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Fast damage detection using max-peak and max-peak-time for PC box girder in vibration experiment
Journal Pre-proofs Fast damage detection using max-peak and max-peak-time for PC box girder in vibration experiment Xiong-Fei Ye, Chul-Woo Kim, Harutoshi Ogai PII: DOI: Reference:
S0263-2241(19)31044-9 https://doi.org/10.1016/j.measurement.2019.107178 MEASUR 107178
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
10 July 2018 25 September 2019 19 October 2019
Please cite this article as: X-F. Ye, C-W. Kim, H. Ogai, Fast damage detection using max-peak and max-peak-time for PC box girder in vibration experiment, Measurement (2019), doi: https://doi.org/10.1016/j.measurement. 2019.107178
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© 2019 Published by Elsevier Ltd.
Fast damage detection using max-peak and max-peak-time for PC box girder in vibration experiment Xiong-Fei YE1*, Chul-Woo KIM2, Harutoshi OGAI3 1 Ph.D. 2
Student, Faculty of Science and Engineering, Waseda University, Tokyo, JAPAN
Professor, Graduate School of Engineering, Kyoto University, Kyoto, JAPAN
3 Professor,
Faculty of Science and Engineering, Waseda University, Tokyo, JAPAN
* Corresponding
author: [email protected], +977-98-1873-9659, FAX: +977-1-5554-450
Abstract: Dynamic features are often used for structural damage diagnosis. Besides the features from the modal analysis, some other features describing the wave propagation in the structure are usually concerned, like the wave’s intensity, speed, etc. In the data processing of experiment on the pre-stressed concrete (PC) box girders with artificial damages, the max-peak and max-peak-time, two of the basic features of time series will be investigated and analyzed in reconstruction combination matrices to investigate the global information of the update of structural status respectively, which are corresponding to the speed and intensity of the wave propagation separately. In all stages of vibration experiment, curve of distance/2-D correlation coefficient of the max-peak-time and max-peak matrices can have very clear tendencies to demonstrate the global damage happened in the PC box girder, which are better than some modal features in modal analysis using the same original data. Keywords: Acceleration; damage detection; vibration experiment; max-peak; max-peak-time; PC box girder.
1. Introduction Kinds of damages occur in the lifetime of a structure, so that the physical characteristics of the structure will gradually wane, and the structure sudden collapses or loses its function eventually. Generally, the daily loads is the main source of damage, and the changing weather [1-3], like change of temperature, humidity, wind etc. erosion caused by the flowing water (the velocity and the height of water flow) as well as freezing and thawing of water, etc. and furthermore, the corrosion [4], fatigue [5] etc. will exacerbate the damage to the structure. Therefore, damage detection and health monitoring are required in practice to prevent this disaster. Non-destructive evaluation (NDE) methods, including magnetic particle [6], acoustic emission [7], ultrasound [8] etc. are widely used in civil engineering, mechanical engineering, and aeronautical engineering etc. Of many methods of NDE proposed [9], measurement of elastic wave, or vibration, is one of the most common methods, where the time series is often the main original data recorded for vibration (wave propagation) in damage assessment. The time and amplitude of the wave are the two basic values of a time series, and every damage detection method will investigate the information from such time series. The damage diagnosis based on structural vibration characteristics, named as modal analysis [10], is the main method used in the experimental analysis and the evaluation in structural health monitoring. The vibration experiment can be enforced to clear the dynamic structural characteristics, and through these characteristics, damage can be evaluated and identified. Some indicators of Eigen parameters in the modal analysis [11-14] , including damping values (e.g. damping ratio), natural frequency, multiple modal assurance criteria (MAC) or coordinate modal assurance criteria (COMAC) for mode shapes associated with each natural frequency, as well as modal participation [15], were proposed. However, it is -1-
still lack of information to satisfy engineers' needs for relatively high precision and confidence [16]. Meanwhile, some other methods [17], like the Computer Tomography (CT) [18], damage detection based on artificial intelligence and revolution algorithm (e.g. particle swarm optimization, PSO) [19, 20], and digital image measurement [21] etc., are sometimes hard to meet the requirement for quick damage determination and identification in time, in which, the application of CT is limited by the size of the structure and can only be used for smaller sized structures. Method of artificial intelligence needs to collect a lot of training material and spend a long time in training for a satisfied model. Digital image measurement cannot be carried out very well under some weather, like rainfall. Therefore, a method to balance the efficiency and precision should be in place for the need of engineering usages. For fast evaluation of the structural deterioration, in this paper, two classes of new indicators inspired by the earthquake engineering and flood control, are investigated from the vibrating time series to simulate the flood of wave caused by the impact loads. In the experiment, there are two kinds of artificial damages adopted into the prestress concrete box girder, which are monitored using the acceleration sensors. In the analysis, both classes of indicators proposed in this research shows the efficiency for deterioration of structure, by comparing the parameters used in the modal analysis with the same time series.
2. Basic Concepts and Indicators for Damage Detection In this research, the wave propagation indicators, max-peak and max-peak-time, are inspired by research in earthquake engineering and flood control. In seismic studies, the epicentre of earthquakes, magnitude, intensity, propagation time, types of vibration waves, etc., are all critical research parameters. In order to understand ground motions, we recognize geophysics, geological structures, fault zones, and even crack rock on slopes or mountains. People understand what is happening inside the earth through the waveform of seismic motion and the propagation of ground motion [22]. In some sense, Acoustic emission (AE) used in damage detection is similar to the Earthquakes, especially the volcanic earthquakes. The wave transferring in the structure is similar to the seismic wave transmission in the Crust, where the characteristics of wave propagation can be sensed to detect damage. On the other hand, when we concern about disasters, we often talk about flood control (flood protection). In the process of flood flowing, to quantitatively analyze the impact of floods on the watercourse, the flood peak is the key parameter [23]. Also, the movement process of peak is another key. When conducting flood prevention work in different rivers, the arrival time of the flood peak is very critical. Through the inflow volumes of every river are different in different years, it can still maintain a relative stability every year. When the watercourse is changed, the transmission time and intensity of water flow will often be very different. On the basis of inspiration from earthquake engineering and flood control projects, we discuss structural damage and hope to recognize the changes of structural from the most basic propagation signals recorded as time series. According to the survey for the experimental analysis, the similar metaphor parameters from earthquake and flood (max-peak, max-peak-time) can also be beneficial to understanding the structural changes in essence. Through a comprehensive analysis on the horizontal and vertical axes of the time series, the changes in the structure caused by the damage as well the damage development in the structure can be quickly found and recognized through comparative analysis with the healthy status. In details, on the one side, the max-peak proposed is investigated through the curve (time series) generated by an impulse wave. In Fig.1, the waves recorded at different locations on the surface of the structure are different, then there is a distribution for intensity of the wave energy at different locations, in which the max-peak can be treated as one kind of extreme value distribution. The max-peak is related to the transmission process of the impulse wave energy. When there is some damages in the structure, the distribution will change as well. -2-
Fig. 1. The power distribution of wave for propagation of the impulse wave in the system On the other side, the max-peak-time is another indicator among features of wave propagation for damage detection of the structure. When the damage is developing in the structure, the route of the wave will change as a large probability event [18]. For example, in Fig.2 at first, it is route B, then it may be route A or route C when the damage of structure exacerbates. Often, the speed of the wave in the structure is constant. In different route, the time used to reach the objective locations from the same location of input (Stress wave by hitting) will be different.
Fig. 2. The wave route for different conditions of the system Moreover, the ratio between max-peak and max-peak-time will be another indicator which is related to the max-peak and max-peak-time at the same time. It shows the speed of transfer between crest and trough of the wave. Furthermore, studies in three different fields have similar research points (Table. 1); the features of wave propagation inspired by the earthquake engineering and the flood control can directly show the change of the structure with damage developing, in which max-peak and max-peak-time are two representative species. Table. 1 The relationship between damage detection and the earthquake engineering, the flood control. Structural damage detection
Earthquake engineering
Flood control
Max-peak
Earthquake magnitude
Flood crest
Max-peak-time
Arriving time of the earthquake
Arriving time of the flood crest
Structural condition (crack etc.)
Geological conditions
River etc.
Non-Destructive
Seismic intensity
Damage evaluation after the flood
Wave route
Fracture zone & epicentral distance etc.
Path of the flood
In the instance in Fig.3, the greatest amplitude of the record of the sensor on the impact hammer (Fig.3 (1)) is obtained at 182 (ms). It is approximately set as the start time of the vibration for every sensor (Fig.3 (2-3)).
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(3) (4) Fig. 3. The sensors’ records. (1) The record of the sensor on the impact hammer’s record of time series, (2) All sensors’ records of the time series, (3) Enlarged view of all sensors’ records of the time series for max-peak and max-peak-time in rectangle in (2), (4) Enlarged view of any single sensor’s record. Here we choose four different types of max-peak (A, A=ΔA) and max-peak-time (t, t=Δt) as well the ratio of max-peak over max-peak-time in Fig. 4 for a single record (Fig.3 (4)) , in which the arrow shows the time range (Time-axis) or value range (Amplitude-axis). Then there are 9 types of indicators in our research, in which the ratio of Max_peak_time over Max_peak is calculated by Δt/ΔA and divided into 4 kinds. Define: ----------------------------------------------------------------------------------------------------------------------------------- Max_peak_1 (MP1): The peak range between positive maximum and negative minimum value; Max_peak_2 (MP2): The positive maximum value; Max_peak_3 (MP3): The negative minimum value; Max_peak_4 (MP4): The maximum absolute value; Max_peak_time_1 (MPT1): The time range of arriving time between positive maximum value and negative minimum value; Max_peak_time_2 (MPT2): The arriving time of positive maximum value; Max_peak_time_3 (MPT3): The arriving time of negative minimum value; Max_peak_time_4 (MPT4): The arriving time of the maximum absolute value; The ratio of Max-peak-time over Max-peak (the time required for unit value change):
Ratio_1: Time range over peak range between positive maximum and negative minimum value;
Ratio_2: The arriving time of positive maximum value over the positive maximum value;
Ratio_3: The arriving time of negative minimum value over the negative minimum value; -4-
Ratio_4: The arriving time of the maximum absolute value over the maximum absolute value.
