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Time–frequency analyses of pile-integrity testing using wavelet transform
Available online at www.sciencedirect.com
Computers and Geotechnics 35 (2008) 600–607 www.elsevier.com/locate/compgeo
Time–frequency analyses of pile-integrity testing using wavelet transform Sheng-Huoo Ni a, Kuo-Feng Lo b
a,*
, Lutz Lehmann b, Yan-Hong Huang
a
a Department of Civil Engineering, National Cheng Kung University, No. 1, University Road, Tainan 70101, Taiwan, ROC Institute of Applied Mechanics, Technical University of Braunschweig, Spielmannstrasse 11, D-38106 Braunschweig, Germany
Received 28 October 2006; received in revised form 6 September 2007; accepted 7 September 2007 Available online 23 October 2007
Abstract Detecting pile-integrity is an important topic for civil engineers and researchers. The pile-integrity testing signals acquired from the receivers on pile heads will tend to be complicated, as defects and undesired background vibrations exist. In this paper the continuous wavelet transform (CWT) method, with time–frequency distribution, is adopted to enhance the characteristics of the testing signal in order to improve the identification ability in both numerical simulations and experimental cases. The results indicate that testing signals can be displayed in the time–frequency domain at the same time and then be explored every single time by CWT, especially for changes in frequency content. In addition, CWT tends to reinforce the traits and makes all the information more visible. The feasibility of combining CWT with the sonic echo (SE) method is demonstrated. This approach is easily and quickly interpreted, and the pile-integrity is proven as well. 2007 Elsevier Ltd. All rights reserved. Keywords: Pile; Integrity test; Continuous wavelet transform; Defect
1. Introduction Non-destructive evaluation (NDE) techniques have been adopted to provide pile-construction quality control for many years. In particular, low-strain, pile-integrity testing methods include both the sonic echo (SE) and impulseresponse methods. Such methods have been used to check the lengths and integrity of newly installed foundations ([12]). These methods can help engineers to judge and identify both the presence and locations of pile defects, as the impact generated by the relatively small device produces a longitudinal wave that then transmits along the pile length, with both low-energy and high-frequency content. The term ‘‘low-strain integrity testing’’ is derived from the fact that the device generates only low-strain stress*
Corresponding author. E-mail addresses: [email protected] (S.-H. Ni), [email protected] (K.-F. Lo), [email protected] (L. Lehmann), [email protected] (Y.-H. Huang). 0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.09.003
waves in the pile. However, in practice, the signals will not always be pure and without noise. A new numerical signal-process tool that can substantially improve and strengthen the signal characteristics substantially is necessary, and that tool is the wavelet transform method (WTM). The wavelet transform application for low-strain integrity testing was first researched at Napier University, Edinburgh ([3,2]). In Addison et al. [1], the authors present finite element-generated responses for an imaginary pile 10 m in length, with and without a 20% impedance increase at midlength. They indicate that the 20% change in impedance is difficult to identify in the time domain, whereas the defect can be observed clearly in the wavelet transform ([14]). The WTM allows the signal to be decomposed so that frequency characteristics, as defined by analyzing the wavelet and the position of particular features in a time series, can be highlighted. The advantage of wavelet-based signalanalysis methods over the traditional Fourier transform (FT) method lies in its capability to produce simultaneous
S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
temporal and scale information ([11]). However, the reflection shape of the wave propagated can only be easily identified if the ratio of the defect area is more than 10% in the laboratory ([6]), which represents a limitation in analyzing the pile-integrity testing results with regard to defects with a smaller area ratio. 2. Theoretical background 2.1. Wavelet transform Fourier Transform (FT) breaks down a signal into its constituent sinusoids with different frequencies. Fourier analysis could also be viewed as a mathematical technique for transforming signals from time based to frequencybased. However, Fourier analysis has a serious drawback: when transforming to the frequency domain time, information disappears ([13]). When inspecting a Fourier-transformed signal, it is almost impossible to tell whether a particular event took place or not. As for characteristics such as abrupt changes, and the beginning and the end of events, Fourier analysis obviously cannot detect them at all. In order to correct this deficiency, Gabor [5] adapted a technique called windowing the signal (or the short time Fourier transform, STFT), which maps a signal into a two-dimensional function of time and frequency. The STFT represents a sort of compromise between the time and frequency-based views of the signal. It provides some information about both when, and at what frequencies, a signal event occurs. However, it can offer this information with limited precision, and this precision is determined only by the size of the window. The drawbacks of STFT are that once we choose a particular size for the time window, it is the same for all frequencies. Many signals require a more flexible approach in which we can vary the size of the window to determine more accurately either time or frequency. Wavelets do this by having a variable window width, which is related to the scale of observation; this flexibility allows for the isolation of the high-frequency features. One major difference is that the STFT has constant time and frequency resolution in contrast to the wavelet transform. Another important distinction between wavelet and Fourier analysis is that wavelet analysis is not limited to using sinusoidal analysis functions, but rather can employ a large selection of localized waveforms as long as they satisfy the predefined mathematical criteria described above. One major advantage afforded by wavelet transform is the capability for performing local analysis – i.e., to analyse a localized section of a long-duration signal. Wavelet analysis allows the choice of long time intervals when more precise low-frequency information is needed, and shorter ones when high-frequency information is desired. It is a windowing technique with variable-sized regions. Wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet so that local features can be described better with wavelets that
601
