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Damage detection of RC beams based on experiment and analysis of nonlinear dynamic characteristics
Construction and Building Materials 29 (2012) 420–427
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Damage detection of RC beams based on experiment and analysis of nonlinear dynamic characteristics Liheng Wang a,b,⇑, Xiyuan Zhou b, He Liu c,1, Weiming Yan b a
National Key Laboratory of Dynamic Measurement and Calibration, Changcheng Institute of Metrology & Measurement, AVIC, P.O. Box 1066, Beijing 100095, PR China Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, No. 100, Pingleyuan, Beijing 100022, PR China c University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA b
a r t i c l e
i n f o
Article history: Received 2 April 2011 Received in revised form 14 October 2011 Accepted 20 October 2011 Available online 29 November 2011 Keywords: Damage detection Time–frequency analysis Nonlinear dynamic characteristics
a b s t r a c t The experimental and analytical investigation on damage detection of reinforced concrete (RC) beams has been carried out using a testing RC beam. The correlations between the damage extent and the nonlinear dynamic characteristics (NDC) of the testing beam are presented in this paper. The NDC of the RC beam were obtained by means of time–frequency analysis (TFA) of displacement responses from the impulsive and harmonic vibration tests. The correlation between the displacement amplitude and the excitation frequency in the case of forced vibration was obtained. Combining the results from the TFA with the relationship of displacement amplitude and corresponding natural vibration frequency (DA-CNVF) from the harmonic vibration tests, a reasonable window width for TFA was suggested. It is observed that the window width of seven times of the fundamental period of the testing structure is a suitable option in the time–frequency analysis. Using the TFA with this appropriate moving window width, the relationship between DA-CNVF at different damage levels in the case of free vibration was used to depict and analyze the NDC of the damaged RC beam. The feasibility of using NDC of a damaged RC beam from either forced harmonic or free vibration tests to detect damage level is discussed. The NDC of the reinforced concrete beam can be used to detect its damage levels if the apparent damage is not significant. Therefore, this damage detection method may be suitable for structures with small damage invisible or hard to be found by naked eyes. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Damage detection technology for structures can be classified into two categories: linear elastic method and nonlinear dynamic approach. The damage detection technology based on linear elastic theory can be sub-categorized as the modal method and the wavemotion method. In the modal method, using general purpose Finite Element software it is convenient to calculate natural frequencies and mode shapes of a structure [1,2]. Therefore, structural natural frequencies and mode shapes and consequently derived results, such as modal curvature [3], modal strain energy [4] and so on, are used as structural damage indexes in many literatures. On the other hand, the wave motion theories are gradually growing in recent years in damage detection because of its suitability in planar structure with multiple damages [5,12–14]. However, most of research ⇑ Corresponding author at: National Key Laboratory of Dynamic Measurement and Calibration, Changcheng Institute of Metrology & Measurement, AVIC, P.O. Box 1066, Beijing 100095, PR China. E-mail addresses: [email protected] (L. Wang), [email protected] (X. Zhou), [email protected] (H. Liu), [email protected] (W. Yan). 1 Tel.: +1 907 786 1971; fax: +1 907 786 1079. 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.10.065
methods in damage detection are limited to or based on theory of linear dynamics. Only a few literatures relate to the nonlinear theory, though RC structures behave nonlinearly in nature [10]. In fact there are several types of stiffness functions available to depict nonlinearity, such as the bilinear model of restoring force vs. displacement simulating fully opening and fully closing of cracked concrete [6–8] or crack’s variation from fully opening to partially opening and partially closing to fully closing during the structural vibration [9]. Because damage detection is mainly based on the identification of structural stiffness varying with time, and the stiffness is correlated to the change of structural natural vibration frequencies, identifying stiffness variation with time is equivalent to identify frequency variation with time using time–frequency analysis (TFA) [10] or empirical mode decomposition analysis [11]. Nevertheless, it was found that there were few documents on nonlinear dynamic characteristics of reinforced concrete (RC) beams using the TFA [10]. In this experimental–analytical study, first, the relationship between displacement amplitudes and its frequencies was obtained from different levels of harmonic excitation tests. And then, the window width for the TFA was adjusted to ensure that the results from TFA are in some degree consistent with the results from harmonic excitation tests. The reasonable window width in this
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study was chosen as seven times of structural fundamental period. Using this window width in TFA, the results of displacement response from impulsive excitation tests could be obtained by short time Fourier transformation (STFT); and the curves of amplitude and frequency could be also obtained prior and after structural damage at each loading level. Utilizing these curves, the correlation between nonlinear dynamic characteristics of a RC beam and its damage level can be detected.
Table 1 Reinforcing steel of the RC beam.
Reinforcing steel Diameter, d (mm) Effective depth, h (mm) Yield strength, fy (MPa) Number of bars, N
Bottom
Top
Shear
12 154 436.45 3
8
6.5
279.5 2
210 44
2. Experimental study 2.1. Beam design and material properties The testing RC beam was designed based on the consideration of the fundamental period of the beam consisting with that of general concrete bridges. The RC beam with dimensions 4350 mm in length, 200 mm in width and 180 mm in depth was simply supported within 4150 mm span as shown in Fig. 1. The fundamental frequency of the beam is 18.75 Hz, the ultimate moment capacity is 21.44 kN-m, the designed mid-span failure load (F) is 21.07 kN, and the maximum shear capacity is 63.95 kN. The 28-day strength of the concrete is 55.1 MPa. The design properties of reinforcing steel of the RC beam are shown in Table 1. The design properties of concrete of the RC beam are shown in Table 2.
2.2. Test set up The stiffness of the supporting structure of the testing beam must be large enough to reduce the measurement error. The supporting structures used in this experimental project are two RC piers with 400 mm in width, 400 mm in depth and 800 mm in length sitting on the laboratory RC basement shear wall with 600 mm depth . Because the interface between the bottom of concrete beam and the top of concrete piers was not smooth, the 1:3 cement mortars were plastered. The 25 mm depth armor plates were bonded to the top of the RC pier with 1:3 cement mortars. The armor plates and the round steels with 50 mm diameter and 200 mm length were welded beforehand as show in Fig. 1. From the results of calculation and measurement, it was indicated that the stiffness of such supporting equipment is large enough to satisfy the requirements of simply supported foundations. It should be noted that only a good contact condition between the bottom of the beam and the top of the round steel can ensure that the supports satisfy the simply supported requirements during the vibrating tests. Two accelerometers at the support of the testing beam were used to monitor acceleration responses at the supports. A 400 Hz low pass filter was used in the data acquisition system for accelerometer channels at the supports. The response acceleration at the mid-span was registered as shown in Fig. 6 when 400 Hz low pass filter was used. In order to ensure the measurement accuracy of acceleration and displacement, an accelerometer and two displacement meters were allocated at mid-span of the testing beam, respectively as show in Figs. 1 and 2.
2.3. Testing procedure The experiments were conducted at Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology. During the testing period the temperature was varied negligible.
Table 2 The concrete properties of the RC beam. Cement Sand Gravel Water fcu
360 kg/m3 736 kg/m3 1104 kg/m3 165 kg/m3 55.1 MPa
fcu is the 28 day cubic strength of the concrete.
