meet Parseval s theorem between time domain and frequency, it does not affect our calculation, and High-Scale with high amplitude becomes more prominent. So we define the wavelet energy band as follows: represents the start scale, s2 represents the start time, e1 represents the end scale, e2 represents the end time. Because the scale and the frequency are related, the wavelet energy value of a certain frequency band in a certain time band can be characterized by this formula. The energy has the resolution ability 5

in frequency and time domains. In wavelet analysis, the way to relate scale to frequency is to determine the center frequencyof the wavelet, Fc, and use the following relationship: Where a is a scale. Fc is the center frequency of the wavelet in Hz. Fa is the pseudo-frequency corresponding to the scale a, in Hz. Assuming that the train has been traveling at a constant speed the time is related to thedistance namely x = vt (12) According to the relationship between frequency and scale and between space and time in formula(11,12), then the wavelet energy band in the time-frequency domain can be transformed into the space-frequency domain, and the damage of the structure can be located by comparing the index of the wavelet energy band before and after the damage. The damage index is as follows (, ) (,,, ) (,,, ) D s s e e U s s e e DI = DI f s = E f s f s E f s f s (13) ED is the wavelet energy band after damage; EU is the wavelet energy band before damage, fs represents the start frequency, ss represents the start location, fe represents the end frequency, se represents the end location. DI is a function of space and frequency. When using this indicator, you can only draw a band, such as only the band. 4. Numerical Simulation 4.1. Calculation parameters In this paper, numerical simulation is used to verify the validity of the wavelet energy band indicator. First, need to determine the TMMM calculation parameters. The length of the double-beam element is the spacing of the sleepers, taken as 0.6 meter, and 200 elements are used to simulate a 180 meters long tunnel. The calculated double-beam model has 603 nodes and 804 degrees of freedom. Assembling the tunnel total matrix with the vehicle s element matrix and handle the boundary conditions, the final model is 812 degrees of freedom. The calculation time step is 0.001 second. The calculation parameters are set unconditionally stable by Newmark implicit integration algorithm. Considering linear elastic wheelrail contact, the contact stiffness is taken as 1.325 109N/m. Other calculation parameters are in Table 1. 6

Table 1. Calculation parameters of TTTM Vehicle parameters Track parameters parameter name Symbol and unit value parameter name Symbol and unit value Mass of car body Mc (kg) 40000 Mass of rail Mr (kg/m) 60 Mass of bogie Mt (kg) 3200 Density of rail (kg/m3) 7800 Mass of wheel Mwi (kg) 1900 Mass moment of inertia of rail Ib(m4) 3.217 10-5 Pitch inertia of carbody Jc (kg.m2) 2.446 106 Young s modulus of rail Eb (MPa) 2.06 105 Pitch inertia of bogie Jt (kg.m2) 3605 Sectional area of rail Ab (cm2) 77.45 Stiffness of primary suspension system ks1 (MN/m) 2.140 106 Stiffness of subgrade kg (MN/m) 65 Stiffness of secondary suspension system ks2 (MN/m) 2.500 106 Damping of subgrade Cg (KN.s/m) 90 Damping of primary suspension cs1 (kn.s/m) 4.900 104 Density of Composite beam (kg/m3) 2500 Damping of secondary suspension cs2 (kn.s/m) 1.960 105 Connect Stiffness between rail and composite beam kt (MN/m) 78 Wheelbase 2l1 (m) 2.5 Connect damp between rail and composite Ct (KN.s/m) 50 Distance between center of front bogie and center of rear bogie 2l2 (m) 18 Half sectional area of composite beam At (m2) 15.44 Contact stiffness kc (N/m) 1.325 109 Half Mass moment of inertia of composite beam Et It(N.m2) 15.25 1010 7

