Determination of Embedment Depth of Timber Polesand Piles Using Wavelet Transform

by

Jianchun Li, Mahbube Subhani and Bijan Samali

Reprinted from

Advances in Structural EngineeringVolume 15 No. 5 2012

MULTI-SCIENCE PUBLISHING CO. LTD.5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

Advances in Structural Engineering Vol. 15 No. 5 2012 759

1. INTRODUCTIONUtility poles represent a significant part of Australia’sinfrastructure. According to Nguyen et al. (2004), thereare nearly 5 million timber poles being used in thecurrent network for distribution of power andcommunications in Australia. The utility pole industryin Australia spends approximately 40–50 millionannually on maintenance and asset management toavoid failure of utility lines, which is very costly andmay cause serious consequences. Each year about300,000 electricity poles are replaced in the easternstates of Australia, despite the fact that up to 80% ofthese poles are still in a very good serviceable condition(Nguyen et al. 2004).

Surface non-destructive testing (NDT) methods suchas Sonic Echo, Bending Waves and Ultraseismicmethods have been considered over the past decade tobe simple and cost-effective tools for identifying thecondition and underground depth of embeddedstructures, such as timber poles or piles in-service (Rix

Determination of Embedment Depth of Timber Poles

and Piles Using Wavelet Transform

Jianchun Li*, Mahbube Subhani and Bijan SamaliCentre for Built Infrastructure Research, Faculty of Engineering, University of Technology, Sydney, NSW 2007, Australia

Abstract: This paper presents an investigation on the wave propagation in timberpoles with Wavelet Transform (WT) analysis for identification of the condition andunderground depth of embedded timber poles in service. Most of non-destructivetesting (NDT) applications for timber poles using wave-based methods consider onlysingle wave mode and no dispersion. However, for wave propagations in timber poles(damaged/undamaged), such simplification may not be correct, especially for broadband excitation using impulse impact. To investigate the problem, a 5m timber polewas investigated numerically and experimentally. A dispersion curve is generatedfrom the numerical results to provide guidance on the velocity and wave modeselection. Continuous wavelet transform (CWT) is applied on the same signal to verifythe presence of modes and to process data from experimental testing. The results arepresented in both time domain and time-frequency domain for comparison. The resultsof the investigation showed that, wavelet transform analysis can be a reliable signalprocessing tool for NDT in terms of condition and embedment length determination.

Key words: timber poles, non-destructive tests (NDT), wavelet transform (WT), wave dispersion, conditionassessment, embedment depth.

et al. 1993; Davis 1994; Lin et al. 1997; Holt 1994; Holtet al. 1994). Despite the wide spread use of thesemethods, the effectiveness and reliability of the methodson determination of embedded length and evaluation ofunderground conditions of poles, especially timberpoles, are not addressed.

When it comes to field applications, thesedeveloped/to be developed NDTs face a significantchallenge due to presence of uncertainties such ascomplex material properties (e.g. timber), environmentalconditions, interaction of soil and structure, defects anddeteriorations as well as coupled nature of unknownlength and condition. Moreover, due to the dispersivenature of the stress wave signal, which is related to thetypes of the wave, many frequency components exist inthe measured signals and each frequency componentcorresponds to an individual velocity.

Sonic echo and Impulse response have been used formany years for different materials and differentstructures, however, applications to timber poles are

*Corresponding author. E-mail address: lij@eng.uts.edu.au; Fax: +61-02-9514-2633; Tel: +61-02-9514-2651.

rarely seen. Vibration based damage detection is verypopular and widely researched these days. Butunfortunately, these approaches are not very suitable forthe pole condition because of complexity of over-hanging electric cables, geotechnical environment anduncertainty of timber material properties. Stress wavebased methods are, therefore, much preferred choice forsuch applications since complexity factorsaforementioned have less impact to the methods.Despite the advantages, propagation of stress wave in afinite media is still very complex in nature. Sounderstanding of the propagation of stress wave intimber poles is essential in development of suitabletechniques for underground pole condition andembedment length determination.

Based on the understanding of wave propagation in atimber pole, advanced signal processing can be utilizedfor data processing to reveal hidden information that iscritical for condition and length determination of poles.Three main groups of signal processing tools are oftenassociated with wave based analysis: time domainanalysis, frequency domain analysis (i.e. Fouriertransform) and time-frequency analysis [such as ShortTime Fourier Transform (STFT), Wigner–VilleDistribution (WVD) and Wavelet Transform (WT)].