The testing RC beam was simply supported. The vertical static loads up to ten levels were applied on the beam prior loading to failure by increasing static loads as shown in Table 3. It was planed that the concrete crack would occur (the crack strain of concrete was 213 le) when level 3 was loaded to its maximum. The residual strain measured after unloading level 4 was 117 le. After the tensile reinforcement yielded, the testing beam reached its ultimate capability when level 10 was loaded to its maximum. The RC beam was tested in its undamaged state and then up to failure in the subsequent loading levels. The tests were split into the following several steps: (1) Step 1 – static loading tests. There were totally ten load levels planned as shown in Table 3. It was expected that the beam reached its failure load at load level 10. The force and the strain were measured during loading listed in Table 3. (2) Step 2 – impulsive vibration tests. After each level of static loading, a hammer shown in Fig. 2 served as an exciting source and three accelerometers and two displacement gauges collected the beam’s response. Using the TFA, the decayed displacement response signals at the mid-span were recorded and analyzed to obtain a group of curves of mid-span displacement amplitude and corresponding natural vibration frequency (DA-CNVF) of the testing beam. (3) Step 3 – harmonic excitation tests. After each impulsive vibration test, an exciter shown in Fig. 2 served as an exciting source, also three accelerometers and two displacement gauges collected the beam’s response. The amplitude of the harmonic force applied at the mid-span point of the beam kept constant and the frequencies of the harmonic force varied within a certain range. The vibration signals of the beam were collected by the accelerometers placed on the beam. Using harmonic tests results the mid-span displacement amplitude and corresponding natural vibration frequency can also be obtained. (4) Step 4 – repeating Step 1 and moving to a new loading level.
Fig. 1. Dimensions, loading and measurement locations of the testing beam.
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Fig. 2. Set up of impulsive and harmonic excitation tests.
Table 3 Loading levels, maximum loads and maximum tensile strains in different loading levels. Load level
Maximum vertical static load (N)
Max. tensile strain at mid-span (le)
1 2 3 4 5 6 7 8 9 10
2107 4214 6321 8428 10,535 12,642 14,749 16,856 18,963 21,070
65 118 264 551 1494 1819 2300 2832 8853
(a) L7
(b) L8
(c) L10 Fig. 3. Observed crack sizes and positions at various loading levels.
Comparing the results of step 3 and step 4 a reasonable window width can be selected. Then, combining the selected reasonable window width the nonlinear dynamic characteristics of the RC beam were drawn using the impact displacement responses. The nonlinear characteristics, DA-CNVF curves are used to damage detection of the RC beam. 2.4. Experimental method 2.4.1. Static loading The static four-point bending tests were performed as shown in Fig. 1. The corresponding load levels are listed in Table 3. When the testing beam was loaded to the designed loading level, the cracks were observed carefully and the cracking traces were drawn in pencil on the surface of the beam. After reaching the designed loading level, the beam was unloaded. The cracking developments are shown in Fig. 3 for several load levels. In Fig. 3, L7, L8 and L10 show the crack distributions when load level 7, 8 and 10 reached their maximums, respectively. The large scale strain gauge, shown in Fig. 1, at mid-span 20 mm above the bottom of the beam was used to measure the strains within the range of 130 mm even after concrete cracking. The measured results are listed in Table 3. 2.4.2. Impulsive excitation tests At each loading level, once the beam had been unloaded, an impulsive excitation vibration test was performed. The hammer used in the test was Lc-04 impulsive hammer. After adding additional mass to the hammer, the total mass used for the impulsive excitation was 4.0 kg. The excitation force was applied at 160 mm away from mid-span with a rubber impacting tip dropped from a height of approximately 232 mm above the testing beam as shown in Fig. 2. The acceleration signal was acquired at the half-span point and two support-points. The displacement data were measured at the half-span point as shown in Fig. 2. Acceleration, displacement and hammer impulsive force were digitized at a sample rate frequency of 15 kHz to ensure that the data were captured accurately. In order to compare the results of impulsive vibration tests, maximum upward acceleration at the mid-span was registered each time after conducting the impulsive vibration test. The same tests must be repeated several times until there were results from at least six tests with relatively less than 5% errors of maximum up-
ward acceleration measured at the mid-span. Within 10 times of tests, there were, at least, 6 times to satisfy the requirements. Results from these six impulsive excitation tests were used for the subsequent TFA. 2.4.3. Harmonic excitation tests After impulsive excitation tests, the harmonic vibration tests began as set up shown in Fig. 2. Under the excitation of the harmonic force, the beam behaved as forced vibration. After the beam vibration reached at steady state, acceleration data were collected by the data acquisition system.
3. Nonlinear dynamic analysis of testing data 3.1. Analysis of testing data from harmonic excitation tests By processing the data from harmonic excitation tests a curve of acceleration amplitudes at the mid-span point vs. the input frequencies was obtained. Each curve shown in Fig. 4 represents a certain level of harmonic excitation and the seven curves in total correspond to different harmonic forces. It can be seen from Fig. 4 that with the increasing the amplitude of harmonic force the corresponding resonance vibration frequencies are slightly changed and the resonance peaks gradually shift leftward. From each of the seven curves, a peak can be detected, and seven peaks are marked by dots in Fig. 4. Furthermore, another data pair was also obtained from the test for measuring fundamental frequency under the same harmonic excitation. These eight data pairs indicate the correlation of acceleration amplitudes and corresponding resonant frequencies of the testing beam. According to the relationship between acceleration and displacement of harmonic
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vibration, the relationship of DA-CNVF can be obtained from the data in Fig. 4, as shown in Fig. 5. It is observed from the test after the loading level 2 that the frequencies decrease as the increasing displacement amplitudes. The fitted curve in Fig. 5 can be expressed as Eq. (1).
f ¼ aebx þ cedx
ð1Þ
where f is the natural frequency of the testing beam; x is the displacement amplitude at the mid-span. The coefficients in Eq. (1) are
a ¼ 0:4066;
b ¼ 4:125;
c ¼ 17:27;
d ¼ 0:03306
3.2. Analysis of testing data from impulsive vibration tests 3.2.1. Time–frequency analysis (TFA) The TFA aims mainly to describe how the frequency spectrum varies with time and then to explain the concepts of variation of frequency spectrum with time. For this purpose it is convenient to perform the TFA using STFT, which is defined as follows:
STFTðt; f Þ ¼
Z
þ1
SðsÞhðs tÞej2pf s ds
Fig. 5. Variation of fundamental natural vibration frequencies with displacement amplitudes, measured after loading level 2.
ð2Þ
1
The signal S(s) in Eq. (2) multiplied by the time window h(s t) can efficiently restrain the signals outside the neighborhood of s = t. Therefore, the results from STFT based on Eq. (2) is just local spectra of S(s) in vicinity of t. When the window h(s t) moves along the t axis, the corresponding time-variable spectra can be obtained using Eq. (2). The total frequency spectra are just the time– frequency distributions. In this paper, the TFA of the displacement response signals at the mid-span was conducted using Eq. (2). 3.2.2. Starting time point for TFA After being impacted, the testing beam behaved with decaying vibration characteristics. Fig. 6 shows the acceleration response at the mid-span below 400 Hz after loading level two. Observing Fig. 6, it can be seen that, at the beginning, the acceleration signal contains the first several natural frequency components. However, after a short period of time (about 0.25 s), the higher frequency components decays away and only the first fundamental frequency component remained. As the displacement amplitude reduces, the first frequency varies. Using the TFA, the decayed signals can be analyzed to obtain the variation of frequencies with time and the variation of frequencies with displacement amplitudes. It can be seen clearly from Fig. 6 that the higher frequency components adding on the lower frequencies components die away quickly. In order to compare the maximum values of acceleration response from six impulsive vibration tests, a 30 Hz low pass filter of the data acquisition system was used for the accelerometer
Fig. 4. The resonance curves measured after loading level 2 for various excitation levels.