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Fig. 4. The max-peak (Amplitude-axis, ΔA) and max-peak-time (time-axis, Δt). (1) the peak range between positive maximum and negative minimum value (ΔA), the time range of arriving time between positive maximum value and negative minimum value (Δt), and time range over peak range between positive maximum and negative minimum value (Δt/ΔA); (2) the positive maximum value (ΔA), the arriving time of positive maximum value (Δt), and the arriving time of positive maximum value over the positive maximum value (Δt/ΔA); (3) the negative minimum value (ΔA), the arriving time of negative minimum value (Δt), and The arriving time of negative minimum value over the negative minimum value (Δt/ΔA); (4) the maximum absolute value (ΔA), the arriving time of the maximum absolute value(Δt) The arriving time of the maximum absolute value over the maximum absolute value (Δt/ΔA). In the following calculations, because the wave intensity is related to the intensity of input pulse, assuming that the input pulse maximum amplitude is U, in calculation, the normalized value is adopted as the value of input for further calculation, i.e.∆A = ∆𝐴/𝑈. As the propagation time is less associated with the intensity of input pulse, the normalization is no need. After all, we need search for the arriving time of maximum peak value for every hit, and survey the max-peak and max-peak-time for all times of hit.
3. Experiment The experiment is conducted on pre-stressed concrete (PC) box-girders. It has length of 8500mm, width of 2300mm, height of 1000mm in Fig.5. The static loading test was conducted at middle of structure. Tendon for pre-stressed concrete is made of seven steel bars twisted together(Fig.19). The type of impact sensors (sensors to detect the -5-
impact signals, using one kind of acceleration sensors) is piezoelectric ONOSOKKI NP-2120 with a maximum acceleration of 8000 m/s2 and a sensitivity of 5pC/(m/s2)+/-2dB (Fig.20 (1)), and the impact hammer is Brüel & Kjær type 8208 with a maximum force compression of 44.4 KN and a sensitivity of 0.225 mV/N (Fig.20 (2)). Then data is stored by the Multi-input Data Acquisition System Keyence Series NR-600 (Fig.20 (3)).
Fig. 5. Sensor arrangement for static-loading experiment and impact hammer experiment (A1 to A10)
(1)
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Fig. 6. (1) the real structure for static loading, (2) the operation of tendon cut, (3) after final static loading, the girder collapses. The variation experiment is conducted through impact hammer hitting the box girder in different locations and the information is recorded by the acceleration sensors. Fig. 5 shows sensors’ locations of the experimental PC box-girder (real structure in Fig.6 (1)). The whole experiment process (Fig.7 (1)) is divided into 11 stages. In the initial stage of the experimental process, only the impact hammer test is conducted. In stage 2 to 9 of the experiment process, firstly conduct the static loading/tendon cut or wait for the recovery of PC box girder after tendon cut, and then conduct the impact hammer test. In stage 10 to 11 only the static loading test is conducted. In the experiment, the hit point is just approximately right above every sensor’s location on the girder. The impact hammer experiment is conducted from hit 1 to hit 10 and sensed by sensor 1 to sensor 10 shown in Fig.8. There are two kinds of damages in this experiment, one is caused by the static loading and the other is caused by the tendon cut. The damage in the experiment is mainly manifested by the expansion of cracks (Fig.21) and the change of structural deflection (comparing Fig.6 (1) and Fig.6 (3), as well as the displacement change in Fig.7 (2)). Because the experimental process is a continuous failure experiment, and there is no maintenance in the experimental process. We can estimate that the damage to the structure continues to increase by data processing. -6-
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(2) Fig. 7. (1) Experimental process of vibration experiment (impact hammer experiment) from stages 1 to 9 and static loading (stage 2, 5, 8, 9, 10, and 11). (2) the process of static loading in detail, and in stage 11, until 1326 kN, the structure collapses. The static loading can be treated as adding artificial damages to the structure. Hit 1
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Fig. 8. Sensors’ locations and hit points with one-to-one correspondence
4. Results and discussion 4.1 Data processing In the analysis damage detection for a concrete structure, time series of acceleration will be reconstructed as
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combination matrices. Some data will be selected from the original x = x1 , x 2 , x3 , , xi , , x n to survey max-peak for the acceleration signal that:
A j = A1, j , A2 , j , A3 , j , , Ai , j , , An , j , j 1, 2, 3, , n
j 1, 2,3, , n
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where Ai , j is the max value of time series recorded by sensor i. And max-peak-time is the time from the hit time point
to the time of the max-peak, set it as t j = t1 , j , t 2 , j , t 3 , j , , ti , j , , t n , j . Using reconstructed combination matrix, the greatest number of every time series is selected for every hit and every sensor’s record. In our experiment, the features (max-peak, A, max-peak-time, t) will be calculated through Eq. (1). Set both as single variable in
M xi , j , i, j 1, 2, , n , where n=10 in this experiment (Fig.8).
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where xi,j means the i-th hit and j-th sensor’ detection of max-peak time. Concerning the physical significance of the matrix, it contains the information of the locations of both sensors and hit points together simultaneously. In the data processing, to distinguish two matrices, the distance often shows the difference, while 2-D correlation coefficient often means the similarity. Both can indicate the change of features for wave propagation in the system, i.e. max-peak and max-peak-time. To calculate the distance between two matrices, suppose M is a square matrix described in Eq. (2). There are many kinds of distance between two status matrices, like the cosine distance, Euclidean distance, etc. Here is another example defined in Eq. (3) for distance between the matrix of initial stage (suppose this stage is the structure in initial stage is healthy) and the stage considered (any stage in the lifetime of the system after initial stage): 2
D 4 Mi, j initial - Mi, j considered Mi, j initial - Mi, j considered j 1 j 1 j 1 i 1
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Also, 2-D correlation coefficient, 2D-CC, between every considered status matrix and initial status matrix is C in Eq. (4). The correlation coefficient is usually to distinguish the status matrix change between the initial state and the state considered in this research. The definition of this technology:
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For static analysis of all samples in the experiment, every line of reconstructed combination matrix is randomly selected from the alternatives of the propagation features (max-peak or max-peak-time) surveyed from different time series recorded by 10 acceleration sensors for any hit at the same location in one stage. Then, there -8-
are distributions of the objective value in every stage which is shown in Fig.9 (1, 2). The evaluation of the damage in the structure will guaranteed by such distributions of different stages by comparing them with the initial health stage according to Fig.9 (3-1, 3-2). However, for the need of quick damage detection, we will use a simplified approach. First, get the expectation of all propagation features (max-peak or max-peak-time) of time series of 10 acceleration sensors for all hits at the same location in one stage, and there is 10 numbers for one location. Then reconstruct the combination matrix by contacting all 10 locations. So, only a single matrix will be obtained for every stage. Here is an example of “Positive maximum value”, though the result of Fig.10 (2) and Fig.13 (2) is more or less different from the result in Fig.9 (3-1, 3-2), but, in both cases, the two corresponding curves’ trends are consistent respectively.
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Fig. 9. (1) The change of the distance of max-peak (“Positive maximum value”) in different stages (stage 1~9), and (2) The change of the 2-D correlation coefficient of max-peak in different stages (stage 1~9) in statistics of Kaplan-Meier estimate [24]. In these figures, FH means “frequency histogram”, KDE means “kernel density estimation” [25], and ND means “normal distribution”. Subfigure (3-1) is the expectation value of the subfigure (1) and Subfigure (3-2) is the expectation value of the subfigure (2) in which, every point of subfigure (3-1) or subfigure (3-2) is the expectation of every subplot in subfigure (1) or subfigure (2). The following figures (Fig.10-15) introduce results of 2 categories (max-peak and max-peak-time), 9 types, or 12 kinds of indicators. The distance between matrix of considered stage and matrix of the initial stage for every indicator in all stages are corresponded to Max_peak_1 in Fig.10 (1), Max_peak_2 in Fig.10 (2), Max_peak_3 in Fig.10 (3), Max_peak_4 in Fig.10 (4), Max_peak_time_1 in Fig.11 (1), Max_peak_time_2 in Fig.11 (2), Max_peak_time_3 in Fig.11 (3), Max_peak_time_4 in Fig.11 (4), and ratio of max-peak-time over max-peak in Fig.12, where Ratio_1 in Fig.12 (1), Ratio_2 in Fig.12 (2), Ratio_3 in Fig.12 (3), and Ratio_4 in Fig.12 (4). 10 -3
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10 5
12
2
8
Distance
Distance
10
6 4
1
2
3
4
5
6
7
8
(1)
2
3
4
(2)
5
6
7
8
9
10 6
2.5
10
2
8
Distance
Distance
1
Stage
10 5
12
6 4
1.5 1 0.5
2
(3)
1
0
9
Stage
0
1.5
0.5
2 0
10 6
2.5
1
2
3
4
5
Stage
6
7
8
0
9
(4)
1
2
3
4
5
6
7
8
9
Stage
Fig. 12. Distance between matrix of considered stage and matrix of the initial stage for ratio of max-peak-time over max-peak of (1) The time range of arriving time between positive maximum value and negative minimum value (3) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value.