Fourier transform breaks down a signal intoconstituent sinusoids of different frequencies andtransforms the signal from a time domain to frequencydomain. As a result all time information is lost. Toovercome this deficiency, Gabor introduced short timeFourier transform (STFT) by introducing thewindowing techniques, that is, by considering a sectionof the signal at a time (Chuis 1997). The STFT maps asignal into a 2-D function of time and frequency.However, the method suffers from the disadvantage thatthe time and frequency information obtained havelimited precision that is influenced by the size of thewindow. A high resolution cannot be obtainedsimultaneously for both time and frequency domainssince once the time window size is chosen; it remainsfixed for all frequencies (Ovanesova and Suarez 2004).Wigner-Ville distribution (WVD) has an excellentfrequency resolution, but suffers from the cross terms(Staszewski and Robertson 2007). This limitation can beovercome by Choi-Williams distribution, but itdecreases the time-frequency resolution (Staszewskiand Robertson 2007). Different from these time-frequency methods, continuous wavelet transform(CWT) allows analysing signals for every frequencywith a different window size. It, therefore, allows choiceof long time intervals when more precise low-frequencyinformation is needed, and shorter ones when highfrequency information is desired. In recent years, time-

frequency analysis has attracted a great deal of interestfrom researchers for data processing of stress wave-based damage detection methods. Some researchershave also applied such techniques on traditional NDTmethods such as sonic echo or impulse response method(Ni et al. 2008).

CWT displays all the frequency components andcorresponding time histories presented in the signal,which can be used advantageously to determine the timehistory associated with particular frequency required foranalysis, as wave generated by impact producesbroadband frequency excitation. To do so, acombination of FFT and DFT is ultilised to find out thedesired frequency range and once the frequency band isselected, CWT is applied to the signal to obtain the timehistory corresponding to the chosen frequency andtherefore to enable the calculation of the embeddedlength of the pole and the presence of damage as well.

2. THEORETICAL BACKGROUNDWaves propagating in infinite media have mainly twoforms, i,e, dilatational wave and distortional wave(Kolsky 1963). Dilatational waves cause a change inthe volume of the medium in which it is propagatingbut no rotation; while distortional waves involverotation but no volume changes. The velocities of thesewaves are merely a function of material properties ofthe medium. On the contrary, wave propagation infinite media is much more complicated and dispersionnature becomes more prominent and more than onemode can be generated. However, under certainconditions, it is possible to simplify the case to producea single mode, large diameter waveguides (Peterson1999). However, for the in-service timber pole, whichhave a fixed length and diameter such simplification isnot applicable. Moreover, it is necessary, in terms ofpracticality and simplicity to generate waves by simpleimpulse which unfortunately will result in a broadbandexcitation.

Under a broadband frequency excitation, it is obviousthat, multimode stress wave will be generated andpropagate through the media. At low frequency, therewill be few modes generated but higher wave modesstart to join in with the increase of excitation frequency.The frequency, at which one particular mode startsgenerating, is called the cut off frequency of that mode.To analyse an output signal, it is necessary to choose aparticular frequency and to know their correspondingmodes and velocities as well.

Selecting a particular frequency in analysis is alsovery important as every mode has a peak stress functioncorresponding to a certain frequency and between twomodes the stress functions have compromised values,

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Determination of Embedment Depth of Timber Poles and Piles Using Wavelet Transform

that is if the chosen frequency is related to the highestpeak of a particular modal frequency, then the signal isnon dispersive at that point and the group velocity isclose to the traditional bar velocity CL = (E /ρ)1/2, (E =modulus of elasticity, (ρ = density). However, the groupvelocity of a signal will be less than the traditional barvelocity and very dispersive, if the chosen frequency isin between the highest peak stress function of twoconsecutive modes (Puckett 2004).

Another important parameter is the length to radiusratio of the pole and if the ratio is very high, the mediumcan be considered as infinite. Thus, the group velocity ofthe modes approaches the velocity in an unboundedsolid, dispersion become more negligible and a highdegree of accuracy can be obtained (Peterson 1999).Moreover, the number of lateral reflection decreases andas a result less transverse waves are generated andlongitudinal waves become dominant. But in the case offield timber poles, it is not adequate to consider them asinfinite media because of their length to radius ratio andin fact, wave reflection is expected as the main principleof the method is reliant on wave reflection to determinedamage and the embedded length. The assumptionssuch as wave propagation at traditional bar velocity andno dispersion are not necessarily correct for wavepropagation in field poles.