Fig. 6. Acceleration response at the mid-span below 400 Hz, measured after loading level 2.
channel at the mid-span to allow only the first frequency component was collected. In this way, it was convenient to find the maximum value of the upward acceleration from the filtered acceleration signals. Comparing the maximum upward accelerations from several impulsive vibration tests, at least six testing results with relatively less than 5% errors of the maximum upward accelerations were chosen for the TFA. Since the first frequency was below 20 Hz and the purpose of TFA was to obtain the variation of fundamental frequency with displacement amplitudes, a 30 Hz low pass filter did not affect the needed results of the TFA. Due to the non-perfect condition at the simple supports, the high frequency shock was induced at the two ends of the testing beam. At the beginning, the acceleration responses at two supports were relatively large, but died away quickly. At the time point of 0.25 s, the acceleration amplitudes at two supports were much less than that at the mid-span. Thus, from this time point on, the beam can be regarded as simply supported beam. Therefore the 0.25 s was chosen as the initial time point for TFA. 3.2.3. Selecting the input signals of the TFA In Neild’s experimental study [10], the variation of frequencies with time was obtained using TFA via the acceleration signals at the mid-span of testing structure. The variation of displacement amplitudes was obtained by computing the acceleration amplitudes and corresponding frequencies, both of which vary with time. Because the error could be introduced into the frequencies in the TFA, when the displacement amplitudes were obtained from the acceleration amplitudes and corresponding frequencies, the errors of the displacement amplitudes might increase. In order to
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decrease such errors, the displacement response signals acquired from the displacement gauge at the mid-span were used. In order to avoid introducing the torsional response signals into the analysis data, two displacement sensors were used on the both sides of the beam at the middle span. The average of the signals from these two displacement gauges was used as the input data for the TFA. 3.2.4. Selecting the Window Width for TFA The relationship DA-CNVF obtained from either the TFA or the harmonic excitation vibration tests are essentially the same, because they both indicate the same structural NDC. The harmonic excitation tests are more time consuming, however, the results are more accurate. By contrast, the TFA is more convenient and quicker. Since the results of TFA are affected by the window width, how to select a suitable window width becomes a critical problem. Currently there are many approaches used in the TFA. For example the STFT is one of the commonly used methods for the TFA. Since the results from these two means are essentially the same in describing the structural NDC, combining the two means together, a reasonable window width for STFT could be identified. The detailed procedures are follows: i. Select a window width for TFA. ii. Obtain the relationship of structural DA-CNVF by the TFA. iii. Compare the results from the TFA and the harmonic excitation tests, the window width for the TFA is adjusted. iv. Repeat steps ii and iii until the results from TFA and the harmonic excitation tests are consistent. The following window widths were selected to compare with the harmonic vibration tests: 0.2, 0.3, 0.4 and 0.6 s. As shown in Fig. 7, using the window width of 0.2 s, the results from the TFA is not good for accurately estimating the natural frequencies; and the width of 0.3 s is a better choice. However, when the displacement amplitude was larger than 0.25 mm, the errors of fundamental natural frequency still existed in the estimation when using TFA with the 0.3 s window. As shown in Fig. 8, the estimation of structural frequency using the window width of 0.4 s is better than 0.3 s. As shown in Fig. 9, the results from the window width of 0.4 s are very close to the results from the 0.6 s; nevertheless the frequencies from 0.6 s window are more stable and below that from the 0.4 s window. Fig. 10 shows the comparison of results from the TFA using 0.4 s window with that from resonant vibration tests after loading level 2. The label of ‘harmonic excitation’ in Fig. 10 represents the relationship of DA-CNVF from harmonic excitation tests; meanwhile, the ‘curve fitting’ gives the result from fitting of the eight data points observed from harmonic excitation tests. The ‘window width
Fig. 7. Relationship of DA-CNVF from TFA using 0.2 and 0.3 s window width, respectively, measured after loading level 2.
Fig. 8. Relationship of DA-CNVF from TFA using 0.3 and 0.4 s window width, respectively, measured after loading level 2.
(0.4 s) time–frequency’ in Fig. 10 represents the results from the 0.4 s window width (equal to seven times of the fundamental period of the testing beam). It can be observed that the results from the 0.4 s window width are consistent with that from the harmonic excitation tests (the relative errors are less than 0.6% as shown in Fig. 10). The estimated frequencies from the 0.6 s window width are lower than that from the 0.4 s window. Comparing Figs. 7–10, the window width of 0.4 s was best chosen in TFA.
3.2.5. The procedure and results of time–frequency analysis Followings were the procedure of TFA using STFT:
Fig. 9. Relationship DA-CNVF from TFA using 0.4 and 0.6 s window widths, respectively, measured after loading level 2.
Fig. 10. Comparison of results from the TFA using 0.4 s window with that from resonant vibration tests – the relationship of DA-CNVF measured after loading level 2.
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i. Define the zero time point: the zero time point was defined as the point at which the impulsive force from the hammer is maximum. ii. Add zeros: according to the window width, zeros were added to the data at both side of the positions prior the first data point and after the last data point. The length of the added zeros was half the window width. iii. Define the sampling rate: in order to collect the vibration signals accurately from the impulsive excitation tests, the sampling rate was set as 15 kHz. In order to increase the calculation efficiencies, every the fiftieth signal point was taken from the collected signal series to form the source data used for TFA. Thus, the sampling rate of signals used for TFA was changed to 15,000/50 = 300 Hz. iv. Perform the TFA: the STFT was carried out using the TFA tools in MATLAB and the main parameters chosen were: the frequency number F = 4096, the window width for the rectangular window = 0.4 s. v. Before loading and after each vertically statics loading level, there were six impulsive vibration tests performed and then six curves of frequency vs. time were obtained by the TFA. The average of the six curves was used at undamaged level and each damaged level as shown in Fig. 11. UL, L2, L4, L5, L7, L9 and L10 in Fig. 11 are the curves of frequency vs. time prior the loading, and after load level 2, 4, 5, 7, 9 and 10, respectively. It can be seen from these curves that the natural frequency decreases as damage increases and slightly increases as time elapses. The reason of natural frequency slightly increasing with time is due to amplitude decaying in free vibration. vi. The duration of the displacement signal that has been windowed is from 0.25 s to 3.0 s as showed in Fig. 12. The positive peaks of the decay curves of displacement response at the mid-span of the testing beam were picked up as shown in Fig. 13. In Fig. 12 the peaks of the mid-span displacement decaying curve were indicated after load level 2. The selected peaks are fitted using polynomial function. The time–displacement amplitude (TDA) curve can be obtained. The mean of the six selected fitting results of the polynomial function is shown in Fig. 13. In Fig. 13, UL, L2, L4, L5, L7, L9 and L10 are the curves of displacement amplitude vs. time prior the loading, and after load level 2, 4, 5, 7, 9 and 10, respectively. vii. According to the relationship between time and the frequencies from the step v and the variation of displacement amplitudes with time from the step vi, the relationship of DA-
Fig. 11. Variation of structural frequencies with time at various levels of damage.