are concentrated on a localized region. Wavelet transforms differ from STFT, as they allow arbitrarily high localization in time of high-frequency-signal features. Their ability to localize in both time and frequency in a distinctly different way from the traditional STFT has spawned a number of sophisticated wavelet-based methods for signal decomposition, manipulation and interrogation. The WTM has been found particularly useful for analyses of signals that can be described as periodic, noisy, intermittent, and transient. There are many different wavelet transforms available, so it is not a unique function. A wavelet must satisfy three conditions: 1. It must have unit energy. 2. It must provide compact support or sufficiently fast decay so that it satisfies the requirement of space (or time) location. 3. It must have a zero mean – i.e., the integral of the wavelet function from 1 to +1 is zero. This is called the admissibility condition, and ensures that there are both positive and negative components to the motherwavelet. Consider a real or complex-value continuous-time function w(t) with the following property ([4,9]): 1. The function integrates to zero. Z 1 wðtÞdt ¼ 0
ð1Þ
1
2. Its square integral, or equivalent, has finite energy. Z 1 2 jf ðtÞj dt < 1 ð2Þ 1
The function w(t) is a mother-wavelet or a wavelet if it satisfies these two properties as well as the admissibility condition as defined above. Let f(t) be any square integral function. The continuous wavelet transform (CWT) with respect to a wavelet w(t) is defined as Z þ1 1 tb W ða; bÞ ¼ pffiffiffi f ðtÞw dt ð3Þ a a 1 where a and b are real, and gate. We set wa,b(t)w as 1 tb wa;b ðtÞ ¼ pffiffiffiffiffiffi w a jaj
*
denotes the complex conju-
Then, combining Eqs. (3) and (4), yields Z þ1 W ða; bÞ ¼ f ðtÞwa;b ðtÞdt
ð4Þ
ð5Þ
1
The variable b represents the time shift or translation, and the variable a determines the amount of time scaling or dilation. Since the CWT is generated using dilates and translates of the signal function w(t), the wavelet for the transform is referred to as the mother-wavelet. Next, we
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S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
generate a set of expansion functions so that any signal in L2(R) (the space of square integral functions) can be represented by the series X pffiffiffiffi aj;k 2j uð2j t kÞ ð6Þ f ðtÞ ¼
1 0.8 0.6 0.4
j;k 0.2
Besides, as with the Fourier transform, the original signal may be reconstructed Z þ1 Z þ1 1 1 tb dadb pffiffiffi w f ðtÞ ¼ ð7Þ W ða; bÞ 2 C g 1 1 a a a
0 -0.2 -0.4 -0.6
where Cg is the admissibility constant Z x _ 2 j wðxÞj dx Cg ¼ x 0
-0.8 -8
ð8Þ
As the wavelet transform given by Eq. (3) is a convolution of the signal with a wavelet function, we can use the convolution theorem to express the integral as a product in Fourier space, i.e., Z 1 _ _ 1 W ða; bÞ ¼ f ðwÞwa;b ðxÞdx ð9Þ 2p 1 _ pffiffiffi_ ð10Þ where wa;b ðxÞ ¼ aw ðaxÞeiwb This is the Fourier spectrum of the analysed wavelet at scale a and location b. In this way a fast Fourier transform (FFT) algorithm can be employed in practice to compute the wavelet transform. For more details of the wavelet with time–frequency, refer to Watson and Addison [15]. To apply the CWT in any application it is important to select the most appropriate (or optimal) mother-wavelet function for the analysis. The selection is usually done by trial and error, and other wavelet forms are possible, each having particular strengths and drawbacks. In general, there is a trade-off between spatial and spectral resolution inherent in the choice of wavelet, and this function is popular in vision analysis. From theory and performance comparison, it is clear that the wavelet transform based on Morlet function is suitable for signal analyses, as it can achieve excellent time and frequency concentration and can track the frequency trend at local time better than other methods ([10]). The truncated Morlet wavelet function, as shown in Fig. 1, is given by ([8]) t2
wðtÞ ¼ e 2 cos 5t
-6
-4
-2
0
2
6
8
Fig. 1. Display of Morlet wavelet function.
2.2. Sonic echo (SE) method The SE method helps engineers to judge the integrity of each pile and whether major discontinuities or defects exist, as shown in Fig. 2. In current practice the low-strain integrity testing of foundation piles involves the interpretation of a sonic echo, a method that has been widely used such that no great preparation or excessively expensive testing equipment is necessary. The technique is an example of impact hammer testing in which a pile head’s response to an instrumented hammer blow is measured while the pile head movement is recorded with an accelerometer (maximum sampling rate of 50 kHz). The resulting time trace is typically made up of transient pulses reflected from the pile integrity as well as the particular condition and features of its boundaries. Both stress-wave inputs and reflections are measured as a function of time by an accelerometer, which is placed on top of the pile, and the acceleration time-history data are integrated to gain the
ð11Þ
In this paper we are especially interested in the relatively higher wavelet coefficients in the time–frequency domain of the signal. The coefficient variation indicates that the signal-frequency content varied for the structure geometry changed (e.g., a defect) in some position. Meanwhile, it could be localized by the information of time-history. Often these discontinuities cannot be observed from the examination of the structure response in the pure time or the frequency domain, but they are detectable from the distribution of the wavelet coefficients (or amplitude) obtained by the CWT in the time–frequency domain.
4
Fig. 2. Sonic echo test of pile.
S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
velocity-history curve. On the basis of the one-dimensional wave propagation theory, the pile length L can be calculated by the number of samples as follows: L¼
1 1 c No: of samples 2 f
ð12Þ
where c = stress-wave velocity; f = sampling rate. In this paper, some numerical and experimental cases of the CWT analysis with SE traces are provided. The analysis is carried out using complete Morlet wavelets on the SE data. For the tests considered, some piles were partly intact, whereas others had defects, and the depth of the pile tip was of specific interest. The location of the tip is found by analyzing the signal-echo reflection from the interface between the tip of the pile and the surrounding earth. The depth of the tip can then be calculated by knowing the speed at which the stress-wave propagates through the pile material (normally assumed to be 4000 m/s). These signals contain obvious components of reflection characteristics (1) initial pulse of pile head, (2) reflection pulse of pile tip and (3) a defect, if present.
Fig. 4. Signal of intact numerical model pile.