- 11 -
2-D correlation coefficient
2-D correlation coefficient
1 0.99 0.98 0.97 0.96 0.95 0.94
1
2
3
4
5
6
7
8
0.98 0.97 0.96
2
3
4
5
6
7
8
9
Stage
(3)
0.94 0.92 0.9
1
2
3
4
5
6
7
8
9
6
7
8
9
Stage 2-D correlation coefficient
2-D correlation coefficient
0.99
1
0.96
(2)
1
0.95
0.98
9
Stage
(1)
1
1 0.99 0.98 0.97 0.96 0.95
1
2
3
4
5
Stage
(4)
Fig. 13. 2-D correlation coefficient between matrix of considered stage and matrix of the initial stage for max-peak of (1) Peak range between positive maximum and negative minimum value (2) Positive maximum value (3) Negative minimum value (4) Maximum absolute value 2-D correlation coefficient
2-D correlation coefficient
1 0.9 0.8 0.7 0.6 0.5
1
2
3
4
5
6
7
8
0.9 0.85 0.8 0.75
9
Stage
(1)
1 0.95
2
3
4
(2)
5
6
7
8
9
1
2-D correlation coefficient
2-D correlation coefficient
(3)
1
Stage
1
0.9
0.8
0.7
0.6
0.7
1
2
3
4
5
Stage
6
7
8
0.95 0.9 0.85 0.8 0.75
9
(4)
1
2
3
4
5
6
7
8
9
Stage
Fig. 14. 2-D correlation coefficient between matrix of considered stage and matrix of the initial stage for max-peak-time of (1) The time range of arriving time between positive maximum value and negative minimum value (2) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value
- 12 -
2-D correlation coefficient
2-D correlation coefficient
1
0.9
0.8
0.7
0.6
1
2
3
4
6
7
8
9
Stage
(1) 1 0.9 0.8 0.7 0.6 0.5
1
2
3
4
5
0.95 0.9 0.85 0.8 0.75 0.7 0.65
1
2
3
4
6
7
8
9
5
6
7
8
9
Stage
(2) 2-D correlation coefficient
2-D correlation coefficient
5
1
1
0.9
0.8
0.7
0.6
1
2
3
4
5
6
7
8
9
Stage
Stage
(3) (4) Fig. 15. 2-D correlation coefficient between matrix of considered stage and matrix of the initial stage for ratio of max-peak-time over max-peak of (1) The time range of arriving time between positive maximum value and negative minimum value (3) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value. The 2D-CC between matrix of considered stage and matrix of the initial stage for every indicator in all stages are corresponded to: Max_peak_1 in Fig.13 (1), Max_peak_2 in Fig.13 (2), Max_peak_3 in Fig.13 (3), Max_peak_4 in Fig.13 (4), Max_peak_time_1 in Fig.14 (1), Max_peak_time_2 in Fig.14 (2), Max_peak_time_3 in Fig.14 (3), Max_peak_time_4 in Fig.14 (4), and ratio of max-peak-time over max-peak in Fig.15, where Ratio_1 in Fig.15 (1), Ratio_2 in Fig.15 (2), Ratio_3 in Fig.15 (3), and Ratio_4 in Fig.15 (4).
4.2 Quantitative analysis and Qualitative analysis From the evaluation results of structural health on basis of max-peak and max-peak-time by the variables of distance and 2D-CC in Fig.10~15, all curves have the clear developing tendency from stage 1 to stage 9 indicating the damage developing in the structure. Here we do not use the fitting curves, since these broken lines are clear enough to show the change of the indicators, and the fitting curve may also mislead the ignorance of the special events like the damage caused by static loadings or tendon cuts in the artificial accelerated failure experiment of structure. In details, both four types of max-peak and max-peak-time, they all show the same tendency that, the distance between the initial stage and the considered stages is become greater and greater, while the 2D-CC between initial stage and the considered stages is become smaller and smaller, which shows a very clear tendency of the indicators. The distance increasing and the 2D-CC decreasing mean the wave in the system changes. Assuredly, the artificial damages of structure change the route of the wave but also the distribution of the wave intensity. However, in some sense, according to the evaluation of difference by calculating the distance, the structure becomes more and more different from the initial stage when the damage was developing in the structure. But it is hard to normalize the evaluating value for the change of the structure. On the other hand, the 2D-CC is much better for the normalization of the similarity where the range of value is [0,1]. But the change of the value is sometimes hard to distinguish especially for the application in structural health monitoring. Further qualitative analysis is - 13 -
applied for deep understanding of the damage developing in the structure. In the qualitative analysis, we just need to know the data intuitional condition updates between every two neighbor stages. For every judgement comparing with the actuality of structural continious continued deterioration, if the derivative of the broken line is positive, the value of change is “+”, if the derivative of the broken line is negative, the value of change is “-”, and if the derivative of the broken line is 0, the value of change is 0.
Table. 2 The qualitative analysis of the result of variable “Distance” in evaluating kinds of indicators Stage
1
MP1 MP2 MP3 MP4 MPT1 MPT2 MPT3 MPT4 Ratio 1 Ratio 2 Ratio 3 Ratio 4 Summary
2
3
4
5
6
7
8
9
Summary
+
-
+
-
+
+
+
+
+
+
-
+
+
+
+
-
+
+
+
-
+
-
+
+
+
+
+
+
-
+
-
-
+
-
+
0
+
-
+
+
+
+
-
-
+
+
+
+
-
+
+
+
+
+
+
+
+
-
+
+
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
+
+
+
+
+
-
+
-
-
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
-
-
+
+
0
+
0
+
+
-
+
+
Table. 3 The qualitative analysis of the result of variable “2D-CC” in evaluating kinds of indicators Stage
MP1 MP2 MP3 MP4 MPT1 MPT2 MPT3 MPT4 Ratio 1 Ratio 2 Ratio 3 Ratio 4 Summary
1
2
3
4
5
6
7
8
9
Summary
-
+
-
+
-
-
+
-
-
-
+
-
-
-
-
+
-
-
-
+
-
+
-
-
-
-
-
-
+
-
+
+
-
+
-
0
-
-
-
-
-
-
+
-
-
-
-
-
+
-
-
-
-
-
-
-
+
-
-
-
+
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
+
-
-
-
-
-
-
-
+
-
-
-
-
-
+
-
-
-
-
-
-
-
-
-
-
-
+
+
-
-
-
-
-
-
-
+
-
-
Concerning the quantitative analysis (Fig. 10-15) and qualitative analysis (Table 2-3) in this research, there are some specific results and discussions for the specific experiment. In details, the dynamic experiment using the propagation features of wave will reveal the damage caused tendon cut clearly. After stage 2, stage 4, stage 7, there are some anomalistic change especially for the stage 7. In stage 8, from the performance of the structure, the big crack is growing quickly that the structure cannot have the same performance resisting almost the same loading as stage 5. Just after the stage 2 and stage 4, the tendon cuts for the pre-stressed box girder were conducted. There are - 14 -
some obvious changes of the max-peak and max-peak-time observed from Fig.10-15. Moreover, after stage 7, the structure cannot have the enough resistance over the loading effects and there may be new internal rebalancing of the structure. The static loading in this stage is great enough, so that the structure may have the crack recompressed together, and the route and intensity of wave at the same location will renew. In other words, the artificial damages caused by both static loading and tendon cut will have different influence on the structure, and the response of the structure in dynamic tests (impact hammer experiment) will also be different. Among all indicators, the most efficient indicators in all indicators are the arriving time of maximum absolute value, the ratio of max-peak-time over max-peak. Also, the arriving time of maximum absolute value can be treated as a mix understanding of the arriving time of positive maximum value and negative minimum value. Randomness (variance of the variable distribution) of the max-peak is greater than the max-peak-time and the ratio of max-peak-time over max-peak. In Table 2-3, regarding distance’s 32 judgments, Max-peak has 10 error judgments, while max-peak-time has only 6 error judgments, and ratio of max-peak-time over max-peak has 9 error judgments; Concerning 2D-CC's 32 judgments, max-peak has 11 error judgments, while max-peak-time has only 4 error judgments, and ratio of max-peak-time over max-peak has 6 error judgments. For different indicators, they have different sensitivity to the damage. In addition, the conflicts of these indicators can show some special event happened in the structure, like the tendon cut. If we look over the loading process and tendon cut, we will find that after the tendon cut in stage 3 and stage 6, the indicators of max-peak increase, while in recovery in stage 4 and 7, the indicators of max-peak decrease. However, the max-peak-time and ratio of the max-peak-time over max-peak will not have such phenomena except Ratio_3. But for the indicator named as ratio of max-peak-time over max-peak would continuous show the damage development as well as the most dangerous case of damage (in this experiment, it is stage 7). According to Table.4-5 (correlation coefficients of the sequence of all max-peak and max-peak-times), generally, the correlation coefficients among four max-peak indicators or among four max-peak-time indicators is greater than the correlation coefficients between max-peak indicators and max-peak-time indicators. In a summary, by comprehensively applying kinds of damage indicators of max-peak and max-peak-time as well as the unsynchronized change or conflicts, it will have the chance to identify the damage developing and the important event happened in the structure.
4.3 Comparison with modal analysis The same data can also be used for the modal analysis. Since the distributions of the frequency are not definite but in a small range, here just chooses the expectation of the parameters (which can be treated as damage indicators in modal analysis) in every subfigure (stage). For example, the expectation of the frequency in Fig.16 can be represented as Fig.18 (3) for frequency ≈29 in all 9 stages. Some other results in Fig.17-18 show the structural characteristics, the mode shape, MAC, and frequency and damping ratio of 1st and 2nd bending mode by using SSI [26]. Comparing modal parameters (indicators) with wave propagation features (indicators) in this paper, the result of the modal analysis cannot provide enough confidence to detect the damages in which some kind of damages cannot easily be described, such as the tendon cut; and the propagation features can indicate a better trend for the developing of the damages in the structure. Furthermore, concerning the limitations of modal analysis that for larger engineering structures, the feasibility of the frequency-based modal analysis method for damage detection are limited for two reasons [27]. First, severe damage can result in small changes in natural frequencies that may be unrecognizable due to measurement or processing errors. Second, the measured natural frequency is much smaller than the number of structural elements, which can lead to erroneous results. Moreover, if we look deep into the features of wave propagation, the wave propagation can describe as the dynamic field (flow) in mathematics which is used widely in fluid mechanics [28]. Suppose the wave is in one kind of field using Eulerian description, kinds of distribution of max-peak (in the surface of the structure) will be - 15 -
fluctuation of intensity (rise and fall). Also, suppose the wave as a motion transferred among the sub-structures using the Lagrangian description, the kinds of distribution (with a lot of vibration tests) of max-peak-time will be obtained. So the method using the features of wave propagation can be named as dynamic field analysis or flow analysis. stage 1
10 5 30 30.2 30.4 Frequency (HZ)
5 30.2 30.4 Frequency (HZ)
30.6
Histogram
10 5 0 30
30.2
30.8
30.6 30.4 Frequency (HZ)
5 30 30.2 30.4 Frequency (HZ)
30.6
30.2 30.4 Frequency (HZ)
10
30.4
30.2 30 29.8 Frequency (HZ)
10
30
30.1 30.2 30.3 Frequency (HZ)
30.4
30.5
29.6
29.8
stage 9
20 10 0 28.6
30.6
30.6
20
30
20
29.6
30
stage 6
0 29.9
30.8
stage 8
0 29.4
31
5
30
10
29.8
10
0 29.8
30.4
15
30
15
30.3
stage 5
0 29.6
30.8
stage 7
20
30 30.1 30.2 Frequency (HZ)
Histogram
Histogram
Histogram
10
30
29.9
20
15
0 29.8
5 0 29.8
30.8
stage 4
20
Histogram
30.6
10
Histogram
29.8
stage 3
15
Histogram
15
0 29.6
stage 2
15
Histogram
Histogram
20
28.8
29.2 29.4 29 Frequency (HZ)
Fig. 16. The histogram of the structural natural frequency (29~31 HZ) in every stage 0.5
1
0.9995
0.3 MAC
MODE
0.4
0.2
0.9985
0.1 0
(1)
0.999
1
2
3
4 Sensor
5
6
0.998
7
(2)
1
2
3
4
5 Stage
2
3
4
5 Stage
6
7
8
9
0.042
30.5
Damping ratio
Frequency: HZ
0.04 30
29.5
0.038 0.036 0.034 0.032
29
(3)
1
2
3
4
5 Stage
6
7
8
9
0.03
(4)
1
6
7
Fig. 17. The mode shape, MAC, frequency and damping ratio of 1st bending mode (29~31 HZ)
- 16 -
8
9
0.5 1
MODE
0.98
MAC
0
0.96
0.94
-0.5
1
2
3
4 Sensor
(1)
5
6
0.92
7
3
4
5 Stage
6
7
8
9
6
7
8
9
0.055
Damping ratio
Frequency: HZ
66 65.5 65
0.05
0.045
64.5
(3)
2
0.06
66.5
64
1
(2)
1
2
3
4
5 Stage
6
7
8
9
0.04
(4)
1
2
3
4
5 Stage
Fig. 18. The mode shape, MAC, frequency and damping ratio of 2nd bending mode (64~66 HZ) In a summary, comparative analysis found that max-peak and max-peak-time demonstrated their advantages in application and achieved better results than conventional modal analysis.