2.1. Stress Wave VelocityAlthough timber pole should be considered as finitemedia, velocity of wave in finite media is related tovelocity of dilatational wave and velocity ofdistortional wave in unbounded media as well asRayleigh surface wave velocity. At low frequency,fewer modes are generated and the velocity reachesclose to the traditional wave velocity or can beconsidered as one dimensional wave problem and athigh frequency, more modes are generated and thevelocity approaches the Rayleigh surface wave velocity(Puckett and Peterson 2005). The equations governingthese velocities are as follows:

(1)

(2)

(3)CE

L =ρ

C2 =µρ

C12

=+λ µρ

where, C1 = velocity of dilatational wave in infinitemedia;

C2 = velocity of distortional wave in infinite media;CL = traditional bar velocity;µ = shear modulus of elasticity;λ = Lame’s constant;ρ = density of the material;E = modulus of elasticity.

(4)

(5)

and Cs = velocity of Rayleigh surface wave.The governing equation for C1 and C2 is given by

(Kolsky 1963)

(6)

where, ∆ is the sum of three normal strains.Lame’s constant and shear modulus can be found

from Poisson’s ratio (ν) and modulus of elasticity byfollowing formula:

(7)

(8)

The properties of the timber poles in Table 1 arechosen from the data provided by the power distributingcompanies and the velocities in Table 2 are calculatedfrom the above equations.

λµ µ

µ=

−

−

( )

( )

E

E

2

3

µν

=+

E

2 1( )

ρ λ µ∂

∂= + ∇

2

222

∆∆

t( )

κ α12

12

1= =

C

Cand

C

Cs

κ κ α κ α16

14

12

12

128 24 16 16 16 0− + − + − =( ) ( )

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Jianchun Li, Mahbube Subhani and Bijan Samali

Table 1. Properties of timber

E(×109Pa) (kg/m3) (×109Pa) (×109Pa)

14 750 0.3 5.3 8.15

ννλλµµρρ

Table 2. Stress wave velocity

C1 (m/s) C2 (m/s) CL (m/s) CS (m/s)

5,021 2,878 4,321 2,500

2.2. Pochhammer-Chree EquationThe Pochhammer-Chree equation for wave in an infinitecylinder is of primary interest for many wave solutions.The equation is derived for an infinite cylinder in a threedimensional space. But if the length to radius ratio ishigh, then it can be considered as a one dimensionalwave and can be determined by traditional bar velocity.The Pochhammer-Chree equation (Rose 2004) is asfollows:

(9)

where, α = (ω2/C12)− k2;

ω = angular frequency;Cp = phase velocity;β = (ω2/C2

2)− k2 ;r = radius of the cylinder;J1 and J0 = Bessel function of the first kind of order

zero and one;The root of the equation is wavenumber, which is

related to the phase velocity of a signal. The relation is,

(10)

where, k = wavenumberThis equation is solved depending on the dimension

of the timber pole case. After putting all the values inEqn 9, the Bessel function can be expanded byfollowing way

(11)

(12)

By considering only the first term of the Besselexpansion and put the values in equation (9), two rootsor two values of wavenumber is found. These two rootsare constant and lead to two velocities of value 4320 m/sand 2682 m/s. The former one is very close to thetraditional bar velocity and latter one is close to theshear wave velocity in infinite media, which means thatthe wave is non-dispersive for which the velocity isunique and depends on material property. The accuracyof the results will increase by considering more termsfrom the expanded series of the Bessel function. If thefirst two terms is considered, then the roots could becomplex (evanescent mode), imaginary (non-

J r r r etc131

2

1

16( ) ( ) ( )β β β= − +L

J r r r02 41

1

4

1

64( ) ( ) ( )α α α= − + −L

kcp

=ω

2 2 21 1

2 2 2

0 1

αβ α β β

α β

rk J r J r k

J r J r

( ) ( ) ( ) ( )

( ) (

+ − −

× )) ( ) ( )− =4 021 0k J r J rαβ α β

propagating mode) and real (propagating mode). Sodispersion occurs and the velocity is closed to thetraditional bar velocity at low frequency. Figure 1 showsthe graphs of considering first two terms of Bessel’sexpansion.

2.3. Phase Velocity due to BroadbandExcitation

To determine the phase velocity, it is necessary to knowthe frequency related wave number and this can be doneby applying Fast Fourier Transform (FFT) to a spatialdata. Moreover, wave number is also related to the wavemode. So in a particular frequency, more than one wavenumber may present. To determine the wave number,location vs axial stress graph is generated. In the case ofa broadband signal, it is also important to determinewhich wave number is related to a particular modalfrequency. The procedure of determining the wavenumbers for a broadband signal is shown in Figure 2.