3. Numerical model cases 3.1. Pile without soil In order to explore the impact force duration, a timehistory curve of hammer impact was recorded and is shown in the top graph of Fig. 3; the loading duration was 1 m/s. Then FFT transformed it into a frequency domain, as shown in the bottom graph of Fig. 3. The signal-frequency bandwidth is 0–3.5 kHz. Hence, this was the main frequency range of the time–frequency analyses used in the numerical cases. Now, we model three types of numerical piles intact, necking and bulb by the finite-element method (FEM) with the PC application ABAQUS. Figs. 4–6 contain finite-ele-
Fig. 5. Signal of numerical model pile with necking.
Normalized Amplitude
1 0.8 0.6 0.4 0.2 0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
2500
3000
3500
Time (sec)
Frequency Amplitude
2500 2000 1500 1000 500
0
500
1000
1500
2000
Frequency (Hz)
Fig. 3. Impact duration and frequency spectrum of numerical model.
Fig. 6. Signal of numerical model pile with bulb.
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Table 1 Three numerical testing pile conditions Label
Pile length-diameter ratio
Defect
P1 P2 P3
18 m/0.9 m = 20 18 m/0.9 m = 20 18 m/0.9 m = 20
Intact Necking area 10% Bulb area 10%
ment test signals generated from an 18 m pile without soil, with a diameter of 0.9 m (length to diameter ratio [LDR] = 20), which were tested without surrounding soil. The geometry and defects of these piles are listed in Table 1. The impact is modeled and generated by a relatively small dynamic loading (10 kg f) at the pile head, which transmits along the pile length, including low-strain energy. The Young’s modulus of pile concrete was assumed to be 3.84 · 107 K N m2, so the propagation wave velocity was set to be 4000 m/s, a Poisson’s ratio of 0.2, a damping ratio of 0.01 and a time increment of 1 · 109 s. We could now find the characteristics of the piles with the SE and CWT methods. The top graph in Fig. 4 shows the time-history curve of the intact pile, P1, detected by the SE method. Obviously, the pile head and tip were identified easily at about 0.5 and 9 m/s, respectively. The model pile length was calculated as L¼
V c Dt 4000 0:009 ¼ ¼ 18:0 2 2
Moreover, two distinct characteristics appeared at the same position of the signal in the wavelet time–frequency domain, as shown in the bottom graph in Fig. 4. The color at each point is associated with the magnitude of the wavelet coefficients, with the lighter color corresponding to larger amplitudes, and the darker color to the smaller ones. These two local peaks not only locate the positions but also indicate frequency contents. The resonant frequency occurs at 3.5 kHz at the pile head and 2 kHz at the pile tip. Furthermore, signal characteristics can be searched for and distinguished easily and quickly by the use of three-dimensional visual models in the time–frequency domain. In Fig. 5 the testing result of pile P2 is shown with the same pile length as P1 and with a necking area. The time-history curve of the SE method is shown in the top graph of Fig. 5. It shows an upward concave trace at about 5 m/s. The distance necking position from the pile head is calculated as 4000 0:005 ¼ 10:0 2 In contrast to Fig. 4, there are not only two characteristics that occur at both pile head and tip but also a slight prominence of a third character, which could be detected at about 5 m/s in the wavelet time–frequency domain. Its resonant frequency extended from 1 to 2 kHz. It could be identified as a signal characteristic of a necking area. The same signal feature also occurs in Fig. 6 of the pile with bulb. In the time-history curve of the SE method, shown in the top graph of Fig. 6, there is clearly a bulb character-
istic of a downward concave trace at about 5 m/s. This feature could be interpreted as a bulb or impedance increase of the pile section area in the one-dimensional wave theory. Similarly, a slight prominence occurs at about 5 m/s in the wavelet time–frequency domain of pile P3. Its resonant frequency ranged from 1 to 2 kHz, which is the same as the necking section. 3.2. Pile with soil In this subsection, pre-stressed concrete (PC) numerical piles were simulated with surrounding soil in three-dimensional analyses, as shown in Fig. 7. The pile is intact, with a length of 6 m and a diameter of 0.3 m. The surrounding soil is 5 times the diameter of pile, which is large enough to avoid stress-waves reflected from the far-field boundary back to the location of the receiver involved in the analysis. The numerical pile is assumed to be embedded 6 m in the soil. If the ratio of the pile’s Young’s modulus (Ep) to that of the surrounding soil (Es) is 50:1 or larger, then the testing method can work clearly ([7]). Herein, the ratios are assumed to be 500 and 1000 for pile P500 and P1000, respectively, as illustrated in Table 2. The results of testing simulation using CWT are shown in Figs. 8a and b. Normally, both positions of the pile heads and tips can be identified easily and clearly, as indicated by the arrowed lines. However, the distributions of the two signals’ contents are different in the time–frequency domain. In the P500 model the frequency contents of the pile head are about 2–2.3 kHz and those of the tip about 2– 2.8 kHz, as shown in Fig. 8a. In Fig. 8b, of the P1000 model, the frequency contents of the pile head are still
Ln ¼
Fig. 7. Three-dimensional finite-element model for hello pile embedded in soil.
Table 2 The Young’s modulus of surrounding of numerical models Pile model
Soil stiffness Es (K N m2)
Ep Es
P500 P1000
3.84 · 104 7.68 · 104
500 1000
S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
605
Fig. 9. CWT of PC intact pile without soil.
Fig. 8. Results of testing simulation of numerical PC piles with surrounding soil for (a) P500 and (b) P1000 using CWT.
about 2–2.3 kHz and 2–2.5 kHz for the pile tip, as shown in Fig. 8a. Obviously, in comparing these two results, the P500 model with harder soil stiffness has the same frequency band range at the pile head as the P1000 model with the softer soil stiffness but with a wider frequency band at the pile tip. This means that the same impact energy produces higher and wider frequency content at the pile tip in the soil with the harder stiffness. In addition, the peak wavelet coefficients are quite close in both models, between 2 · 107 and 8 · 107. In summary, both the pile head and tip can be located clearly in the traditional time domain. Nevertheless, only the frequency content difference can be recognized in the wavelet time– frequency domain. This works as an advantage, which we will discuss with regard to the variations in differing levels of stiffness for the surrounding soil when analyzing the signal. 4. Experimental cases 4.1. Pre-stressed concrete pile without soil-structure interaction Because the numerical case studies can prove the feasibility of the SE method with CWT time frequency, some experimental cases will be discussed in this section. Figs. 9–12 show the intact pile and the pile with a defect at full scale, which have been tested in situ. These two real PC piles are 6 m in length and 0.3 m in diameter. The defective
Fig. 10. CWT of the PC pile with defect and without soil.