5. Conclusion In this work, vibration test (impact hammer experiment) is conducted on an almost full-scale experimental PC box girder. This paper examined indicators of two categories (max-peak/max-peak-time), 9 types, or 12 kinds for damage diagnosis for this PC box girder. In all 9 types of damage indicators, they all show the same tendency that, the distance between the initial stage and the considered stages is approximately become greater and greater, while the 2D-CC between the initial stage and the considered stages is approximately become smaller and smaller from stage 1 to stage 9. These indicators have very successful performance in distinguishing the change of propagation of wave due to damages in the PC box girder. The investigations in this study shows that the proposed structural damage indicators are more sensitive than modal parameters. Meanwhile, these indicators are convenient to use and have lower calculating complexity than the indicators of structural characteristics in modal analysis, which can be used for early warning of structural collapse, and to improve the effectiveness of the risk diagnosis in structural health monitoring.
Acknowledgement This study was partly supported by a Japanese Society for Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (B) under Project No. 16H04398. That financial support is gratefully acknowledged. The authors would like to thank Dr. Yoshinobu Oshima of Public Works Research Institute (PWRI), Mr. O.S. Luna Vera and Dr. Kai-Chun Chang from Kyoto University for their great support in research and experiment, and also the first author expresses his appreciation to the support from China Scholarship Council (No. 201508050019) in both life and research. - 17 -
Appendix
(1)
(2) Figure 19. The (1) Top-view and (2) Cross-section of Tendon
(1) (2) (3) Figure 20. (1) Real photo of impact sensors (2) Real photo of impact hammer (3) Real photo of impact recorder.
(1)
(2)
(3)
Figure 21. Real photo of crack development at (1) stage 2, (2) stage 5, and (3) stage 9. Table. 4 Correlation coefficient of sequence for indicators evaluated by Distance MP1
MP2
MP3
MP4
MPT1
MPT2
MPT3
MPT4
Ratio
MP1
1.000000
0.986643
0.992609
0.992273
0.852312
0.944691
0.847274
0.921048
1.000000
MP2
0.986643
1.000000
0.992375
0.984743
0.852741
0.945297
0.848017
0.933594
0.986643
MP3
0.992609
0.992375
1.000000
0.986410
0.867518
0.959260
0.860454
0.946556
0.992609
MP4
0.992273
0.984743
0.986410
1.000000
0.886370
0.960028
0.885421
0.942951
0.992273
MPT1
0.852312
0.852741
0.867518
0.886370
1.000000
0.948550
0.997793
0.954290
0.852312
MPT2
0.944691
0.945297
0.959260
0.960028
0.948550
1.000000
0.940992
0.991225
0.944691
MPT3
0.847274
0.848017
0.860454
0.885421
0.997793
0.940992
1.000000
0.946831
0.847274
MPT4
0.921048
0.933594
0.946556
0.942951
0.954290
0.991225
0.946831
1.000000
0.921048
- 18 -
Table. 5 Correlation coefficient of sequence for indicators evaluated by 2D-CC MP1
MP2
MP3
MP4
MPT1
MPT2
MPT3
MPT4
Ratio
MP1
1.000000
0.962783
0.985198
0.993568
0.748786
0.942596
0.742114
0.902833
0.676918
MP2
0.962783
1.000000
0.944206
0.961098
0.819343
0.963251
0.816928
0.940240
0.760928
MP3
0.985198
0.944206
1.000000
0.968443
0.728367
0.940932
0.716600
0.902576
0.648681
MP4
0.993568
0.961098
0.968443
1.000000
0.749722
0.937905
0.745849
0.904109
0.695776
MPT1
0.748786
0.819343
0.728367
0.749722
1.000000
0.878692
0.996330
0.917019
0.977540
MPT2
0.942596
0.963251
0.940932
0.937905
0.878692
1.000000
0.862623
0.989811
0.836405
MPT3
0.742114
0.816928
0.716600
0.745849
0.996330
0.862623
1.000000
0.905928
0.977418
MPT4
0.902833
0.940240
0.902576
0.904109
0.917019
0.989811
0.905928
1.000000
0.891719
Table. 6 Result summary evaluated by Distance for max-peak and max-peak-time Stage
1
2
3
4
5
6
7
8
9
MP1
0
0.000576
0.000494
0.000690
0.000599
0.000630
0.000850
0.000863
0.001182
MP2
0
0.000377
0.000317
0.000413
0.000416
0.000453
0.000644
0.000636
0.000787
MP3
0
0.000378
0.000319
0.000426
0.000407
0.000477
0.000566
0.000598
0.000770
MP4
0
0.000339
0.000279
0.000425
0.000391
0.000375
0.000525
0.000501
0.000651
MPT1
0
305.5607
297.7434
326.3063
335.5642
348.1499
376.4213
367.9830
360.0092
MPT2
0
134.3950
150.6795
188.1290
171.2952
205.8821
221.1182
217.7436
249.2824
MPT3
0
304.5348
292.1395
319.9625
347.5510
338.8140
379.5912
352.8451
357.7611
MPT4
0
77.1812
80.4722
101.5852
104.7501
120.2043
122.4137
131.5810
134.4012
Ratio 1
0
867207.2
872317.7
981897.8
1039417.5
1047314.9
1137674.9
1100496.7
1014381.2
Ratio 2
0
1491516.1
1557482.1
1763758.3
1970326.2
1938812.5
2118270.5
2085593.9
2007596.2
Ratio 3
0
607325.2
626080.5
824765.5
655658.3
812752.2
989527.1
999994.3
1027064.4
Ratio 4
0
1607927.2
1600738.3
1819452.9
1863511.4
2023422.9
2298116.4
2288468.9
2096467.7
Table. 7 Result summary evaluated by 2D-CC for max-peak and max-peak-time Stage
1
2
3
4
5
6
7
8
9
MP1 MP2 MP3 MP4 MPT1 MPT2 MPT3 MPT4
1
0.985437
0.988265
0.976988
0.983386
0.981674
0.966725
0.969597
0.948408
1
0.975530
0.981439
0.966262
0.963659
0.958008
0.923644
0.937675
0.911380
1
0.986012
0.990991
0.981462
0.985301
0.979813
0.973794
0.970486
0.950737
1
0.987778
0.990733
0.976138
0.982564
0.984011
0.968239
0.971860
0.953849
1
0.745019
0.735617
0.732977
0.703831
0.648306
0.593666
0.643080
0.640561
1
0.911431
0.894119
0.852230
0.871261
0.820614
0.777031
0.760675
0.717110
1
0.762787
0.757673
0.762550
0.708545
0.689182
0.621951
0.681015
0.669921
1
0.912831
0.898957
0.867947
0.860723
0.834807
0.805826
0.783466
0.767931
Ratio 1
1
0.793628
0.742149
0.716534
0.672273
0.651199
0.609036
0.653072
0.681106
Ratio 2
1
0.838395
0.786682
0.782271
0.715611
0.701887
0.671919
0.699583
0.697371
Ratio 3
1
0.897570
0.889142
0.803037
0.893578
0.840601
0.651598
0.603234
0.575296
Ratio 4
1
0.829242
0.778332
0.772797
0.727889
0.697815
0.638751
0.652126
0.699300
- 19 -
References [1]
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[2]
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Highlights
1. Max-peak and max-peak-time are proposed as the indicators for damage detection. 2. The use of combination matrix guarantees a wider use of data information. 3. Distance and the 2D-CC are used to identify the indicator matrix updating.
- 23 -
S0263-2241(19)31044-9 https://doi.org/10.1016/j.measurement.2019.107178 MEASUR 107178
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
10 July 2018 25 September 2019 19 October 2019
Please cite this article as: X-F. Ye, C-W. Kim, H. Ogai, Fast damage detection using max-peak and max-peak-time for PC box girder in vibration experiment, Measurement (2019), doi: https://doi.org/10.1016/j.measurement. 2019.107178
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© 2019 Published by Elsevier Ltd.