In this paper, axial stresses at both ends areconsidered to determine the presence of modes. From

762 Advances in Structural Engineering Vol. 15 No. 5 2012

Determination of Embedment Depth of Timber Poles and Piles Using Wavelet Transform

4400

4200

4000

0 2Frequency

× 104

4 6

Pha

se v

eloc

ity

3800

Dispersive curve

Figure 1. Dispersive curve (considering first two terms of

Bessel expansion)

5000Time stress 0.05 m

Time stress 4.95 m

Time (s)···

···

Time (s)

0 0.02 0.04

0 0.02 0.04

FFT

× 106 Frequency 0.05 m

0500 1000

Frequency f = f1

1500

42

× 106 Frequency 4.95 m

0500 1000

Frequency f = f1

Forf = f1

1500

42

0

X −5000

50000

−5000

× 106 Location stress

Location (cm)

0 100 200 300 400 500 0 0.01 0.02 0.03

× 108 Wave number, f = f1

Wave number

1050

210

FFT

FFT

Figure 2. Graphical representation of getting wave number for

broadband signal

the results of the analysis, dispersive curve is generatedand the presence of modes at different frequencies isdetermined by this procedure and verified by analysingthe same signal using wavelet transform. Finally, thenumerical model is compared with the experimentalresults.

2.4. Wavelet Transform (WT)For dispersive wave, using wavelet transform forsignal processing is necessary. Both ContinuousWavelet Transform (CWT) and Discrete WaveletTransform (DWT) have their own merits. UsuallyCWT is used to analyse the signal. As it has a hightime resolution for high frequency component andhigh frequency resolution for low frequencycomponents, it is easier to choose the desiredfrequency band and to determine the velocity fromthat frequency only, since each frequency correspondsto a different velocity.

On the other hand, DWT is also useful todecompose and recompose measured signals and isoften used for de-noising the data. This is especiallyattractive for processing experimental data or fieldtesting data where noise often causes a great degree ofproblem.

There are many different wavelet transformsavailable, so it is not a unique function. A wavelet mustsatisfy three conditions:

(1) It must have unit energy.(2) It must provide compact support or sufficiently

fast decay so that it satisfies the requirement ofspace (or time) location.

(3) It must have a zero mean so the integral of thewavelet function from −∞ to +∞ is zero. This iscalled admissibility condition and ensures thatthere are both positive and negative componentsto the mother wavelet.

If f(t) be any square integral function, the definitionof the CWT (Mallat 1998) is given by:

(13)

where, u and s are real, u represent the time shift ortranslation and the variable s determines the amount oftime scaling or dilation, t is the time, * denotes thecomplex conjugate and

(14)ψ ψ( , ) ( )u s ts

t u

s=

−

1

Wf u s f t u s t dt

f ts

t

( , ) ( ) ,* ( )

( ) *

=

=−

−∞

+∞

−∞

+∞

∫

∫

ψ

ψ1 uu

sdt

The function ψ(t) is called mother wavelet satisfyingthe admissibility condition

(15)

where, ψ~(ω) is the Fourier transform of ψ(t).To satisfy the admissibility condition the

requirements are

(16)

The mother wavelet also has to satisfy that, its squareintegral, or equivalent, has finite energy,

(17)

From the equation of CWT it is found that the motherwavelet ψ(t) is translated by the translation parameter uand dilated by the scaling parameter s when a signal isanalyzed.

Figure 3 shows the tiling of CWT, often referred asHeisenberg box, which indicates that for a lowfrequency, time resolution is low and vice versa. If thecentre frequency of ψ~(ω) is η and the time andfrequency spread of ψ(t) are σt and σw, respectively,then the time and frequency spread of ψ (u,s) are sσt andσw/s respectively. So the centre of the correspondingHeisenberg box is located at (u, η /s).