Fig. 11. CWT of PC intact pile with soil.
pile is attached with a defect area on a single side 4 m from the pile head. After detection, the results of the time-history curve and the CWT plot are shown in Fig. 9. The pile head can be localized at about 0.4 m/s, and the tip location at about 3.3 m/s. There are clear characteristics of the pile head and tip at the same position as in the time domain. In addition, the main frequency bandwidth of the whole signal exists at 1.2 kHz. The frequency content maintains the same amplitude at the pile head, and decays from the lower frequency, 1.2 kHz, to the higher one, 2 kHz, at the pile tip.
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S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
5. Conclusions Wavelet analysis is a mathematical technique and a new signal-process tool that allows the signal to be viewed in the wavelet time–frequency domain. This is a new approach to expanding the signal for the SE method. The main goal of this paper is to discuss the feasibility of coupling the SE and CWT methods in the time–frequency domain. The results show that such an approach may provide a more complete way of viewing the signal. The following conclusions can be drawn:
Fig. 12. CWT of the PC pile with 10% defect and soil.
Furthermore, the testing result of another defective pile in the wavelet domain graph is shown at the bottom of Fig. 10. Obviously, in addition to both the pile head and tip, there is a distinct signal characteristic that occurs at about 2 m/s. The frequency bandwidth of the defect is extended from 1.2 to 1.6 kHz, and the amplitude gradually decays. In summary, we can obtain the same results in pileintegrity evaluation both in the time and wavelet domains. Anyone can identify the signal characteristics visually without requiring professional knowledge of the wave-propagation theory, and thus we can avoid misjudging the data by a confused and unclear signal. These two cases also prove that (1) the defect and boundary of a pile can be detected and identified clearly and accurately by coupling the SE and wavelet time–frequency analyses, and (2) the whole signal-frequency contents, including the localization of the defect, can be realized every single time. 4.2. Pre-stressed concrete pile with soil-structure interaction If the pile is surrounded with soil, it will complicate the analysis of the reflected signal and lead to waveform decay because of the soil-pile interaction. In order to distinguish the difference between the pile with and without the surrounding soil, the same two PC piles, as mentioned above, were driven into the soil, and Figs. 11 and 12 show the SE testing results. In the top graph of Fig. 11 the pile head was localized at about 0.3 m/s and the tip reflected at about 3.2 m/s. The frequency content of the whole testing signal, between 1.3 and 2 kHz, extended without decay. Moreover, in the top graph of Fig. 12 the testing signal of the defective pile was able to detect this characteristic at 2 m/s in the time domain, which is identified as a defect area. But the defect characteristic seems to be obfuscated by the pile tip characteristic in the time–frequency domain. This means that a defect cannot be identified where the defect position is too close to the pile tip in the wavelet domain.
1. The testing signal of the SE method can highlight the detailed characteristics, including the pile tip and visualized defects in the time–frequency domain of the CWT method. A clear characteristic phenomenon occurs if a defect exists. This is a great advantage for assistance in defect identification using the raw signal with noise. Hence, SE testing results can be judged more easily, effectively and reliably than in traditional methods. The numerical or experimental cases show that it does not matter whether the pile is surrounded with soil or not. 2. In the past, the time and frequency relationships of signals could not be presented and interpreted by the traditional FT method. The SE testing signal can now be displayed in the time–frequency domain at the same time by using the CWT method. The frequency change can be localized at any time in the whole signal. Such transformation not only offers frequency change information in every instance but also provides advantages for researchers in detecting the defect and/or boundary conditions directly. 3. Based on wavelet transformation in the time–frequency domain, the difference in the testing signal between differing levels of surrounding soil stiffness can be distinguished in the frequency content. In this study the result of numerical simulation indicates that harder surrounding soil produces higher and wider frequency content at the pile tip under the same impact energy. References [1] Addison PS, Sibbald A, Watson JN. Wavelet analysis: a mathematical microscope with civil engineering applications. Insight 1997;39(7):493–7. [2] Addison PS, Watson JN. Wavelet analysis for low-strain integrity testing of foundation piles. In: 5th International conference on inspection, appraisal, repairs, maintenance of buildings and structures, Singapore; 1997. p. 15–6. [3] Addison PS, Watson JN, Feng T. Low-oscillation complex wavelets. J Sound Vib 2002;254(4):733–62. [4] Daubechies I. Ten lectures on wavelets. Philadelphia (PA): Society for Industrial and Applied Mathematics; 1992. p. 129–31. [5] Gabor D. Theory of communication. J IEEE, London 1946: 429–57. [6] Hartung M, Meier K, Rodatz W. Integrity testing on model pile. In: The 4th international conference on the application of stress-wave theory to piles. Netherlands, Rotterdam; 1992. p. 265–71.
S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607 [7] Liao ST, Roesset JM. Dynamic Response of intact piles to impulse loads. Int J Numer Anal Meth Geomech 1997;21:255–75. [8] Math Works Inc. Wavelet-toolbox for use with Matlab user’s guide Version I; 1996. p. 1–99. [9] Morlet J, Grossmann A. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM, J Math Anal 1984;15:723–36. [10] Ma S, Sasaki T, Yoshihisa E, Honda T. Time–frequency analysis of ultrasonic echoes and its application to nondestructive evaluation of thermal damage of steel. Research Reports NIIS-RR-2002; 2003. [11] Ni SH, Lo KF. A study of wavelet transformation for pulse echo applied to piles. NDT Sci Technol 2001;19(1):4–15 [in Chinese].