Fast damage detection using max-peak and max-peak-time for PC box girder in vibration experiment Xiong-Fei YE1*, Chul-Woo KIM2, Harutoshi OGAI3 1 Ph.D. 2
Student, Faculty of Science and Engineering, Waseda University, Tokyo, JAPAN
Professor, Graduate School of Engineering, Kyoto University, Kyoto, JAPAN
3 Professor,
Faculty of Science and Engineering, Waseda University, Tokyo, JAPAN
* Corresponding
author: [email protected], +977-98-1873-9659, FAX: +977-1-5554-450
Abstract: Dynamic features are often used for structural damage diagnosis. Besides the features from the modal analysis, some other features describing the wave propagation in the structure are usually concerned, like the wave’s intensity, speed, etc. In the data processing of experiment on the pre-stressed concrete (PC) box girders with artificial damages, the max-peak and max-peak-time, two of the basic features of time series will be investigated and analyzed in reconstruction combination matrices to investigate the global information of the update of structural status respectively, which are corresponding to the speed and intensity of the wave propagation separately. In all stages of vibration experiment, curve of distance/2-D correlation coefficient of the max-peak-time and max-peak matrices can have very clear tendencies to demonstrate the global damage happened in the PC box girder, which are better than some modal features in modal analysis using the same original data. Keywords: Acceleration; damage detection; vibration experiment; max-peak; max-peak-time; PC box girder.
1. Introduction Kinds of damages occur in the lifetime of a structure, so that the physical characteristics of the structure will gradually wane, and the structure sudden collapses or loses its function eventually. Generally, the daily loads is the main source of damage, and the changing weather [1-3], like change of temperature, humidity, wind etc. erosion caused by the flowing water (the velocity and the height of water flow) as well as freezing and thawing of water, etc. and furthermore, the corrosion [4], fatigue [5] etc. will exacerbate the damage to the structure. Therefore, damage detection and health monitoring are required in practice to prevent this disaster. Non-destructive evaluation (NDE) methods, including magnetic particle [6], acoustic emission [7], ultrasound [8] etc. are widely used in civil engineering, mechanical engineering, and aeronautical engineering etc. Of many methods of NDE proposed [9], measurement of elastic wave, or vibration, is one of the most common methods, where the time series is often the main original data recorded for vibration (wave propagation) in damage assessment. The time and amplitude of the wave are the two basic values of a time series, and every damage detection method will investigate the information from such time series. The damage diagnosis based on structural vibration characteristics, named as modal analysis [10], is the main method used in the experimental analysis and the evaluation in structural health monitoring. The vibration experiment can be enforced to clear the dynamic structural characteristics, and through these characteristics, damage can be evaluated and identified. Some indicators of Eigen parameters in the modal analysis [11-14] , including damping values (e.g. damping ratio), natural frequency, multiple modal assurance criteria (MAC) or coordinate modal assurance criteria (COMAC) for mode shapes associated with each natural frequency, as well as modal participation [15], were proposed. However, it is -1-
still lack of information to satisfy engineers' needs for relatively high precision and confidence [16]. Meanwhile, some other methods [17], like the Computer Tomography (CT) [18], damage detection based on artificial intelligence and revolution algorithm (e.g. particle swarm optimization, PSO) [19, 20], and digital image measurement [21] etc., are sometimes hard to meet the requirement for quick damage determination and identification in time, in which, the application of CT is limited by the size of the structure and can only be used for smaller sized structures. Method of artificial intelligence needs to collect a lot of training material and spend a long time in training for a satisfied model. Digital image measurement cannot be carried out very well under some weather, like rainfall. Therefore, a method to balance the efficiency and precision should be in place for the need of engineering usages. For fast evaluation of the structural deterioration, in this paper, two classes of new indicators inspired by the earthquake engineering and flood control, are investigated from the vibrating time series to simulate the flood of wave caused by the impact loads. In the experiment, there are two kinds of artificial damages adopted into the prestress concrete box girder, which are monitored using the acceleration sensors. In the analysis, both classes of indicators proposed in this research shows the efficiency for deterioration of structure, by comparing the parameters used in the modal analysis with the same time series.
2. Basic Concepts and Indicators for Damage Detection In this research, the wave propagation indicators, max-peak and max-peak-time, are inspired by research in earthquake engineering and flood control. In seismic studies, the epicentre of earthquakes, magnitude, intensity, propagation time, types of vibration waves, etc., are all critical research parameters. In order to understand ground motions, we recognize geophysics, geological structures, fault zones, and even crack rock on slopes or mountains. People understand what is happening inside the earth through the waveform of seismic motion and the propagation of ground motion [22]. In some sense, Acoustic emission (AE) used in damage detection is similar to the Earthquakes, especially the volcanic earthquakes. The wave transferring in the structure is similar to the seismic wave transmission in the Crust, where the characteristics of wave propagation can be sensed to detect damage. On the other hand, when we concern about disasters, we often talk about flood control (flood protection). In the process of flood flowing, to quantitatively analyze the impact of floods on the watercourse, the flood peak is the key parameter [23]. Also, the movement process of peak is another key. When conducting flood prevention work in different rivers, the arrival time of the flood peak is very critical. Through the inflow volumes of every river are different in different years, it can still maintain a relative stability every year. When the watercourse is changed, the transmission time and intensity of water flow will often be very different. On the basis of inspiration from earthquake engineering and flood control projects, we discuss structural damage and hope to recognize the changes of structural from the most basic propagation signals recorded as time series. According to the survey for the experimental analysis, the similar metaphor parameters from earthquake and flood (max-peak, max-peak-time) can also be beneficial to understanding the structural changes in essence. Through a comprehensive analysis on the horizontal and vertical axes of the time series, the changes in the structure caused by the damage as well the damage development in the structure can be quickly found and recognized through comparative analysis with the healthy status. In details, on the one side, the max-peak proposed is investigated through the curve (time series) generated by an impulse wave. In Fig.1, the waves recorded at different locations on the surface of the structure are different, then there is a distribution for intensity of the wave energy at different locations, in which the max-peak can be treated as one kind of extreme value distribution. The max-peak is related to the transmission process of the impulse wave energy. When there is some damages in the structure, the distribution will change as well. -2-
Fig. 1. The power distribution of wave for propagation of the impulse wave in the system On the other side, the max-peak-time is another indicator among features of wave propagation for damage detection of the structure. When the damage is developing in the structure, the route of the wave will change as a large probability event [18]. For example, in Fig.2 at first, it is route B, then it may be route A or route C when the damage of structure exacerbates. Often, the speed of the wave in the structure is constant. In different route, the time used to reach the objective locations from the same location of input (Stress wave by hitting) will be different.
Fig. 2. The wave route for different conditions of the system Moreover, the ratio between max-peak and max-peak-time will be another indicator which is related to the max-peak and max-peak-time at the same time. It shows the speed of transfer between crest and trough of the wave. Furthermore, studies in three different fields have similar research points (Table. 1); the features of wave propagation inspired by the earthquake engineering and the flood control can directly show the change of the structure with damage developing, in which max-peak and max-peak-time are two representative species. Table. 1 The relationship between damage detection and the earthquake engineering, the flood control. Structural damage detection
Earthquake engineering
Flood control
Max-peak
Earthquake magnitude
Flood crest
Max-peak-time
Arriving time of the earthquake
Arriving time of the flood crest
Structural condition (crack etc.)
Geological conditions
River etc.
Non-Destructive
Seismic intensity
Damage evaluation after the flood
Wave route
Fracture zone & epicentral distance etc.
Path of the flood
In the instance in Fig.3, the greatest amplitude of the record of the sensor on the impact hammer (Fig.3 (1)) is obtained at 182 (ms). It is approximately set as the start time of the vibration for every sensor (Fig.3 (2-3)).
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(3) (4) Fig. 3. The sensors’ records. (1) The record of the sensor on the impact hammer’s record of time series, (2) All sensors’ records of the time series, (3) Enlarged view of all sensors’ records of the time series for max-peak and max-peak-time in rectangle in (2), (4) Enlarged view of any single sensor’s record. Here we choose four different types of max-peak (A, A=ΔA) and max-peak-time (t, t=Δt) as well the ratio of max-peak over max-peak-time in Fig. 4 for a single record (Fig.3 (4)) , in which the arrow shows the time range (Time-axis) or value range (Amplitude-axis). Then there are 9 types of indicators in our research, in which the ratio of Max_peak_time over Max_peak is calculated by Δt/ΔA and divided into 4 kinds. Define: ----------------------------------------------------------------------------------------------------------------------------------- Max_peak_1 (MP1): The peak range between positive maximum and negative minimum value; Max_peak_2 (MP2): The positive maximum value; Max_peak_3 (MP3): The negative minimum value; Max_peak_4 (MP4): The maximum absolute value; Max_peak_time_1 (MPT1): The time range of arriving time between positive maximum value and negative minimum value; Max_peak_time_2 (MPT2): The arriving time of positive maximum value; Max_peak_time_3 (MPT3): The arriving time of negative minimum value; Max_peak_time_4 (MPT4): The arriving time of the maximum absolute value; The ratio of Max-peak-time over Max-peak (the time required for unit value change):
Ratio_1: Time range over peak range between positive maximum and negative minimum value;
Ratio_2: The arriving time of positive maximum value over the positive maximum value;
Ratio_3: The arriving time of negative minimum value over the negative minimum value; -4-
Ratio_4: The arriving time of the maximum absolute value over the maximum absolute value.
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Fig. 4. The max-peak (Amplitude-axis, ΔA) and max-peak-time (time-axis, Δt). (1) the peak range between positive maximum and negative minimum value (ΔA), the time range of arriving time between positive maximum value and negative minimum value (Δt), and time range over peak range between positive maximum and negative minimum value (Δt/ΔA); (2) the positive maximum value (ΔA), the arriving time of positive maximum value (Δt), and the arriving time of positive maximum value over the positive maximum value (Δt/ΔA); (3) the negative minimum value (ΔA), the arriving time of negative minimum value (Δt), and The arriving time of negative minimum value over the negative minimum value (Δt/ΔA); (4) the maximum absolute value (ΔA), the arriving time of the maximum absolute value(Δt) The arriving time of the maximum absolute value over the maximum absolute value (Δt/ΔA). In the following calculations, because the wave intensity is related to the intensity of input pulse, assuming that the input pulse maximum amplitude is U, in calculation, the normalized value is adopted as the value of input for further calculation, i.e.∆A = ∆𝐴/𝑈. As the propagation time is less associated with the intensity of input pulse, the normalization is no need. After all, we need search for the arriving time of maximum peak value for every hit, and survey the max-peak and max-peak-time for all times of hit.