4.2. Calculation results 4.2.1. Self-programming verification TTTM model is calculated according to the finite element program of moving element method by MATLAB. Based on the above calculation parameters, first calculate the results without track irregularities. The direction of the train is to the right. When taking the vehicle speed at 20m/s, it can be seen that the displacement of the rail at 67.2m from the left end as shown in the Figure3. Although the calculated structure of railways and tunnels is different, the laws are similar. This calculation result is very similar to reference [12]. This is enough to prove the correctness of the self-compiled calculation program. Figure 3. The vertical displacement of the rail at 67.2m from the left end, V=20m/s 4.2.2. The results without track irregularity The finite element model of a train has 10 degrees of freedom, representing different positions. First, it needs to figure out which degrees of freedom are sensitive to damage and which are dull. Based on this question, we first analyze te sensitivity of the ten degrees of freedom to the damage through continuous wavelet transform. In order to save the paper space, taking into account a certain degree of symmetry, therefore, only the sensitivity of the body, the front bogie, and the first wheel s vertical acceleration response signals to damage has been analyzed here. On the 98th, 99th, 100th, 101th and 102th element of the tunnel finite model, the stiffness of the composite beam subunit of the double- beam tunnel is reduced by 10%, the damage range is 3 meters wide and the distance from the left end is from 58.2 to 61.2 meters. The speed at which the car travels is 20m/s and is also the normal speed of subway operation. In order to eliminate the adverse effect of the initial boundary during the calculation, the signal processing data starts from 10m. 8 First observe the time domain results. Figure 4 to Figure 7, is the first wheel, the first bogie and the body of the accelerated map without damage and with damage 10%. As you can see from the acceleration time histories at three different locations, the amplitudes are quite different. When the damage is 10%, there is only a slight change in the time-domain graph. Figure 4 has two mutations because the whole

vehicle has two bogies and bogies one after the run through the injury segment. A similar phenomenon can be found in Figure 5. Figure 6 Acceleration time history curve(a) no damage; (b) 10% loss stiffness The distance between the first wheel and the damage segment is 37.7 ~ 40.7m. According to the speed converted into time should be 1.855s ~ 2.035s. It shows that the damage location is basically accurate. Wavelet Toolbox in MATLAB2017b is used to calculate continuous wavelet transform. The default Morse wavelet is the default wavelet basis function. If interested, see MATLAB help documentation for more detailed instructions on Morse wavelets. 9

The results shown in Figure 7 to Figure 9 indicates that from the wheel to bogie to car vibration energy transfer from high frequency to low frequency. After the making wavelet transform directly, the body and the bogie can still correctly locate the damage, but not very obvious. Where in wheels it almost hard to detect the damage. The damage is located at 1.35 ~ 1.535 seconds because cut off the first 0.5s data. Figure 7 Acceleration wavelet transform:(a) no damage; (b) 10% loss stiffness Figure 8 acceleration wavelet transform:(a) no damage; (b) 10% loss stiffness Figure 9 acceleration wavelet transform:(a) no damage; (b) 10% loss stiffness 10

Body and wheel energy are too concentrated to possibly cover up minor damage. It is proposed to use bogies vibration acceleration signal to detection damages. 4.2.3. The results with track irregularity Based on the above conclusions, this section investigates the effect of wavelet energy band identification of tunnel damage in the presence of orbital irregularities. The damage setting is similar to the previous section. On the 98th, 99th, 100th, 101th and 102th element of the tunnel finite model, the stiffness of the composite beam subunit of the doublebeam tunnel is reduced by 5%,10%, 15% and 20% the damage range is 3 meters wide and the distance from the left end is from 58.2 to 61.2 meters. Using a periodic sine function to simulate the irregularity. y!! and λc respectively represent amplitude and wavelength, the sine wave equation is as follows ( ) 0 ( ) sin 2 / c c c y x = y π x λ Take equal to 0.0005m and λc equal to 12.5m.The vehicle speed is still equal to 20m/s. Because the damages on tunnel composite beam causes changes in the low frequency band. Here using DI index, only the frequency band is adopted. Take the frequency from10 to 60HZ. The result is shown in Figure 10 to Figure11. Figure 10: acceleration DI (a) 5% loss stiffness; (b) 10% loss stiffness Figure 11: acceleration DI (a) 15% loss stiffness; (b) 20% loss stiffness 11