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[12] Olson LD, Wright CC. Non-destructive testing of deep foundations with sonic methods. In: Proceedings found engineering congress: current principles and practice, vol. 2, Reston; ASCE: 1989. p. 1173– 83. [13] Ovanesova AV, Suarez LE. Application of wavelet transform to damage detection in frame structures. Eng Struct 2004;26:39–49. [14] Sedel JP. Presentation of low-strain integrity testing in the time– frequency domain. The application of stress-wave theory to piles, Rotterdam; 2000. p. 193–200. [15] Watson JN, Addison PS. Spectral-temporal filtering of NDT data using wavelet transform modulus maxima. Mech Res Commun 2002;29(2–3):99–106.
Computers and Geotechnics 35 (2008) 600–607 www.elsevier.com/locate/compgeo
Time–frequency analyses of pile-integrity testing using wavelet transform Sheng-Huoo Ni a, Kuo-Feng Lo b
a,*
, Lutz Lehmann b, Yan-Hong Huang
a
a Department of Civil Engineering, National Cheng Kung University, No. 1, University Road, Tainan 70101, Taiwan, ROC Institute of Applied Mechanics, Technical University of Braunschweig, Spielmannstrasse 11, D-38106 Braunschweig, Germany
Received 28 October 2006; received in revised form 6 September 2007; accepted 7 September 2007 Available online 23 October 2007
Abstract Detecting pile-integrity is an important topic for civil engineers and researchers. The pile-integrity testing signals acquired from the receivers on pile heads will tend to be complicated, as defects and undesired background vibrations exist. In this paper the continuous wavelet transform (CWT) method, with time–frequency distribution, is adopted to enhance the characteristics of the testing signal in order to improve the identification ability in both numerical simulations and experimental cases. The results indicate that testing signals can be displayed in the time–frequency domain at the same time and then be explored every single time by CWT, especially for changes in frequency content. In addition, CWT tends to reinforce the traits and makes all the information more visible. The feasibility of combining CWT with the sonic echo (SE) method is demonstrated. This approach is easily and quickly interpreted, and the pile-integrity is proven as well. 2007 Elsevier Ltd. All rights reserved. Keywords: Pile; Integrity test; Continuous wavelet transform; Defect
1. Introduction Non-destructive evaluation (NDE) techniques have been adopted to provide pile-construction quality control for many years. In particular, low-strain, pile-integrity testing methods include both the sonic echo (SE) and impulseresponse methods. Such methods have been used to check the lengths and integrity of newly installed foundations ([12]). These methods can help engineers to judge and identify both the presence and locations of pile defects, as the impact generated by the relatively small device produces a longitudinal wave that then transmits along the pile length, with both low-energy and high-frequency content. The term ‘‘low-strain integrity testing’’ is derived from the fact that the device generates only low-strain stress*
Corresponding author. E-mail addresses: [email protected] (S.-H. Ni), [email protected] (K.-F. Lo), [email protected] (L. Lehmann), [email protected] (Y.-H. Huang). 0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.09.003
waves in the pile. However, in practice, the signals will not always be pure and without noise. A new numerical signal-process tool that can substantially improve and strengthen the signal characteristics substantially is necessary, and that tool is the wavelet transform method (WTM). The wavelet transform application for low-strain integrity testing was first researched at Napier University, Edinburgh ([3,2]). In Addison et al. [1], the authors present finite element-generated responses for an imaginary pile 10 m in length, with and without a 20% impedance increase at midlength. They indicate that the 20% change in impedance is difficult to identify in the time domain, whereas the defect can be observed clearly in the wavelet transform ([14]). The WTM allows the signal to be decomposed so that frequency characteristics, as defined by analyzing the wavelet and the position of particular features in a time series, can be highlighted. The advantage of wavelet-based signalanalysis methods over the traditional Fourier transform (FT) method lies in its capability to produce simultaneous
S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
temporal and scale information ([11]). However, the reflection shape of the wave propagated can only be easily identified if the ratio of the defect area is more than 10% in the laboratory ([6]), which represents a limitation in analyzing the pile-integrity testing results with regard to defects with a smaller area ratio. 2. Theoretical background 2.1. Wavelet transform Fourier Transform (FT) breaks down a signal into its constituent sinusoids with different frequencies. Fourier analysis could also be viewed as a mathematical technique for transforming signals from time based to frequencybased. However, Fourier analysis has a serious drawback: when transforming to the frequency domain time, information disappears ([13]). When inspecting a Fourier-transformed signal, it is almost impossible to tell whether a particular event took place or not. As for characteristics such as abrupt changes, and the beginning and the end of events, Fourier analysis obviously cannot detect them at all. In order to correct this deficiency, Gabor [5] adapted a technique called windowing the signal (or the short time Fourier transform, STFT), which maps a signal into a two-dimensional function of time and frequency. The STFT represents a sort of compromise between the time and frequency-based views of the signal. It provides some information about both when, and at what frequencies, a signal event occurs. However, it can offer this information with limited precision, and this precision is determined only by the size of the window. The drawbacks of STFT are that once we choose a particular size for the time window, it is the same for all frequencies. Many signals require a more flexible approach in which we can vary the size of the window to determine more accurately either time or frequency. Wavelets do this by having a variable window width, which is related to the scale of observation; this flexibility allows for the isolation of the high-frequency features. One major difference is that the STFT has constant time and frequency resolution in contrast to the wavelet transform. Another important distinction between wavelet and Fourier analysis is that wavelet analysis is not limited to using sinusoidal analysis functions, but rather can employ a large selection of localized waveforms as long as they satisfy the predefined mathematical criteria described above. One major advantage afforded by wavelet transform is the capability for performing local analysis – i.e., to analyse a localized section of a long-duration signal. Wavelet analysis allows the choice of long time intervals when more precise low-frequency information is needed, and shorter ones when high-frequency information is desired. It is a windowing technique with variable-sized regions. Wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet so that local features can be described better with wavelets that