3. Experiment The experiment is conducted on pre-stressed concrete (PC) box-girders. It has length of 8500mm, width of 2300mm, height of 1000mm in Fig.5. The static loading test was conducted at middle of structure. Tendon for pre-stressed concrete is made of seven steel bars twisted together(Fig.19). The type of impact sensors (sensors to detect the -5-
impact signals, using one kind of acceleration sensors) is piezoelectric ONOSOKKI NP-2120 with a maximum acceleration of 8000 m/s2 and a sensitivity of 5pC/(m/s2)+/-2dB (Fig.20 (1)), and the impact hammer is Brüel & Kjær type 8208 with a maximum force compression of 44.4 KN and a sensitivity of 0.225 mV/N (Fig.20 (2)). Then data is stored by the Multi-input Data Acquisition System Keyence Series NR-600 (Fig.20 (3)).
Fig. 5. Sensor arrangement for static-loading experiment and impact hammer experiment (A1 to A10)
(1)
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Fig. 6. (1) the real structure for static loading, (2) the operation of tendon cut, (3) after final static loading, the girder collapses. The variation experiment is conducted through impact hammer hitting the box girder in different locations and the information is recorded by the acceleration sensors. Fig. 5 shows sensors’ locations of the experimental PC box-girder (real structure in Fig.6 (1)). The whole experiment process (Fig.7 (1)) is divided into 11 stages. In the initial stage of the experimental process, only the impact hammer test is conducted. In stage 2 to 9 of the experiment process, firstly conduct the static loading/tendon cut or wait for the recovery of PC box girder after tendon cut, and then conduct the impact hammer test. In stage 10 to 11 only the static loading test is conducted. In the experiment, the hit point is just approximately right above every sensor’s location on the girder. The impact hammer experiment is conducted from hit 1 to hit 10 and sensed by sensor 1 to sensor 10 shown in Fig.8. There are two kinds of damages in this experiment, one is caused by the static loading and the other is caused by the tendon cut. The damage in the experiment is mainly manifested by the expansion of cracks (Fig.21) and the change of structural deflection (comparing Fig.6 (1) and Fig.6 (3), as well as the displacement change in Fig.7 (2)). Because the experimental process is a continuous failure experiment, and there is no maintenance in the experimental process. We can estimate that the damage to the structure continues to increase by data processing. -6-
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(2) Fig. 7. (1) Experimental process of vibration experiment (impact hammer experiment) from stages 1 to 9 and static loading (stage 2, 5, 8, 9, 10, and 11). (2) the process of static loading in detail, and in stage 11, until 1326 kN, the structure collapses. The static loading can be treated as adding artificial damages to the structure. Hit 1
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4. Results and discussion 4.1 Data processing In the analysis damage detection for a concrete structure, time series of acceleration will be reconstructed as
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combination matrices. Some data will be selected from the original x = x1 , x 2 , x3 , , xi , , x n to survey max-peak for the acceleration signal that:
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to the time of the max-peak, set it as t j = t1 , j , t 2 , j , t 3 , j , , ti , j , , t n , j . Using reconstructed combination matrix, the greatest number of every time series is selected for every hit and every sensor’s record. In our experiment, the features (max-peak, A, max-peak-time, t) will be calculated through Eq. (1). Set both as single variable in
M xi , j , i, j 1, 2, , n , where n=10 in this experiment (Fig.8).
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where xi,j means the i-th hit and j-th sensor’ detection of max-peak time. Concerning the physical significance of the matrix, it contains the information of the locations of both sensors and hit points together simultaneously. In the data processing, to distinguish two matrices, the distance often shows the difference, while 2-D correlation coefficient often means the similarity. Both can indicate the change of features for wave propagation in the system, i.e. max-peak and max-peak-time. To calculate the distance between two matrices, suppose M is a square matrix described in Eq. (2). There are many kinds of distance between two status matrices, like the cosine distance, Euclidean distance, etc. Here is another example defined in Eq. (3) for distance between the matrix of initial stage (suppose this stage is the structure in initial stage is healthy) and the stage considered (any stage in the lifetime of the system after initial stage): 2
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Also, 2-D correlation coefficient, 2D-CC, between every considered status matrix and initial status matrix is C in Eq. (4). The correlation coefficient is usually to distinguish the status matrix change between the initial state and the state considered in this research. The definition of this technology:
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For static analysis of all samples in the experiment, every line of reconstructed combination matrix is randomly selected from the alternatives of the propagation features (max-peak or max-peak-time) surveyed from different time series recorded by 10 acceleration sensors for any hit at the same location in one stage. Then, there -8-
are distributions of the objective value in every stage which is shown in Fig.9 (1, 2). The evaluation of the damage in the structure will guaranteed by such distributions of different stages by comparing them with the initial health stage according to Fig.9 (3-1, 3-2). However, for the need of quick damage detection, we will use a simplified approach. First, get the expectation of all propagation features (max-peak or max-peak-time) of time series of 10 acceleration sensors for all hits at the same location in one stage, and there is 10 numbers for one location. Then reconstruct the combination matrix by contacting all 10 locations. So, only a single matrix will be obtained for every stage. Here is an example of “Positive maximum value”, though the result of Fig.10 (2) and Fig.13 (2) is more or less different from the result in Fig.9 (3-1, 3-2), but, in both cases, the two corresponding curves’ trends are consistent respectively.
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FH KDE ND
3000
Frequency
3000
0.6
0
Frequency
Frequency
9
2000
4000 FH KDE ND
(1)
Frequency
8
FH KDE ND
10 -4
Distance Stage7
Distance
Frequency
7
Distance Stage5
3000
13
12
11
10
9
8
7
6
5
4
0 0.4
Frequency
6
1000
4000
(2)
5
4000
1000
0 0.86
4
5000 FH KDE ND
2000
0 0.88
3
10 -4
Distance Stage4
3000
0 0.91
2000
Frequency
2
4000
0
3000
1000
5000
Frequency
Frequency
2000
Stage3
4000 FH KDE ND
4000
Frequency
Frequency
3000
0
Stage2
5000 FH KDE ND
Frequency
Stage1
4000
20 15 10 5
0.9
0.92
0.94
0.96
2-D correlation coefficient
-9-
0.98
1
0 0.85
0.9
0.95
2-D correlation coefficient
1
10 -4
10.5
0.975
2-D correlation coefficient
10 9.5
Distance
9 8.5 8 7.5 7 6.5 6
(3-1)
0.965
0.96
0.955
0.95
0.945
0.94
9
8
7
6
5
4
3
2
1
0.97
1
2
3
4
(3-2)
Stage
5
6
7
8
9
Stage
Fig. 9. (1) The change of the distance of max-peak (“Positive maximum value”) in different stages (stage 1~9), and (2) The change of the 2-D correlation coefficient of max-peak in different stages (stage 1~9) in statistics of Kaplan-Meier estimate [24]. In these figures, FH means “frequency histogram”, KDE means “kernel density estimation” [25], and ND means “normal distribution”. Subfigure (3-1) is the expectation value of the subfigure (1) and Subfigure (3-2) is the expectation value of the subfigure (2) in which, every point of subfigure (3-1) or subfigure (3-2) is the expectation of every subplot in subfigure (1) or subfigure (2). The following figures (Fig.10-15) introduce results of 2 categories (max-peak and max-peak-time), 9 types, or 12 kinds of indicators. The distance between matrix of considered stage and matrix of the initial stage for every indicator in all stages are corresponded to Max_peak_1 in Fig.10 (1), Max_peak_2 in Fig.10 (2), Max_peak_3 in Fig.10 (3), Max_peak_4 in Fig.10 (4), Max_peak_time_1 in Fig.11 (1), Max_peak_time_2 in Fig.11 (2), Max_peak_time_3 in Fig.11 (3), Max_peak_time_4 in Fig.11 (4), and ratio of max-peak-time over max-peak in Fig.12, where Ratio_1 in Fig.12 (1), Ratio_2 in Fig.12 (2), Ratio_3 in Fig.12 (3), and Ratio_4 in Fig.12 (4). 10 -3
1.2
10 -4
8
1
Distance
Distance
6 0.8 0.6 0.4
4
2 0.2 0
1
2
3
4
(1)
5
6
7
8
0
9
2
3
4
5
6
7
8
9
6
7
8
9
Stage
10 -4
8
1
(2)
Stage
10 -4
7 6 5
Distance
Distance
6
4
4 3 2
2
1 0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
Stage Stage (3) (4) Fig. 10. Distance between matrix of considered stage and matrix of the initial stage for max-peak of (1) Peak range between positive maximum and negative minimum value (2) Positive maximum value (3) Negative minimum value (4) Maximum absolute value
- 10 -
250
400
200
Distance
Distance
300
200
150 100
100
0
50
(1)
6
5
4
3
2
1
7
8
0
9
1
2
3
4
5
(2)
Stage
6
7
8
9
Stage 140
400
120 100
Distance
Distance
300
200
80 60 40
100
20 0
1
2
3
4
5
6
7
8
0
9
2
1
3
4
5
9
8
7
6
Stage Stage (3) (4) Fig. 11. Distance between matrix of considered stage and matrix of the initial stage for max-peak-time of (1) The time range of arriving time between positive maximum value and negative minimum value (3) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value.
10 5
12
2
8
Distance
Distance
10
6 4
1
2
3
4
5
6
7
8
(1)
2
3
4
(2)
5
6
7
8
9
10 6
2.5
10
2
8
Distance
Distance
1
Stage
10 5
12
6 4
1.5 1 0.5
2
(3)
1
0
9
Stage
0
1.5
0.5
2 0
10 6
2.5
1
2
3
4
5
Stage
6
7
8
0
9
(4)
1
2
3
4
5
6
7
8
9
Stage
Fig. 12. Distance between matrix of considered stage and matrix of the initial stage for ratio of max-peak-time over max-peak of (1) The time range of arriving time between positive maximum value and negative minimum value (3) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value.