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Fig. 12. Peaks of the displacement after load level 2.
Fig. 13. Variation of structural displacement amplitude with time at various levels of damage.
CNVF could be obtained. As shown in Fig. 14, the curves of DA-CNVF are given at various levels of damage. UL, L2, L4, L5, L7, L9 and L10 are the curves of DA-CNVF observed in states of unloading and after load level 2, 4, 5, 7, 9 and 10, which represent intact and different damage level, respectively. It is noted that the group of curves of the frequencies vs. time have the same inceptive time point, namely 0.25 s, as shown in Fig. 11. Although there was the same starting time point in Fig. 11, the displacement amplitudes were different in the state of unloading and these measured after different loading levels at this time point (which also can be seen in Figs. 13 and 14). In fact the ends of several curves in Fig. 14 have different displacement amplitudes as well (the starting points in Fig. 11 are corresponding to the end points in Fig. 14 at the same loading level). There were two main reasons for the differences. The first reason is that the maximums of the displacement response decay curves are different in the states of different damaging loading level, which could be observed clearly from the results of the impulsive excitation tests. The second reason is that the damping ratios were also different corresponding to different loading levels; and the variation of displacement amplitudes within time intervals of 0.25 s were also different under the circumstances of different loading levels as shown in Fig. 13. 3.2.6. Discussions of the results from time–frequency analysis As shown in Fig. 14, the relationship of DA-CNVF is nonlinear for the beam from the states of intact to different damaging loading levels. It can be found that the descending degree of the curves increases gradually from unloading intact state up to the curves
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Fig. 14. Variation of structural frequencies with displacement amplitudes at various levels of damage.
corresponding to the damaging loading level 4. However, the descending gradient of the curves measured after damaging loading level 4 to after level 10 decreases gradually. It can be concluded that the structural damage levels increase with increasing vertical load level, and the nonlinear dynamic strength (indicated by the curves’ descending degree) increases initially and then gradually diminishing. In order to gain insight into the correlation between the structural NDC and its damage levels, the curves in Fig. 14 were processed this way: the maximum frequency of each curve was subtracted from the same curve, and the results are shown in Figs. 15 and 16.
Fig. 15. Comparison of nonlinear amplitude–frequency relationships, measured after UL to L4 damage.
As shown in Fig. 15, the curve measured after loading level 4 is the lowest compared with the other three curves and the gradient of the curve is the greatest. The gradients of the curves decrease gradually from loading levels 4, 3, and 2 to prior loading. As shown in Fig. 16, the gradient decreases gradually from loading levels 4, 6, 7, 9 to level 10. From Figs. 15 and 16, it can be concluded that the damage degree increases with increasing vertically loading levels or damages; and the structural NDC vary with increasing damage levels as well. At the beginning, the nonlinearity of the tested beam increases with increasing damage levels. As the damage level reaches to a certain level, the nonlinear dynamic strength decreases with further increased damage levels. That means there is a moderate damage level, namely L4, the relationship between differential frequency and displacement amplitude has greatest descending gradient and then gradually reduces as damage further increases. 4. The application of structural NDC in damage detection From the analysis results of the impulsive vibration tests, it could be observed that the measured change of NDC of the testing beam was the most sensitive after loading level 4. In fact up to loading level 3, there were only very thin cracks on the surface of the RC beam and indicated that the structural damage was not significant. When static vertical load reached the maximum of loading level 4, the strain within the range from 65 mm left of the mid-span to 65 mm right of the mid-span and 20 mm above the bottom of the beam was 551 le. After unloading at this stage, the measured residual strain became 117 le (indicated by test results). Using 28 day cubic strength, the concrete cracking strain was calculated as about 213 le. Obviously this crack level was close to that measured just after unloading of load level 4, which was only 40% of the failure load. In fact, in this particular test, the nonlinear strength continuously increasing up to approximately 40% of failure load. With further increases of damage beyond the approximate 40% of failure load, the increment of nonlinearity of the testing beam decreased. Thus, it is impossible to use the variation of nonlinear dynamic characteristics of the RC beam to detect its damage levels when the structural performance near failure. However, for most of the structures that require a very high level of structural integrity and its damage levels are normally not significant, and within such a low damage level, the change of structural nonlinearity monotonously increases with increasing damage levels. Therefore, the structural nonlinear dynamic characteristics can be used to detect its slight damage levels, especially to moderate ones. 5. Conclusions In this paper, the nonlinear dynamic characteristics of the reinforced concrete beam at various levels of damage were studied based on Neild’s previous work [10]. The followings can be concluded from the experimental-analytical study:
Fig. 16. Comparison of nonlinear amplitude–frequency relationships, which were measured after damaging levels from L4 to Ll0.
(1) The curves of the DA-CNVF can be obtained either from harmonic resonant tests or the time–frequency analysis for the test under impulsive excitation. Combining the analysis results from these two tests, the window width for the time–frequency analysis can be optimized. It is observed that the window width of seven times of the fundamental period of the testing structure is a suitable option in the time–frequency analysis, because in this case the results from both harmonic resonant tests and the time–frequency analysis are consistent.
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(2) Using the window width of seven times of the fundamental period of the testing structure, the curves of displacement amplitudes vs. corresponding vibration frequencies can be obtained by the time–frequency analysis. Through analyzing groups of these curves, the different changing patterns of nonlinear dynamic characteristics of the tested RC beam are able to be observed for various levels of damage. These different changing patterns can generate a foundation for damage detection. (3) The NDC of the reinforced concrete beam can be used to detect its damage levels if the apparent damage is not significant, because the NDC increases monotonically with increasing of damaging levels when damage levels are low. Therefore this damage detection method may be suitable for structures with small damage invisible or hard to be found by naked eyes. But for detecting large damage, caution should be paid when use these relations between differential frequency and displacement amplitude, because the structural NDC may not increase or decrease monotonically with increasing levels of damage within the range of large damage levels.
Acknowledgements The authors wish to express their sincere thanks to Professor Haohua Huang of Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, China for providing his many valuable suggestions and technical supports. The research work reported in this article was supported by Project
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‘‘973’’ of Department of Science and Technology of China under grant of 2007CB714200.This support is greatly appreciated. References [1] Geng Hao, Wang Rui, Gao Fangqing. Study on damage test of the steel truss girder model bridge. Bridge Build 1998;106(2):67–70. [2] Gao Fangqing, Jin Jianming, Gao Shuying. Damage detection in structures by modal analysis. J Southwest Jiao Tong Univ 1998;33(1):108–13. [3] Dong Cong, Ding Hui, Gao Song. The basic principle and method for recognition and location of structural damage. China Rail Sci 1999;20(3):89–94. [4] Shi ZY, Slaw S. Structural damage localization from modal strain energy change. J Sound Vib 1998;218(5):825–44. [5] Lin X, Yuan FG. Detection of multiple damages by prestack reverse-time migration. AIAA J 2001;39(11):2206–15. [6] Pugno Nicola, Surace Cecilia. Evaluation of the nonlinear dynamic response to harmonic excitation of a beam with several breathing cracks. J Sound Vib 2000;235(5):749–62. [7] Tsyfansky SL, Beresnevich VI. Nonlinear vibration method for detection of fatigue cracks in aircraft wings. J Sound Vib 2000;236(1):49–60. [8] Choi YS, Noah ST. Forced periodic vibration of unsymmetric piecevise-linear systems. J Sound Vib 1988;121(1):117–26. [9] Léonard François, Lanteigne Jacques, Lalonde Serge, Turcotte Yvon. Free vibration behaviour of a cracked cantilever beam and crack detection. Mech Syst Signal Process 2001;15(3):529–48. [10] Neild SA, William MS, McFadden PD. Nonlinear vibration characteristics of damaged concrete beams. J struct Eng 2003;129(2):260–8. [11] Loutridis SJ. Damage detection in gear system using empirical mode decomposition. Eng Struct 2004;26(12):1833–41. [12] Lanza di Scalea F, Matt H, Bartoli I, Coccia S, Park G, Farrar C. Health monitoring of UAV wing skin-to-spar joints using guided waves and macro fiber composite transducers. J Intell Mater Syst Struct 2007;18(4):373–88. [13] Lanza di Scalea F, Salamone S. Temperature effects in ultrasonic lamb wave structural health monitoring systems. J Acoust Soc Am 2008;124(1):161–74. [14] Sabra K, Srivastava A, Lanza di Scalea F, Bartoli I, Rizzo P, Conti S. Structural health monitoring by extraction of coherent guided waves from diffuse fields. J Acoust Soc Am 2007;123(1):EL8–13.