As can be seen from the figure above, as the degree of damage increases, the peak value of DI is also increasing. Although the rail has some irregularities, it does not affect the ability of the DI to locate the damaged position. The exact location of the damage is 58.2 to 61.2m. The damage location detected by the bogie acceleration DI is slightly delayed. Figure10~Figure11 show that the center value is about 61.3m, and taking into account the half width of the bogie is 1.25m, the two subtraction results to 60.05m.Therefore, it can be said that this positioning is very accurate 5. CONCLUSION The key of structural damage identification of shield tunnel is damage localization. Because once identified the location of the damage occurred interval, the professional testing equipment can reach the damage detection of the type of injury and the level of detailed testing. In this paper, the finite element model TTTM is established by the moving element method. The wavelet energy band damage index is constructed based on the principle of wavelet transform. In this way, the damage index can be measured from two scales of time and space. The numerical results show that the DI value has good robustness and can locate the damage location very accurately. From the simulation results, if the sensor is installed on the bogie of the train, the damage can be well identified. 6. ACKNOWLEDGEMENTS This research is supported by the National Natural Science Foundation of China under the grants 51608379 and 51778476, and Shanghai Science and Technology Innovation Plan Funds under the grant 15DZ1203903, 15DZ1204102, 15DZ1203806, 17DZ1204203. Those supports are greatly appreciated. References: Fu, Y. Z., Lu, Z. R., & Liu, J. K. (2013). Damage identification in plates using finite element model updating in time domain. JOURNAL OF SOUND AND VIBRATION, 332(26), 7018-7032. doi: 10.1016/j.jsv.2013.08.028 Huang, Q., Gardoni, P., & Hurlebaus, S. (2012). A probabilistic damage detection approach using vibrationbased nondestructive testing. STRUCTURAL SAFETY, 38, 11-21. doi: 10.1016/j.strusafe.2012.01.004 Hussein, M. F. M., & Hunt, H. E. M. (2007). A numerical model for calculating vibration from a railway tunnel embedded in a full-space. JOURNAL OF SOUND AND VIBRATION, 305(3), 401-431. doi: 10.1016/j.jsv.2007.03.068 Koh, C. G., Ong, J., Chua, D., & Feng, J. (2003). Moving element method for traintrack dynamics. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 56(11), 1549-1567. doi: 10.1002/nme.624 Lam, H. F., Wong, M. T., & Yang, Y. B. (2012). A feasibility study on railway ballast damage detection utilizing measured vibration of in situ concrete sleeper. 12

ENGINEERING STRUCTURES, 45, 284-298. doi: 10.1016/j.engstruct.2012.06.022 Berggren EG. (2009) Railway track stiffness dynamic measurement sand evaluation for efficient maintenance. PhDThesis, Royal Institute of Technology. Kolbe T., & Kratochwille R(2015). ICE- S-Vehicle reaction measurement and track geometry measurement on the same measuring train. Results of the comparison of the two different track inspection methods. In: IMech E Stephenson conference for railways, London. Paper: C1408/068, 2015. Lei, X., & Zhang, B. (2011). Analyses of dynamic behavior of track transition with finite elements. JOURNAL OF VIBRATION AND CONTROL, 17(11), 1733-1747. doi: 10.1177/1077546310385035 Link, M., & Weiland, M. (2009). Damage identification by multi-model updating in the modal and in the time domain. MECHANICAL SYSTEMS AND SIGNAL PROCESSING, 23(6), 1734-1746. doi: 10.1016/j.ymssp.2008.11.009 Zapico-Valle, J. L., Alonso-Camblor, R., Gonzalez-Martinez, M. P., & Garcia- Dieguez, M. (2010). A new method for finite element model updating in structural dynamics. MECHANICAL SYSTEMS AND SIGNAL PROCESSING, 24(7), 2137-2159. doi: 10.1016/j.ymssp.2010.03.011 13

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