- 11 -
2-D correlation coefficient
2-D correlation coefficient
1 0.99 0.98 0.97 0.96 0.95 0.94
1
2
3
4
5
6
7
8
0.98 0.97 0.96
2
3
4
5
6
7
8
9
Stage
(3)
0.94 0.92 0.9
1
2
3
4
5
6
7
8
9
6
7
8
9
Stage 2-D correlation coefficient
2-D correlation coefficient
0.99
1
0.96
(2)
1
0.95
0.98
9
Stage
(1)
1
1 0.99 0.98 0.97 0.96 0.95
1
2
3
4
5
Stage
(4)
Fig. 13. 2-D correlation coefficient between matrix of considered stage and matrix of the initial stage for max-peak of (1) Peak range between positive maximum and negative minimum value (2) Positive maximum value (3) Negative minimum value (4) Maximum absolute value 2-D correlation coefficient
2-D correlation coefficient
1 0.9 0.8 0.7 0.6 0.5
1
2
3
4
5
6
7
8
0.9 0.85 0.8 0.75
9
Stage
(1)
1 0.95
2
3
4
(2)
5
6
7
8
9
1
2-D correlation coefficient
2-D correlation coefficient
(3)
1
Stage
1
0.9
0.8
0.7
0.6
0.7
1
2
3
4
5
Stage
6
7
8
0.95 0.9 0.85 0.8 0.75
9
(4)
1
2
3
4
5
6
7
8
9
Stage
Fig. 14. 2-D correlation coefficient between matrix of considered stage and matrix of the initial stage for max-peak-time of (1) The time range of arriving time between positive maximum value and negative minimum value (2) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value
- 12 -
2-D correlation coefficient
2-D correlation coefficient
1
0.9
0.8
0.7
0.6
1
2
3
4
6
7
8
9
Stage
(1) 1 0.9 0.8 0.7 0.6 0.5
1
2
3
4
5
0.95 0.9 0.85 0.8 0.75 0.7 0.65
1
2
3
4
6
7
8
9
5
6
7
8
9
Stage
(2) 2-D correlation coefficient
2-D correlation coefficient
5
1
1
0.9
0.8
0.7
0.6
1
2
3
4
5
6
7
8
9
Stage
Stage
(3) (4) Fig. 15. 2-D correlation coefficient between matrix of considered stage and matrix of the initial stage for ratio of max-peak-time over max-peak of (1) The time range of arriving time between positive maximum value and negative minimum value (3) The arriving time of positive maximum value (3) The arriving time of negative minimum value (4) The arriving time of maximum absolute value. The 2D-CC between matrix of considered stage and matrix of the initial stage for every indicator in all stages are corresponded to: Max_peak_1 in Fig.13 (1), Max_peak_2 in Fig.13 (2), Max_peak_3 in Fig.13 (3), Max_peak_4 in Fig.13 (4), Max_peak_time_1 in Fig.14 (1), Max_peak_time_2 in Fig.14 (2), Max_peak_time_3 in Fig.14 (3), Max_peak_time_4 in Fig.14 (4), and ratio of max-peak-time over max-peak in Fig.15, where Ratio_1 in Fig.15 (1), Ratio_2 in Fig.15 (2), Ratio_3 in Fig.15 (3), and Ratio_4 in Fig.15 (4).
4.2 Quantitative analysis and Qualitative analysis From the evaluation results of structural health on basis of max-peak and max-peak-time by the variables of distance and 2D-CC in Fig.10~15, all curves have the clear developing tendency from stage 1 to stage 9 indicating the damage developing in the structure. Here we do not use the fitting curves, since these broken lines are clear enough to show the change of the indicators, and the fitting curve may also mislead the ignorance of the special events like the damage caused by static loadings or tendon cuts in the artificial accelerated failure experiment of structure. In details, both four types of max-peak and max-peak-time, they all show the same tendency that, the distance between the initial stage and the considered stages is become greater and greater, while the 2D-CC between initial stage and the considered stages is become smaller and smaller, which shows a very clear tendency of the indicators. The distance increasing and the 2D-CC decreasing mean the wave in the system changes. Assuredly, the artificial damages of structure change the route of the wave but also the distribution of the wave intensity. However, in some sense, according to the evaluation of difference by calculating the distance, the structure becomes more and more different from the initial stage when the damage was developing in the structure. But it is hard to normalize the evaluating value for the change of the structure. On the other hand, the 2D-CC is much better for the normalization of the similarity where the range of value is [0,1]. But the change of the value is sometimes hard to distinguish especially for the application in structural health monitoring. Further qualitative analysis is - 13 -
applied for deep understanding of the damage developing in the structure. In the qualitative analysis, we just need to know the data intuitional condition updates between every two neighbor stages. For every judgement comparing with the actuality of structural continious continued deterioration, if the derivative of the broken line is positive, the value of change is “+”, if the derivative of the broken line is negative, the value of change is “-”, and if the derivative of the broken line is 0, the value of change is 0.
Table. 2 The qualitative analysis of the result of variable “Distance” in evaluating kinds of indicators Stage
1
MP1 MP2 MP3 MP4 MPT1 MPT2 MPT3 MPT4 Ratio 1 Ratio 2 Ratio 3 Ratio 4 Summary
2
3
4
5
6
7
8
9
Summary
+
-
+
-
+
+
+
+
+
+
-
+
+
+
+
-
+
+
+
-
+
-
+
+
+
+
+
+
-
+
-
-
+
-
+
0
+
-
+
+
+
+
-
-
+
+
+
+
-
+
+
+
+
+
+
+
+
-
+
+
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
+
+
+
+
+
-
+
-
-
+
+
+
+
-
+
+
+
+
+
+
-
+
+
+
+
-
-
+
+
0
+
0
+
+
-
+
+
Table. 3 The qualitative analysis of the result of variable “2D-CC” in evaluating kinds of indicators Stage
MP1 MP2 MP3 MP4 MPT1 MPT2 MPT3 MPT4 Ratio 1 Ratio 2 Ratio 3 Ratio 4 Summary
1
2
3
4
5
6
7
8
9
Summary
-
+
-
+
-
-
+
-
-
-
+
-
-
-
-
+
-
-
-
+
-
+
-
-
-
-
-
-
+
-
+
+
-
+
-
0
-
-
-
-
-
-
+
-
-
-
-
-
+
-
-
-
-
-
-
-
+
-
-
-
+
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
+
-
-
-
-
-
-
-
+
-
-
-
-
-
+
-
-
-
-
-
-
-
-
-
-
-
+
+
-
-
-
-
-
-
-
+
-
-
Concerning the quantitative analysis (Fig. 10-15) and qualitative analysis (Table 2-3) in this research, there are some specific results and discussions for the specific experiment. In details, the dynamic experiment using the propagation features of wave will reveal the damage caused tendon cut clearly. After stage 2, stage 4, stage 7, there are some anomalistic change especially for the stage 7. In stage 8, from the performance of the structure, the big crack is growing quickly that the structure cannot have the same performance resisting almost the same loading as stage 5. Just after the stage 2 and stage 4, the tendon cuts for the pre-stressed box girder were conducted. There are - 14 -
some obvious changes of the max-peak and max-peak-time observed from Fig.10-15. Moreover, after stage 7, the structure cannot have the enough resistance over the loading effects and there may be new internal rebalancing of the structure. The static loading in this stage is great enough, so that the structure may have the crack recompressed together, and the route and intensity of wave at the same location will renew. In other words, the artificial damages caused by both static loading and tendon cut will have different influence on the structure, and the response of the structure in dynamic tests (impact hammer experiment) will also be different. Among all indicators, the most efficient indicators in all indicators are the arriving time of maximum absolute value, the ratio of max-peak-time over max-peak. Also, the arriving time of maximum absolute value can be treated as a mix understanding of the arriving time of positive maximum value and negative minimum value. Randomness (variance of the variable distribution) of the max-peak is greater than the max-peak-time and the ratio of max-peak-time over max-peak. In Table 2-3, regarding distance’s 32 judgments, Max-peak has 10 error judgments, while max-peak-time has only 6 error judgments, and ratio of max-peak-time over max-peak has 9 error judgments; Concerning 2D-CC's 32 judgments, max-peak has 11 error judgments, while max-peak-time has only 4 error judgments, and ratio of max-peak-time over max-peak has 6 error judgments. For different indicators, they have different sensitivity to the damage. In addition, the conflicts of these indicators can show some special event happened in the structure, like the tendon cut. If we look over the loading process and tendon cut, we will find that after the tendon cut in stage 3 and stage 6, the indicators of max-peak increase, while in recovery in stage 4 and 7, the indicators of max-peak decrease. However, the max-peak-time and ratio of the max-peak-time over max-peak will not have such phenomena except Ratio_3. But for the indicator named as ratio of max-peak-time over max-peak would continuous show the damage development as well as the most dangerous case of damage (in this experiment, it is stage 7). According to Table.4-5 (correlation coefficients of the sequence of all max-peak and max-peak-times), generally, the correlation coefficients among four max-peak indicators or among four max-peak-time indicators is greater than the correlation coefficients between max-peak indicators and max-peak-time indicators. In a summary, by comprehensively applying kinds of damage indicators of max-peak and max-peak-time as well as the unsynchronized change or conflicts, it will have the chance to identify the damage developing and the important event happened in the structure.