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Damage detection of RC beams based on experiment and analysis of nonlinear dynamic characteristics Liheng Wang a,b,⇑, Xiyuan Zhou b, He Liu c,1, Weiming Yan b a
National Key Laboratory of Dynamic Measurement and Calibration, Changcheng Institute of Metrology & Measurement, AVIC, P.O. Box 1066, Beijing 100095, PR China Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, No. 100, Pingleyuan, Beijing 100022, PR China c University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA b
a r t i c l e
i n f o
Article history: Received 2 April 2011 Received in revised form 14 October 2011 Accepted 20 October 2011 Available online 29 November 2011 Keywords: Damage detection Time–frequency analysis Nonlinear dynamic characteristics
a b s t r a c t The experimental and analytical investigation on damage detection of reinforced concrete (RC) beams has been carried out using a testing RC beam. The correlations between the damage extent and the nonlinear dynamic characteristics (NDC) of the testing beam are presented in this paper. The NDC of the RC beam were obtained by means of time–frequency analysis (TFA) of displacement responses from the impulsive and harmonic vibration tests. The correlation between the displacement amplitude and the excitation frequency in the case of forced vibration was obtained. Combining the results from the TFA with the relationship of displacement amplitude and corresponding natural vibration frequency (DA-CNVF) from the harmonic vibration tests, a reasonable window width for TFA was suggested. It is observed that the window width of seven times of the fundamental period of the testing structure is a suitable option in the time–frequency analysis. Using the TFA with this appropriate moving window width, the relationship between DA-CNVF at different damage levels in the case of free vibration was used to depict and analyze the NDC of the damaged RC beam. The feasibility of using NDC of a damaged RC beam from either forced harmonic or free vibration tests to detect damage level is discussed. The NDC of the reinforced concrete beam can be used to detect its damage levels if the apparent damage is not significant. Therefore, this damage detection method may be suitable for structures with small damage invisible or hard to be found by naked eyes. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Damage detection technology for structures can be classified into two categories: linear elastic method and nonlinear dynamic approach. The damage detection technology based on linear elastic theory can be sub-categorized as the modal method and the wavemotion method. In the modal method, using general purpose Finite Element software it is convenient to calculate natural frequencies and mode shapes of a structure [1,2]. Therefore, structural natural frequencies and mode shapes and consequently derived results, such as modal curvature [3], modal strain energy [4] and so on, are used as structural damage indexes in many literatures. On the other hand, the wave motion theories are gradually growing in recent years in damage detection because of its suitability in planar structure with multiple damages [5,12–14]. However, most of research ⇑ Corresponding author at: National Key Laboratory of Dynamic Measurement and Calibration, Changcheng Institute of Metrology & Measurement, AVIC, P.O. Box 1066, Beijing 100095, PR China. E-mail addresses: [email protected] (L. Wang), [email protected] (X. Zhou), [email protected] (H. Liu), [email protected] (W. Yan). 1 Tel.: +1 907 786 1971; fax: +1 907 786 1079. 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.10.065
methods in damage detection are limited to or based on theory of linear dynamics. Only a few literatures relate to the nonlinear theory, though RC structures behave nonlinearly in nature [10]. In fact there are several types of stiffness functions available to depict nonlinearity, such as the bilinear model of restoring force vs. displacement simulating fully opening and fully closing of cracked concrete [6–8] or crack’s variation from fully opening to partially opening and partially closing to fully closing during the structural vibration [9]. Because damage detection is mainly based on the identification of structural stiffness varying with time, and the stiffness is correlated to the change of structural natural vibration frequencies, identifying stiffness variation with time is equivalent to identify frequency variation with time using time–frequency analysis (TFA) [10] or empirical mode decomposition analysis [11]. Nevertheless, it was found that there were few documents on nonlinear dynamic characteristics of reinforced concrete (RC) beams using the TFA [10]. In this experimental–analytical study, first, the relationship between displacement amplitudes and its frequencies was obtained from different levels of harmonic excitation tests. And then, the window width for the TFA was adjusted to ensure that the results from TFA are in some degree consistent with the results from harmonic excitation tests. The reasonable window width in this
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study was chosen as seven times of structural fundamental period. Using this window width in TFA, the results of displacement response from impulsive excitation tests could be obtained by short time Fourier transformation (STFT); and the curves of amplitude and frequency could be also obtained prior and after structural damage at each loading level. Utilizing these curves, the correlation between nonlinear dynamic characteristics of a RC beam and its damage level can be detected.
Table 1 Reinforcing steel of the RC beam.
Reinforcing steel Diameter, d (mm) Effective depth, h (mm) Yield strength, fy (MPa) Number of bars, N
Bottom
Top
Shear
12 154 436.45 3
8
6.5
279.5 2
210 44
2. Experimental study 2.1. Beam design and material properties The testing RC beam was designed based on the consideration of the fundamental period of the beam consisting with that of general concrete bridges. The RC beam with dimensions 4350 mm in length, 200 mm in width and 180 mm in depth was simply supported within 4150 mm span as shown in Fig. 1. The fundamental frequency of the beam is 18.75 Hz, the ultimate moment capacity is 21.44 kN-m, the designed mid-span failure load (F) is 21.07 kN, and the maximum shear capacity is 63.95 kN. The 28-day strength of the concrete is 55.1 MPa. The design properties of reinforcing steel of the RC beam are shown in Table 1. The design properties of concrete of the RC beam are shown in Table 2.