601
are concentrated on a localized region. Wavelet transforms differ from STFT, as they allow arbitrarily high localization in time of high-frequency-signal features. Their ability to localize in both time and frequency in a distinctly different way from the traditional STFT has spawned a number of sophisticated wavelet-based methods for signal decomposition, manipulation and interrogation. The WTM has been found particularly useful for analyses of signals that can be described as periodic, noisy, intermittent, and transient. There are many different wavelet transforms available, so it is not a unique function. A wavelet must satisfy three conditions: 1. It must have unit energy. 2. It must provide compact support or sufficiently fast decay so that it satisfies the requirement of space (or time) location. 3. It must have a zero mean – i.e., the integral of the wavelet function from 1 to +1 is zero. This is called the admissibility condition, and ensures that there are both positive and negative components to the motherwavelet. Consider a real or complex-value continuous-time function w(t) with the following property ([4,9]): 1. The function integrates to zero. Z 1 wðtÞdt ¼ 0
ð1Þ
1
2. Its square integral, or equivalent, has finite energy. Z 1 2 jf ðtÞj dt < 1 ð2Þ 1
The function w(t) is a mother-wavelet or a wavelet if it satisfies these two properties as well as the admissibility condition as defined above. Let f(t) be any square integral function. The continuous wavelet transform (CWT) with respect to a wavelet w(t) is defined as Z þ1 1 tb W ða; bÞ ¼ pffiffiffi f ðtÞw dt ð3Þ a a 1 where a and b are real, and gate. We set wa,b(t)w as 1 tb wa;b ðtÞ ¼ pffiffiffiffiffiffi w a jaj
*
denotes the complex conju-
Then, combining Eqs. (3) and (4), yields Z þ1 W ða; bÞ ¼ f ðtÞwa;b ðtÞdt
ð4Þ
ð5Þ
1
The variable b represents the time shift or translation, and the variable a determines the amount of time scaling or dilation. Since the CWT is generated using dilates and translates of the signal function w(t), the wavelet for the transform is referred to as the mother-wavelet. Next, we
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generate a set of expansion functions so that any signal in L2(R) (the space of square integral functions) can be represented by the series X pffiffiffiffi aj;k 2j uð2j t kÞ ð6Þ f ðtÞ ¼
1 0.8 0.6 0.4
j;k 0.2
Besides, as with the Fourier transform, the original signal may be reconstructed Z þ1 Z þ1 1 1 tb dadb pffiffiffi w f ðtÞ ¼ ð7Þ W ða; bÞ 2 C g 1 1 a a a
0 -0.2 -0.4 -0.6
where Cg is the admissibility constant Z x _ 2 j wðxÞj dx Cg ¼ x 0
-0.8 -8
ð8Þ
As the wavelet transform given by Eq. (3) is a convolution of the signal with a wavelet function, we can use the convolution theorem to express the integral as a product in Fourier space, i.e., Z 1 _ _ 1 W ða; bÞ ¼ f ðwÞwa;b ðxÞdx ð9Þ 2p 1 _ pffiffiffi_ ð10Þ where wa;b ðxÞ ¼ aw ðaxÞeiwb This is the Fourier spectrum of the analysed wavelet at scale a and location b. In this way a fast Fourier transform (FFT) algorithm can be employed in practice to compute the wavelet transform. For more details of the wavelet with time–frequency, refer to Watson and Addison [15]. To apply the CWT in any application it is important to select the most appropriate (or optimal) mother-wavelet function for the analysis. The selection is usually done by trial and error, and other wavelet forms are possible, each having particular strengths and drawbacks. In general, there is a trade-off between spatial and spectral resolution inherent in the choice of wavelet, and this function is popular in vision analysis. From theory and performance comparison, it is clear that the wavelet transform based on Morlet function is suitable for signal analyses, as it can achieve excellent time and frequency concentration and can track the frequency trend at local time better than other methods ([10]). The truncated Morlet wavelet function, as shown in Fig. 1, is given by ([8]) t2
wðtÞ ¼ e 2 cos 5t
-6
-4
-2
0
2
6
8
Fig. 1. Display of Morlet wavelet function.
2.2. Sonic echo (SE) method The SE method helps engineers to judge the integrity of each pile and whether major discontinuities or defects exist, as shown in Fig. 2. In current practice the low-strain integrity testing of foundation piles involves the interpretation of a sonic echo, a method that has been widely used such that no great preparation or excessively expensive testing equipment is necessary. The technique is an example of impact hammer testing in which a pile head’s response to an instrumented hammer blow is measured while the pile head movement is recorded with an accelerometer (maximum sampling rate of 50 kHz). The resulting time trace is typically made up of transient pulses reflected from the pile integrity as well as the particular condition and features of its boundaries. Both stress-wave inputs and reflections are measured as a function of time by an accelerometer, which is placed on top of the pile, and the acceleration time-history data are integrated to gain the
ð11Þ
In this paper we are especially interested in the relatively higher wavelet coefficients in the time–frequency domain of the signal. The coefficient variation indicates that the signal-frequency content varied for the structure geometry changed (e.g., a defect) in some position. Meanwhile, it could be localized by the information of time-history. Often these discontinuities cannot be observed from the examination of the structure response in the pure time or the frequency domain, but they are detectable from the distribution of the wavelet coefficients (or amplitude) obtained by the CWT in the time–frequency domain.
4
Fig. 2. Sonic echo test of pile.
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velocity-history curve. On the basis of the one-dimensional wave propagation theory, the pile length L can be calculated by the number of samples as follows: L¼
1 1 c No: of samples 2 f
ð12Þ
where c = stress-wave velocity; f = sampling rate. In this paper, some numerical and experimental cases of the CWT analysis with SE traces are provided. The analysis is carried out using complete Morlet wavelets on the SE data. For the tests considered, some piles were partly intact, whereas others had defects, and the depth of the pile tip was of specific interest. The location of the tip is found by analyzing the signal-echo reflection from the interface between the tip of the pile and the surrounding earth. The depth of the tip can then be calculated by knowing the speed at which the stress-wave propagates through the pile material (normally assumed to be 4000 m/s). These signals contain obvious components of reflection characteristics (1) initial pulse of pile head, (2) reflection pulse of pile tip and (3) a defect, if present.