4.3 Comparison with modal analysis The same data can also be used for the modal analysis. Since the distributions of the frequency are not definite but in a small range, here just chooses the expectation of the parameters (which can be treated as damage indicators in modal analysis) in every subfigure (stage). For example, the expectation of the frequency in Fig.16 can be represented as Fig.18 (3) for frequency ≈29 in all 9 stages. Some other results in Fig.17-18 show the structural characteristics, the mode shape, MAC, and frequency and damping ratio of 1st and 2nd bending mode by using SSI [26]. Comparing modal parameters (indicators) with wave propagation features (indicators) in this paper, the result of the modal analysis cannot provide enough confidence to detect the damages in which some kind of damages cannot easily be described, such as the tendon cut; and the propagation features can indicate a better trend for the developing of the damages in the structure. Furthermore, concerning the limitations of modal analysis that for larger engineering structures, the feasibility of the frequency-based modal analysis method for damage detection are limited for two reasons [27]. First, severe damage can result in small changes in natural frequencies that may be unrecognizable due to measurement or processing errors. Second, the measured natural frequency is much smaller than the number of structural elements, which can lead to erroneous results. Moreover, if we look deep into the features of wave propagation, the wave propagation can describe as the dynamic field (flow) in mathematics which is used widely in fluid mechanics [28]. Suppose the wave is in one kind of field using Eulerian description, kinds of distribution of max-peak (in the surface of the structure) will be - 15 -
fluctuation of intensity (rise and fall). Also, suppose the wave as a motion transferred among the sub-structures using the Lagrangian description, the kinds of distribution (with a lot of vibration tests) of max-peak-time will be obtained. So the method using the features of wave propagation can be named as dynamic field analysis or flow analysis. stage 1
10 5 30 30.2 30.4 Frequency (HZ)
5 30.2 30.4 Frequency (HZ)
30.6
Histogram
10 5 0 30
30.2
30.8
30.6 30.4 Frequency (HZ)
5 30 30.2 30.4 Frequency (HZ)
30.6
30.2 30.4 Frequency (HZ)
10
30.4
30.2 30 29.8 Frequency (HZ)
10
30
30.1 30.2 30.3 Frequency (HZ)
30.4
30.5
29.6
29.8
stage 9
20 10 0 28.6
30.6
30.6
20
30
20
29.6
30
stage 6
0 29.9
30.8
stage 8
0 29.4
31
5
30
10
29.8
10
0 29.8
30.4
15
30
15
30.3
stage 5
0 29.6
30.8
stage 7
20
30 30.1 30.2 Frequency (HZ)
Histogram
Histogram
Histogram
10
30
29.9
20
15
0 29.8
5 0 29.8
30.8
stage 4
20
Histogram
30.6
10
Histogram
29.8
stage 3
15
Histogram
15
0 29.6
stage 2
15
Histogram
Histogram
20
28.8
29.2 29.4 29 Frequency (HZ)
Fig. 16. The histogram of the structural natural frequency (29~31 HZ) in every stage 0.5
1
0.9995
0.3 MAC
MODE
0.4
0.2
0.9985
0.1 0
(1)
0.999
1
2
3
4 Sensor
5
6
0.998
7
(2)
1
2
3
4
5 Stage
2
3
4
5 Stage
6
7
8
9
0.042
30.5
Damping ratio
Frequency: HZ
0.04 30
29.5
0.038 0.036 0.034 0.032
29
(3)
1
2
3
4
5 Stage
6
7
8
9
0.03
(4)
1
6
7
Fig. 17. The mode shape, MAC, frequency and damping ratio of 1st bending mode (29~31 HZ)
- 16 -
8
9
0.5 1
MODE
0.98
MAC
0
0.96
0.94
-0.5
1
2
3
4 Sensor
(1)
5
6
0.92
7
3
4
5 Stage
6
7
8
9
6
7
8
9
0.055
Damping ratio
Frequency: HZ
66 65.5 65
0.05
0.045
64.5
(3)
2
0.06
66.5
64
1
(2)
1
2
3
4
5 Stage
6
7
8
9
0.04
(4)
1
2
3
4
5 Stage
Fig. 18. The mode shape, MAC, frequency and damping ratio of 2nd bending mode (64~66 HZ) In a summary, comparative analysis found that max-peak and max-peak-time demonstrated their advantages in application and achieved better results than conventional modal analysis.
5. Conclusion In this work, vibration test (impact hammer experiment) is conducted on an almost full-scale experimental PC box girder. This paper examined indicators of two categories (max-peak/max-peak-time), 9 types, or 12 kinds for damage diagnosis for this PC box girder. In all 9 types of damage indicators, they all show the same tendency that, the distance between the initial stage and the considered stages is approximately become greater and greater, while the 2D-CC between the initial stage and the considered stages is approximately become smaller and smaller from stage 1 to stage 9. These indicators have very successful performance in distinguishing the change of propagation of wave due to damages in the PC box girder. The investigations in this study shows that the proposed structural damage indicators are more sensitive than modal parameters. Meanwhile, these indicators are convenient to use and have lower calculating complexity than the indicators of structural characteristics in modal analysis, which can be used for early warning of structural collapse, and to improve the effectiveness of the risk diagnosis in structural health monitoring.
Acknowledgement This study was partly supported by a Japanese Society for Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (B) under Project No. 16H04398. That financial support is gratefully acknowledged. The authors would like to thank Dr. Yoshinobu Oshima of Public Works Research Institute (PWRI), Mr. O.S. Luna Vera and Dr. Kai-Chun Chang from Kyoto University for their great support in research and experiment, and also the first author expresses his appreciation to the support from China Scholarship Council (No. 201508050019) in both life and research. - 17 -
Appendix
(1)
(2) Figure 19. The (1) Top-view and (2) Cross-section of Tendon
(1) (2) (3) Figure 20. (1) Real photo of impact sensors (2) Real photo of impact hammer (3) Real photo of impact recorder.
(1)
(2)
(3)
Figure 21. Real photo of crack development at (1) stage 2, (2) stage 5, and (3) stage 9. Table. 4 Correlation coefficient of sequence for indicators evaluated by Distance MP1
MP2
MP3
MP4
MPT1
MPT2
MPT3
MPT4
Ratio
MP1
1.000000
0.986643
0.992609
0.992273
0.852312
0.944691
0.847274
0.921048
1.000000
MP2
0.986643
1.000000
0.992375
0.984743
0.852741
0.945297
0.848017
0.933594
0.986643
MP3
0.992609
0.992375
1.000000
0.986410
0.867518
0.959260
0.860454
0.946556
0.992609
MP4
0.992273
0.984743
0.986410
1.000000
0.886370
0.960028
0.885421
0.942951
0.992273
MPT1
0.852312
0.852741
0.867518
0.886370
1.000000
0.948550
0.997793
0.954290
0.852312
MPT2
0.944691
0.945297
0.959260
0.960028
0.948550
1.000000
0.940992
0.991225
0.944691
MPT3
0.847274
0.848017
0.860454
0.885421
0.997793
0.940992
1.000000
0.946831
0.847274
MPT4
0.921048
0.933594
0.946556
0.942951
0.954290
0.991225
0.946831
1.000000
0.921048
- 18 -
Table. 5 Correlation coefficient of sequence for indicators evaluated by 2D-CC MP1
MP2
MP3
MP4
MPT1
MPT2
MPT3
MPT4
Ratio
MP1
1.000000
0.962783
0.985198
0.993568
0.748786
0.942596
0.742114
0.902833
0.676918
MP2
0.962783
1.000000
0.944206
0.961098
0.819343
0.963251
0.816928
0.940240
0.760928
MP3
0.985198
0.944206
1.000000
0.968443
0.728367
0.940932
0.716600
0.902576
0.648681
MP4
0.993568
0.961098
0.968443
1.000000
0.749722
0.937905
0.745849
0.904109
0.695776
MPT1
0.748786
0.819343
0.728367
0.749722
1.000000
0.878692
0.996330
0.917019
0.977540
MPT2
0.942596
0.963251
0.940932
0.937905
0.878692
1.000000
0.862623
0.989811
0.836405
MPT3
0.742114
0.816928
0.716600
0.745849
0.996330
0.862623
1.000000
0.905928
0.977418
MPT4
0.902833
0.940240
0.902576
0.904109
0.917019
0.989811
0.905928
1.000000
0.891719
Table. 6 Result summary evaluated by Distance for max-peak and max-peak-time Stage
1
2
3
4
5
6
7
8
9
MP1
0
0.000576
0.000494
0.000690
0.000599
0.000630
0.000850
0.000863
0.001182
MP2
0
0.000377
0.000317
0.000413
0.000416
0.000453
0.000644
0.000636
0.000787
MP3
0
0.000378
0.000319
0.000426
0.000407
0.000477
0.000566
0.000598
0.000770
MP4
0
0.000339
0.000279
0.000425
0.000391
0.000375
0.000525
0.000501
0.000651
MPT1
0
305.5607
297.7434
326.3063
335.5642
348.1499
376.4213
367.9830
360.0092
MPT2
0
134.3950
150.6795
188.1290
171.2952
205.8821
221.1182
217.7436
249.2824
MPT3
0
304.5348
292.1395
319.9625
347.5510
338.8140
379.5912
352.8451
357.7611
MPT4
0
77.1812
80.4722
101.5852
104.7501
120.2043
122.4137
131.5810
134.4012
Ratio 1
0
867207.2
872317.7
981897.8
1039417.5
1047314.9
1137674.9
1100496.7
1014381.2
Ratio 2
0
1491516.1
1557482.1
1763758.3
1970326.2
1938812.5
2118270.5
2085593.9
2007596.2
Ratio 3
0
607325.2
626080.5
824765.5
655658.3
812752.2
989527.1
999994.3
1027064.4
Ratio 4
0
1607927.2
1600738.3
1819452.9
1863511.4
2023422.9
2298116.4
2288468.9
2096467.7
Table. 7 Result summary evaluated by 2D-CC for max-peak and max-peak-time Stage
1
2
3
4
5
6
7
8
9
MP1 MP2 MP3 MP4 MPT1 MPT2 MPT3 MPT4
1
0.985437
0.988265
0.976988
0.983386
0.981674
0.966725
0.969597
0.948408
1
0.975530
0.981439
0.966262
0.963659
0.958008
0.923644
0.937675
0.911380
1
0.986012
0.990991
0.981462
0.985301
0.979813
0.973794
0.970486
0.950737
1
0.987778
0.990733
0.976138
0.982564
0.984011
0.968239
0.971860
0.953849
1
0.745019
0.735617
0.732977
0.703831
0.648306
0.593666
0.643080
0.640561
1
0.911431
0.894119
0.852230
0.871261
0.820614
0.777031
0.760675
0.717110
1
0.762787
0.757673
0.762550
0.708545
0.689182
0.621951
0.681015
0.669921
1
0.912831
0.898957
0.867947
0.860723
0.834807
0.805826
0.783466
0.767931
Ratio 1
1
0.793628
0.742149
0.716534
0.672273
0.651199
0.609036
0.653072
0.681106
Ratio 2
1
0.838395
0.786682
0.782271
0.715611
0.701887
0.671919
0.699583
0.697371
Ratio 3
1
0.897570
0.889142
0.803037
0.893578
0.840601
0.651598
0.603234
0.575296
Ratio 4
1
0.829242
0.778332
0.772797
0.727889
0.697815
0.638751
0.652126
0.699300
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Highlights
1. Max-peak and max-peak-time are proposed as the indicators for damage detection. 2. The use of combination matrix guarantees a wider use of data information. 3. Distance and the 2D-CC are used to identify the indicator matrix updating.
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