2.2. Test set up The stiffness of the supporting structure of the testing beam must be large enough to reduce the measurement error. The supporting structures used in this experimental project are two RC piers with 400 mm in width, 400 mm in depth and 800 mm in length sitting on the laboratory RC basement shear wall with 600 mm depth . Because the interface between the bottom of concrete beam and the top of concrete piers was not smooth, the 1:3 cement mortars were plastered. The 25 mm depth armor plates were bonded to the top of the RC pier with 1:3 cement mortars. The armor plates and the round steels with 50 mm diameter and 200 mm length were welded beforehand as show in Fig. 1. From the results of calculation and measurement, it was indicated that the stiffness of such supporting equipment is large enough to satisfy the requirements of simply supported foundations. It should be noted that only a good contact condition between the bottom of the beam and the top of the round steel can ensure that the supports satisfy the simply supported requirements during the vibrating tests. Two accelerometers at the support of the testing beam were used to monitor acceleration responses at the supports. A 400 Hz low pass filter was used in the data acquisition system for accelerometer channels at the supports. The response acceleration at the mid-span was registered as shown in Fig. 6 when 400 Hz low pass filter was used. In order to ensure the measurement accuracy of acceleration and displacement, an accelerometer and two displacement meters were allocated at mid-span of the testing beam, respectively as show in Figs. 1 and 2.
2.3. Testing procedure The experiments were conducted at Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology. During the testing period the temperature was varied negligible.
Table 2 The concrete properties of the RC beam. Cement Sand Gravel Water fcu
360 kg/m3 736 kg/m3 1104 kg/m3 165 kg/m3 55.1 MPa
fcu is the 28 day cubic strength of the concrete.
The testing RC beam was simply supported. The vertical static loads up to ten levels were applied on the beam prior loading to failure by increasing static loads as shown in Table 3. It was planed that the concrete crack would occur (the crack strain of concrete was 213 le) when level 3 was loaded to its maximum. The residual strain measured after unloading level 4 was 117 le. After the tensile reinforcement yielded, the testing beam reached its ultimate capability when level 10 was loaded to its maximum. The RC beam was tested in its undamaged state and then up to failure in the subsequent loading levels. The tests were split into the following several steps: (1) Step 1 – static loading tests. There were totally ten load levels planned as shown in Table 3. It was expected that the beam reached its failure load at load level 10. The force and the strain were measured during loading listed in Table 3. (2) Step 2 – impulsive vibration tests. After each level of static loading, a hammer shown in Fig. 2 served as an exciting source and three accelerometers and two displacement gauges collected the beam’s response. Using the TFA, the decayed displacement response signals at the mid-span were recorded and analyzed to obtain a group of curves of mid-span displacement amplitude and corresponding natural vibration frequency (DA-CNVF) of the testing beam. (3) Step 3 – harmonic excitation tests. After each impulsive vibration test, an exciter shown in Fig. 2 served as an exciting source, also three accelerometers and two displacement gauges collected the beam’s response. The amplitude of the harmonic force applied at the mid-span point of the beam kept constant and the frequencies of the harmonic force varied within a certain range. The vibration signals of the beam were collected by the accelerometers placed on the beam. Using harmonic tests results the mid-span displacement amplitude and corresponding natural vibration frequency can also be obtained. (4) Step 4 – repeating Step 1 and moving to a new loading level.
Fig. 1. Dimensions, loading and measurement locations of the testing beam.
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Fig. 2. Set up of impulsive and harmonic excitation tests.
Table 3 Loading levels, maximum loads and maximum tensile strains in different loading levels. Load level
Maximum vertical static load (N)
Max. tensile strain at mid-span (le)
1 2 3 4 5 6 7 8 9 10
2107 4214 6321 8428 10,535 12,642 14,749 16,856 18,963 21,070
65 118 264 551 1494 1819 2300 2832 8853
(a) L7
(b) L8
(c) L10 Fig. 3. Observed crack sizes and positions at various loading levels.
Comparing the results of step 3 and step 4 a reasonable window width can be selected. Then, combining the selected reasonable window width the nonlinear dynamic characteristics of the RC beam were drawn using the impact displacement responses. The nonlinear characteristics, DA-CNVF curves are used to damage detection of the RC beam. 2.4. Experimental method 2.4.1. Static loading The static four-point bending tests were performed as shown in Fig. 1. The corresponding load levels are listed in Table 3. When the testing beam was loaded to the designed loading level, the cracks were observed carefully and the cracking traces were drawn in pencil on the surface of the beam. After reaching the designed loading level, the beam was unloaded. The cracking developments are shown in Fig. 3 for several load levels. In Fig. 3, L7, L8 and L10 show the crack distributions when load level 7, 8 and 10 reached their maximums, respectively. The large scale strain gauge, shown in Fig. 1, at mid-span 20 mm above the bottom of the beam was used to measure the strains within the range of 130 mm even after concrete cracking. The measured results are listed in Table 3. 2.4.2. Impulsive excitation tests At each loading level, once the beam had been unloaded, an impulsive excitation vibration test was performed. The hammer used in the test was Lc-04 impulsive hammer. After adding additional mass to the hammer, the total mass used for the impulsive excitation was 4.0 kg. The excitation force was applied at 160 mm away from mid-span with a rubber impacting tip dropped from a height of approximately 232 mm above the testing beam as shown in Fig. 2. The acceleration signal was acquired at the half-span point and two support-points. The displacement data were measured at the half-span point as shown in Fig. 2. Acceleration, displacement and hammer impulsive force were digitized at a sample rate frequency of 15 kHz to ensure that the data were captured accurately. In order to compare the results of impulsive vibration tests, maximum upward acceleration at the mid-span was registered each time after conducting the impulsive vibration test. The same tests must be repeated several times until there were results from at least six tests with relatively less than 5% errors of maximum up-
ward acceleration measured at the mid-span. Within 10 times of tests, there were, at least, 6 times to satisfy the requirements. Results from these six impulsive excitation tests were used for the subsequent TFA. 2.4.3. Harmonic excitation tests After impulsive excitation tests, the harmonic vibration tests began as set up shown in Fig. 2. Under the excitation of the harmonic force, the beam behaved as forced vibration. After the beam vibration reached at steady state, acceleration data were collected by the data acquisition system.
3. Nonlinear dynamic analysis of testing data 3.1. Analysis of testing data from harmonic excitation tests By processing the data from harmonic excitation tests a curve of acceleration amplitudes at the mid-span point vs. the input frequencies was obtained. Each curve shown in Fig. 4 represents a certain level of harmonic excitation and the seven curves in total correspond to different harmonic forces. It can be seen from Fig. 4 that with the increasing the amplitude of harmonic force the corresponding resonance vibration frequencies are slightly changed and the resonance peaks gradually shift leftward. From each of the seven curves, a peak can be detected, and seven peaks are marked by dots in Fig. 4. Furthermore, another data pair was also obtained from the test for measuring fundamental frequency under the same harmonic excitation. These eight data pairs indicate the correlation of acceleration amplitudes and corresponding resonant frequencies of the testing beam. According to the relationship between acceleration and displacement of harmonic
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vibration, the relationship of DA-CNVF can be obtained from the data in Fig. 4, as shown in Fig. 5. It is observed from the test after the loading level 2 that the frequencies decrease as the increasing displacement amplitudes. The fitted curve in Fig. 5 can be expressed as Eq. (1).
f ¼ aebx þ cedx
ð1Þ
where f is the natural frequency of the testing beam; x is the displacement amplitude at the mid-span. The coefficients in Eq. (1) are
a ¼ 0:4066;
b ¼ 4:125;
c ¼ 17:27;
d ¼ 0:03306
3.2. Analysis of testing data from impulsive vibration tests 3.2.1. Time–frequency analysis (TFA) The TFA aims mainly to describe how the frequency spectrum varies with time and then to explain the concepts of variation of frequency spectrum with time. For this purpose it is convenient to perform the TFA using STFT, which is defined as follows:
STFTðt; f Þ ¼
Z
þ1
SðsÞhðs tÞej2pf s ds
Fig. 5. Variation of fundamental natural vibration frequencies with displacement amplitudes, measured after loading level 2.