Fig. 4. Signal of intact numerical model pile.
3. Numerical model cases 3.1. Pile without soil In order to explore the impact force duration, a timehistory curve of hammer impact was recorded and is shown in the top graph of Fig. 3; the loading duration was 1 m/s. Then FFT transformed it into a frequency domain, as shown in the bottom graph of Fig. 3. The signal-frequency bandwidth is 0–3.5 kHz. Hence, this was the main frequency range of the time–frequency analyses used in the numerical cases. Now, we model three types of numerical piles intact, necking and bulb by the finite-element method (FEM) with the PC application ABAQUS. Figs. 4–6 contain finite-ele-
Fig. 5. Signal of numerical model pile with necking.
Normalized Amplitude
1 0.8 0.6 0.4 0.2 0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
2500
3000
3500
Time (sec)
Frequency Amplitude
2500 2000 1500 1000 500
0
500
1000
1500
2000
Frequency (Hz)
Fig. 3. Impact duration and frequency spectrum of numerical model.
Fig. 6. Signal of numerical model pile with bulb.
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Table 1 Three numerical testing pile conditions Label
Pile length-diameter ratio
Defect
P1 P2 P3
18 m/0.9 m = 20 18 m/0.9 m = 20 18 m/0.9 m = 20
Intact Necking area 10% Bulb area 10%
ment test signals generated from an 18 m pile without soil, with a diameter of 0.9 m (length to diameter ratio [LDR] = 20), which were tested without surrounding soil. The geometry and defects of these piles are listed in Table 1. The impact is modeled and generated by a relatively small dynamic loading (10 kg f) at the pile head, which transmits along the pile length, including low-strain energy. The Young’s modulus of pile concrete was assumed to be 3.84 · 107 K N m2, so the propagation wave velocity was set to be 4000 m/s, a Poisson’s ratio of 0.2, a damping ratio of 0.01 and a time increment of 1 · 109 s. We could now find the characteristics of the piles with the SE and CWT methods. The top graph in Fig. 4 shows the time-history curve of the intact pile, P1, detected by the SE method. Obviously, the pile head and tip were identified easily at about 0.5 and 9 m/s, respectively. The model pile length was calculated as L¼
V c Dt 4000 0:009 ¼ ¼ 18:0 2 2
Moreover, two distinct characteristics appeared at the same position of the signal in the wavelet time–frequency domain, as shown in the bottom graph in Fig. 4. The color at each point is associated with the magnitude of the wavelet coefficients, with the lighter color corresponding to larger amplitudes, and the darker color to the smaller ones. These two local peaks not only locate the positions but also indicate frequency contents. The resonant frequency occurs at 3.5 kHz at the pile head and 2 kHz at the pile tip. Furthermore, signal characteristics can be searched for and distinguished easily and quickly by the use of three-dimensional visual models in the time–frequency domain. In Fig. 5 the testing result of pile P2 is shown with the same pile length as P1 and with a necking area. The time-history curve of the SE method is shown in the top graph of Fig. 5. It shows an upward concave trace at about 5 m/s. The distance necking position from the pile head is calculated as 4000 0:005 ¼ 10:0 2 In contrast to Fig. 4, there are not only two characteristics that occur at both pile head and tip but also a slight prominence of a third character, which could be detected at about 5 m/s in the wavelet time–frequency domain. Its resonant frequency extended from 1 to 2 kHz. It could be identified as a signal characteristic of a necking area. The same signal feature also occurs in Fig. 6 of the pile with bulb. In the time-history curve of the SE method, shown in the top graph of Fig. 6, there is clearly a bulb character-
istic of a downward concave trace at about 5 m/s. This feature could be interpreted as a bulb or impedance increase of the pile section area in the one-dimensional wave theory. Similarly, a slight prominence occurs at about 5 m/s in the wavelet time–frequency domain of pile P3. Its resonant frequency ranged from 1 to 2 kHz, which is the same as the necking section. 3.2. Pile with soil In this subsection, pre-stressed concrete (PC) numerical piles were simulated with surrounding soil in three-dimensional analyses, as shown in Fig. 7. The pile is intact, with a length of 6 m and a diameter of 0.3 m. The surrounding soil is 5 times the diameter of pile, which is large enough to avoid stress-waves reflected from the far-field boundary back to the location of the receiver involved in the analysis. The numerical pile is assumed to be embedded 6 m in the soil. If the ratio of the pile’s Young’s modulus (Ep) to that of the surrounding soil (Es) is 50:1 or larger, then the testing method can work clearly ([7]). Herein, the ratios are assumed to be 500 and 1000 for pile P500 and P1000, respectively, as illustrated in Table 2. The results of testing simulation using CWT are shown in Figs. 8a and b. Normally, both positions of the pile heads and tips can be identified easily and clearly, as indicated by the arrowed lines. However, the distributions of the two signals’ contents are different in the time–frequency domain. In the P500 model the frequency contents of the pile head are about 2–2.3 kHz and those of the tip about 2– 2.8 kHz, as shown in Fig. 8a. In Fig. 8b, of the P1000 model, the frequency contents of the pile head are still
Ln ¼
Fig. 7. Three-dimensional finite-element model for hello pile embedded in soil.
Table 2 The Young’s modulus of surrounding of numerical models Pile model
Soil stiffness Es (K N m2)
Ep Es
P500 P1000
3.84 · 104 7.68 · 104
500 1000
S.-H. Ni et al. / Computers and Geotechnics 35 (2008) 600–607
605
Fig. 9. CWT of PC intact pile without soil.
Fig. 8. Results of testing simulation of numerical PC piles with surrounding soil for (a) P500 and (b) P1000 using CWT.
about 2–2.3 kHz and 2–2.5 kHz for the pile tip, as shown in Fig. 8a. Obviously, in comparing these two results, the P500 model with harder soil stiffness has the same frequency band range at the pile head as the P1000 model with the softer soil stiffness but with a wider frequency band at the pile tip. This means that the same impact energy produces higher and wider frequency content at the pile tip in the soil with the harder stiffness. In addition, the peak wavelet coefficients are quite close in both models, between 2 · 107 and 8 · 107. In summary, both the pile head and tip can be located clearly in the traditional time domain. Nevertheless, only the frequency content difference can be recognized in the wavelet time– frequency domain. This works as an advantage, which we will discuss with regard to the variations in differing levels of stiffness for the surrounding soil when analyzing the signal. 4. Experimental cases 4.1. Pre-stressed concrete pile without soil-structure interaction Because the numerical case studies can prove the feasibility of the SE method with CWT time frequency, some experimental cases will be discussed in this section. Figs. 9–12 show the intact pile and the pile with a defect at full scale, which have been tested in situ. These two real PC piles are 6 m in length and 0.3 m in diameter. The defective
Fig. 10. CWT of the PC pile with defect and without soil.
Fig. 11. CWT of PC intact pile with soil.
pile is attached with a defect area on a single side 4 m from the pile head. After detection, the results of the time-history curve and the CWT plot are shown in Fig. 9. The pile head can be localized at about 0.4 m/s, and the tip location at about 3.3 m/s. There are clear characteristics of the pile head and tip at the same position as in the time domain. In addition, the main frequency bandwidth of the whole signal exists at 1.2 kHz. The frequency content maintains the same amplitude at the pile head, and decays from the lower frequency, 1.2 kHz, to the higher one, 2 kHz, at the pile tip.
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5. Conclusions Wavelet analysis is a mathematical technique and a new signal-process tool that allows the signal to be viewed in the wavelet time–frequency domain. This is a new approach to expanding the signal for the SE method. The main goal of this paper is to discuss the feasibility of coupling the SE and CWT methods in the time–frequency domain. The results show that such an approach may provide a more complete way of viewing the signal. The following conclusions can be drawn:
Fig. 12. CWT of the PC pile with 10% defect and soil.
Furthermore, the testing result of another defective pile in the wavelet domain graph is shown at the bottom of Fig. 10. Obviously, in addition to both the pile head and tip, there is a distinct signal characteristic that occurs at about 2 m/s. The frequency bandwidth of the defect is extended from 1.2 to 1.6 kHz, and the amplitude gradually decays. In summary, we can obtain the same results in pileintegrity evaluation both in the time and wavelet domains. Anyone can identify the signal characteristics visually without requiring professional knowledge of the wave-propagation theory, and thus we can avoid misjudging the data by a confused and unclear signal. These two cases also prove that (1) the defect and boundary of a pile can be detected and identified clearly and accurately by coupling the SE and wavelet time–frequency analyses, and (2) the whole signal-frequency contents, including the localization of the defect, can be realized every single time. 4.2. Pre-stressed concrete pile with soil-structure interaction If the pile is surrounded with soil, it will complicate the analysis of the reflected signal and lead to waveform decay because of the soil-pile interaction. In order to distinguish the difference between the pile with and without the surrounding soil, the same two PC piles, as mentioned above, were driven into the soil, and Figs. 11 and 12 show the SE testing results. In the top graph of Fig. 11 the pile head was localized at about 0.3 m/s and the tip reflected at about 3.2 m/s. The frequency content of the whole testing signal, between 1.3 and 2 kHz, extended without decay. Moreover, in the top graph of Fig. 12 the testing signal of the defective pile was able to detect this characteristic at 2 m/s in the time domain, which is identified as a defect area. But the defect characteristic seems to be obfuscated by the pile tip characteristic in the time–frequency domain. This means that a defect cannot be identified where the defect position is too close to the pile tip in the wavelet domain.
1. The testing signal of the SE method can highlight the detailed characteristics, including the pile tip and visualized defects in the time–frequency domain of the CWT method. A clear characteristic phenomenon occurs if a defect exists. This is a great advantage for assistance in defect identification using the raw signal with noise. Hence, SE testing results can be judged more easily, effectively and reliably than in traditional methods. The numerical or experimental cases show that it does not matter whether the pile is surrounded with soil or not. 2. In the past, the time and frequency relationships of signals could not be presented and interpreted by the traditional FT method. The SE testing signal can now be displayed in the time–frequency domain at the same time by using the CWT method. The frequency change can be localized at any time in the whole signal. Such transformation not only offers frequency change information in every instance but also provides advantages for researchers in detecting the defect and/or boundary conditions directly. 3. Based on wavelet transformation in the time–frequency domain, the difference in the testing signal between differing levels of surrounding soil stiffness can be distinguished in the frequency content. In this study the result of numerical simulation indicates that harder surrounding soil produces higher and wider frequency content at the pile tip under the same impact energy. References [1] Addison PS, Sibbald A, Watson JN. Wavelet analysis: a mathematical microscope with civil engineering applications. Insight 1997;39(7):493–7. [2] Addison PS, Watson JN. Wavelet analysis for low-strain integrity testing of foundation piles. In: 5th International conference on inspection, appraisal, repairs, maintenance of buildings and structures, Singapore; 1997. p. 15–6. [3] Addison PS, Watson JN, Feng T. Low-oscillation complex wavelets. J Sound Vib 2002;254(4):733–62. [4] Daubechies I. Ten lectures on wavelets. Philadelphia (PA): Society for Industrial and Applied Mathematics; 1992. p. 129–31. [5] Gabor D. Theory of communication. J IEEE, London 1946: 429–57. [6] Hartung M, Meier K, Rodatz W. Integrity testing on model pile. In: The 4th international conference on the application of stress-wave theory to piles. Netherlands, Rotterdam; 1992. p. 265–71.
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