ð2Þ
1
The signal S(s) in Eq. (2) multiplied by the time window h(s t) can efficiently restrain the signals outside the neighborhood of s = t. Therefore, the results from STFT based on Eq. (2) is just local spectra of S(s) in vicinity of t. When the window h(s t) moves along the t axis, the corresponding time-variable spectra can be obtained using Eq. (2). The total frequency spectra are just the time– frequency distributions. In this paper, the TFA of the displacement response signals at the mid-span was conducted using Eq. (2). 3.2.2. Starting time point for TFA After being impacted, the testing beam behaved with decaying vibration characteristics. Fig. 6 shows the acceleration response at the mid-span below 400 Hz after loading level two. Observing Fig. 6, it can be seen that, at the beginning, the acceleration signal contains the first several natural frequency components. However, after a short period of time (about 0.25 s), the higher frequency components decays away and only the first fundamental frequency component remained. As the displacement amplitude reduces, the first frequency varies. Using the TFA, the decayed signals can be analyzed to obtain the variation of frequencies with time and the variation of frequencies with displacement amplitudes. It can be seen clearly from Fig. 6 that the higher frequency components adding on the lower frequencies components die away quickly. In order to compare the maximum values of acceleration response from six impulsive vibration tests, a 30 Hz low pass filter of the data acquisition system was used for the accelerometer
Fig. 4. The resonance curves measured after loading level 2 for various excitation levels.
Fig. 6. Acceleration response at the mid-span below 400 Hz, measured after loading level 2.
channel at the mid-span to allow only the first frequency component was collected. In this way, it was convenient to find the maximum value of the upward acceleration from the filtered acceleration signals. Comparing the maximum upward accelerations from several impulsive vibration tests, at least six testing results with relatively less than 5% errors of the maximum upward accelerations were chosen for the TFA. Since the first frequency was below 20 Hz and the purpose of TFA was to obtain the variation of fundamental frequency with displacement amplitudes, a 30 Hz low pass filter did not affect the needed results of the TFA. Due to the non-perfect condition at the simple supports, the high frequency shock was induced at the two ends of the testing beam. At the beginning, the acceleration responses at two supports were relatively large, but died away quickly. At the time point of 0.25 s, the acceleration amplitudes at two supports were much less than that at the mid-span. Thus, from this time point on, the beam can be regarded as simply supported beam. Therefore the 0.25 s was chosen as the initial time point for TFA. 3.2.3. Selecting the input signals of the TFA In Neild’s experimental study [10], the variation of frequencies with time was obtained using TFA via the acceleration signals at the mid-span of testing structure. The variation of displacement amplitudes was obtained by computing the acceleration amplitudes and corresponding frequencies, both of which vary with time. Because the error could be introduced into the frequencies in the TFA, when the displacement amplitudes were obtained from the acceleration amplitudes and corresponding frequencies, the errors of the displacement amplitudes might increase. In order to
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decrease such errors, the displacement response signals acquired from the displacement gauge at the mid-span were used. In order to avoid introducing the torsional response signals into the analysis data, two displacement sensors were used on the both sides of the beam at the middle span. The average of the signals from these two displacement gauges was used as the input data for the TFA. 3.2.4. Selecting the Window Width for TFA The relationship DA-CNVF obtained from either the TFA or the harmonic excitation vibration tests are essentially the same, because they both indicate the same structural NDC. The harmonic excitation tests are more time consuming, however, the results are more accurate. By contrast, the TFA is more convenient and quicker. Since the results of TFA are affected by the window width, how to select a suitable window width becomes a critical problem. Currently there are many approaches used in the TFA. For example the STFT is one of the commonly used methods for the TFA. Since the results from these two means are essentially the same in describing the structural NDC, combining the two means together, a reasonable window width for STFT could be identified. The detailed procedures are follows: i. Select a window width for TFA. ii. Obtain the relationship of structural DA-CNVF by the TFA. iii. Compare the results from the TFA and the harmonic excitation tests, the window width for the TFA is adjusted. iv. Repeat steps ii and iii until the results from TFA and the harmonic excitation tests are consistent. The following window widths were selected to compare with the harmonic vibration tests: 0.2, 0.3, 0.4 and 0.6 s. As shown in Fig. 7, using the window width of 0.2 s, the results from the TFA is not good for accurately estimating the natural frequencies; and the width of 0.3 s is a better choice. However, when the displacement amplitude was larger than 0.25 mm, the errors of fundamental natural frequency still existed in the estimation when using TFA with the 0.3 s window. As shown in Fig. 8, the estimation of structural frequency using the window width of 0.4 s is better than 0.3 s. As shown in Fig. 9, the results from the window width of 0.4 s are very close to the results from the 0.6 s; nevertheless the frequencies from 0.6 s window are more stable and below that from the 0.4 s window. Fig. 10 shows the comparison of results from the TFA using 0.4 s window with that from resonant vibration tests after loading level 2. The label of ‘harmonic excitation’ in Fig. 10 represents the relationship of DA-CNVF from harmonic excitation tests; meanwhile, the ‘curve fitting’ gives the result from fitting of the eight data points observed from harmonic excitation tests. The ‘window width
Fig. 7. Relationship of DA-CNVF from TFA using 0.2 and 0.3 s window width, respectively, measured after loading level 2.
Fig. 8. Relationship of DA-CNVF from TFA using 0.3 and 0.4 s window width, respectively, measured after loading level 2.
(0.4 s) time–frequency’ in Fig. 10 represents the results from the 0.4 s window width (equal to seven times of the fundamental period of the testing beam). It can be observed that the results from the 0.4 s window width are consistent with that from the harmonic excitation tests (the relative errors are less than 0.6% as shown in Fig. 10). The estimated frequencies from the 0.6 s window width are lower than that from the 0.4 s window. Comparing Figs. 7–10, the window width of 0.4 s was best chosen in TFA.
3.2.5. The procedure and results of time–frequency analysis Followings were the procedure of TFA using STFT:
Fig. 9. Relationship DA-CNVF from TFA using 0.4 and 0.6 s window widths, respectively, measured after loading level 2.
Fig. 10. Comparison of results from the TFA using 0.4 s window with that from resonant vibration tests – the relationship of DA-CNVF measured after loading level 2.
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i. Define the zero time point: the zero time point was defined as the point at which the impulsive force from the hammer is maximum. ii. Add zeros: according to the window width, zeros were added to the data at both side of the positions prior the first data point and after the last data point. The length of the added zeros was half the window width. iii. Define the sampling rate: in order to collect the vibration signals accurately from the impulsive excitation tests, the sampling rate was set as 15 kHz. In order to increase the calculation efficiencies, every the fiftieth signal point was taken from the collected signal series to form the source data used for TFA. Thus, the sampling rate of signals used for TFA was changed to 15,000/50 = 300 Hz. iv. Perform the TFA: the STFT was carried out using the TFA tools in MATLAB and the main parameters chosen were: the frequency number F = 4096, the window width for the rectangular window = 0.4 s. v. Before loading and after each vertically statics loading level, there were six impulsive vibration tests performed and then six curves of frequency vs. time were obtained by the TFA. The average of the six curves was used at undamaged level and each damaged level as shown in Fig. 11. UL, L2, L4, L5, L7, L9 and L10 in Fig. 11 are the curves of frequency vs. time prior the loading, and after load level 2, 4, 5, 7, 9 and 10, respectively. It can be seen from these curves that the natural frequency decreases as damage increases and slightly increases as time elapses. The reason of natural frequency slightly increasing with time is due to amplitude decaying in free vibration. vi. The duration of the displacement signal that has been windowed is from 0.25 s to 3.0 s as showed in Fig. 12. The positive peaks of the decay curves of displacement response at the mid-span of the testing beam were picked up as shown in Fig. 13. In Fig. 12 the peaks of the mid-span displacement decaying curve were indicated after load level 2. The selected peaks are fitted using polynomial function. The time–displacement amplitude (TDA) curve can be obtained. The mean of the six selected fitting results of the polynomial function is shown in Fig. 13. In Fig. 13, UL, L2, L4, L5, L7, L9 and L10 are the curves of displacement amplitude vs. time prior the loading, and after load level 2, 4, 5, 7, 9 and 10, respectively. vii. According to the relationship between time and the frequencies from the step v and the variation of displacement amplitudes with time from the step vi, the relationship of DA-
Fig. 11. Variation of structural frequencies with time at various levels of damage.
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Fig. 12. Peaks of the displacement after load level 2.
Fig. 13. Variation of structural displacement amplitude with time at various levels of damage.
CNVF could be obtained. As shown in Fig. 14, the curves of DA-CNVF are given at various levels of damage. UL, L2, L4, L5, L7, L9 and L10 are the curves of DA-CNVF observed in states of unloading and after load level 2, 4, 5, 7, 9 and 10, which represent intact and different damage level, respectively. It is noted that the group of curves of the frequencies vs. time have the same inceptive time point, namely 0.25 s, as shown in Fig. 11. Although there was the same starting time point in Fig. 11, the displacement amplitudes were different in the state of unloading and these measured after different loading levels at this time point (which also can be seen in Figs. 13 and 14). In fact the ends of several curves in Fig. 14 have different displacement amplitudes as well (the starting points in Fig. 11 are corresponding to the end points in Fig. 14 at the same loading level). There were two main reasons for the differences. The first reason is that the maximums of the displacement response decay curves are different in the states of different damaging loading level, which could be observed clearly from the results of the impulsive excitation tests. The second reason is that the damping ratios were also different corresponding to different loading levels; and the variation of displacement amplitudes within time intervals of 0.25 s were also different under the circumstances of different loading levels as shown in Fig. 13. 3.2.6. Discussions of the results from time–frequency analysis As shown in Fig. 14, the relationship of DA-CNVF is nonlinear for the beam from the states of intact to different damaging loading levels. It can be found that the descending degree of the curves increases gradually from unloading intact state up to the curves
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Fig. 14. Variation of structural frequencies with displacement amplitudes at various levels of damage.
corresponding to the damaging loading level 4. However, the descending gradient of the curves measured after damaging loading level 4 to after level 10 decreases gradually. It can be concluded that the structural damage levels increase with increasing vertical load level, and the nonlinear dynamic strength (indicated by the curves’ descending degree) increases initially and then gradually diminishing. In order to gain insight into the correlation between the structural NDC and its damage levels, the curves in Fig. 14 were processed this way: the maximum frequency of each curve was subtracted from the same curve, and the results are shown in Figs. 15 and 16.
Fig. 15. Comparison of nonlinear amplitude–frequency relationships, measured after UL to L4 damage.
As shown in Fig. 15, the curve measured after loading level 4 is the lowest compared with the other three curves and the gradient of the curve is the greatest. The gradients of the curves decrease gradually from loading levels 4, 3, and 2 to prior loading. As shown in Fig. 16, the gradient decreases gradually from loading levels 4, 6, 7, 9 to level 10. From Figs. 15 and 16, it can be concluded that the damage degree increases with increasing vertically loading levels or damages; and the structural NDC vary with increasing damage levels as well. At the beginning, the nonlinearity of the tested beam increases with increasing damage levels. As the damage level reaches to a certain level, the nonlinear dynamic strength decreases with further increased damage levels. That means there is a moderate damage level, namely L4, the relationship between differential frequency and displacement amplitude has greatest descending gradient and then gradually reduces as damage further increases. 4. The application of structural NDC in damage detection From the analysis results of the impulsive vibration tests, it could be observed that the measured change of NDC of the testing beam was the most sensitive after loading level 4. In fact up to loading level 3, there were only very thin cracks on the surface of the RC beam and indicated that the structural damage was not significant. When static vertical load reached the maximum of loading level 4, the strain within the range from 65 mm left of the mid-span to 65 mm right of the mid-span and 20 mm above the bottom of the beam was 551 le. After unloading at this stage, the measured residual strain became 117 le (indicated by test results). Using 28 day cubic strength, the concrete cracking strain was calculated as about 213 le. Obviously this crack level was close to that measured just after unloading of load level 4, which was only 40% of the failure load. In fact, in this particular test, the nonlinear strength continuously increasing up to approximately 40% of failure load. With further increases of damage beyond the approximate 40% of failure load, the increment of nonlinearity of the testing beam decreased. Thus, it is impossible to use the variation of nonlinear dynamic characteristics of the RC beam to detect its damage levels when the structural performance near failure. However, for most of the structures that require a very high level of structural integrity and its damage levels are normally not significant, and within such a low damage level, the change of structural nonlinearity monotonously increases with increasing damage levels. Therefore, the structural nonlinear dynamic characteristics can be used to detect its slight damage levels, especially to moderate ones. 5. Conclusions In this paper, the nonlinear dynamic characteristics of the reinforced concrete beam at various levels of damage were studied based on Neild’s previous work [10]. The followings can be concluded from the experimental-analytical study:
Fig. 16. Comparison of nonlinear amplitude–frequency relationships, which were measured after damaging levels from L4 to Ll0.
(1) The curves of the DA-CNVF can be obtained either from harmonic resonant tests or the time–frequency analysis for the test under impulsive excitation. Combining the analysis results from these two tests, the window width for the time–frequency analysis can be optimized. It is observed that the window width of seven times of the fundamental period of the testing structure is a suitable option in the time–frequency analysis, because in this case the results from both harmonic resonant tests and the time–frequency analysis are consistent.
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(2) Using the window width of seven times of the fundamental period of the testing structure, the curves of displacement amplitudes vs. corresponding vibration frequencies can be obtained by the time–frequency analysis. Through analyzing groups of these curves, the different changing patterns of nonlinear dynamic characteristics of the tested RC beam are able to be observed for various levels of damage. These different changing patterns can generate a foundation for damage detection. (3) The NDC of the reinforced concrete beam can be used to detect its damage levels if the apparent damage is not significant, because the NDC increases monotonically with increasing of damaging levels when damage levels are low. Therefore this damage detection method may be suitable for structures with small damage invisible or hard to be found by naked eyes. But for detecting large damage, caution should be paid when use these relations between differential frequency and displacement amplitude, because the structural NDC may not increase or decrease monotonically with increasing levels of damage within the range of large damage levels.
Acknowledgements The authors wish to express their sincere thanks to Professor Haohua Huang of Beijing Key Laboratory of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, China for providing his many valuable suggestions and technical supports. The research work reported in this article was supported by Project
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