The choice of mother wavelet is very important forCWT analysis. Depending on data characteristics andanalysis purposes, the choice of mother wavelet may bevaried. Gabor wavelet (which is a complex valuedmodulated Gaussian function) has the smallestHeisenberg box and can be adjusted to have a shortertime support as seen in Figure 3 (Kim and Kim 2001).

f t dt( ) 2 0<−∞

+∞

∫

%ψ ω ψ( ) ( )= =−∞

+∞

∫0 0and t dt

%ψ ω

ωω

( ) 2

0−∞

+∞

∫ <d

Advances in Structural Engineering Vol. 15 No. 5 2012 763

Jianchun Li, Mahbube Subhani and Bijan Samali

Fre

quen

cy

Time

Figure 3. The tiling of CWT in the time-frequency plane

These characteristics make Gabor wavelet very suitablefor damage detection. Although a Gabor wavelet ofGS = 5 (also called Morlet wavelet, GS = ησ) is used bymany researchers (Inoue et al. 1996), lower values of GS

may sometimes give better results. However, it shouldnot be too low, as then it may breach the admissibilitycondition (Kim and Kim 2001). In this paper, Gaussianfunction of order 3 has been used.

3. NUMERICAL MODELING3.1. Choosing the Adequate FrequencyWave propagation in timber is complicated due to thenature and the properties of timber pole. Therefore,numerical study will begin with considering onlytimber without soil to gain a basic understanding ofwave propagation in timber itself. Numerical modelingis accomplished by Finite Element (FEM) analysisusing ANSYS. Free end boundary conditions is realisedby suspending the pole with two pinned rigid links.Impact location is vital for the generation of variouswaves (longitudinal, bending etc). In the numericalmodeling, impact is imparted at the centre of the crosssection at top of the pole, producing mainlylongitudinal waves. A 5 m pole is considered with adiameter of 300 mm at the bottom and tapered to260 mm to the top. This configuration is the same as thetimber pole experimentally tested in the laboratory.Figure 4 shows the mesh set up for the numericalsimulation of the timber pole. Transient analysis (i.e.time history analysis) is conducted on the numericalmodel and recorded impulse loading from the hammertest during the experimental test was used as the loadinginput. Figure 5 shows the time history and frequencycontents of the loading where the input frequency ofvalues 0 to 3000 Hz. To obtain a needed frequencyresolution and wave number, zero padding technique isused.

Figure 6 shows the temporal and spectral signal at alocation of 3 m from the top (impact location) of thepole. It is clearly seen that there are mainly fourdominating frequencies present in the signal, i.e.

437 Hz, 868 Hz, 1,288 Hz and 1,720 Hz. Near both endsof the pole, some additional frequencies exist such as,2,156 Hz, 2,552 Hz (Figure 7) and the stress value at theend of the pole is very small compared to the stresses atthe mid position.

Along the 5 m surface of the timber pole, a total of100 locations (i.e. with an interval of 50 mm) are chosenfor calculation of temporal and spectral signals and up tosix frequencies are chosen to determine thewavenumbers and number of modes corresponding tothe frequency.

764 Advances in Structural Engineering Vol. 15 No. 5 2012

Determination of Embedment Depth of Timber Poles and Piles Using Wavelet Transform

Impact

X

x

Figure 4. Numerical modeling of the timber pole (cross section and

front elevation)

Spectral data hammer× 10415

10

5

00 1000 2000 3000

Frequency

Temporal data

Time (s)

4000 5000

0

−1000

−20000 0.002 0.004 0.006 0.008 0.01

Figure 5. Temporal & spectral signal of hammer

Spectral signal - 3 m× 107

× 104

6

8

4

2

00 1000 2000 3000

Frequency (Hz)

Temporal signal

Time (sec)

4000 5000

4

2

0

−2

−40 0.01 0.02 0.03 0.04 0.05

Figure 6. Temporal & spectral signal of at 3 m

Figure 8 shows wave numbers and number of modesat frequencies of 437 Hz, 1,288 Hz, 1,720 Hz and2,156 Hz. It is clear from the wave number graph that,the first mode is present at the frequency of 437 Hz andthe second to forth modes are present at 1,288, 1,720and 2,156 Hz, respectively. In relation to practicalapplication to field testing, it is noted that accelerationsare often captured rather than stresses. Therefore, itneeds to be verified that the same processing can beconducted using continuous wavelet transform (CWT)on time acceleration data. From the CWT, fourfrequencies are then selected from FFT results asaforementioned to determine the presence of modes andthe results will be used for comparison with those fromstress data.

Figure 9 shows the coefficient plot of the signal atthese four frequencies obtained by using CWT analysisof time displacement data at the location of 0 m. It can beseen clearly that the increase of frequency resulted in the

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Jianchun Li, Mahbube Subhani and Bijan Samali

Spectral signal - 5 m× 1062

1

00 1000 2000 3000

Frequency (Hz)

Time-stress history

Time (sec)

4000 5000

1000

0

−1000

0 0.01 0.02 0.03 0.04 0.05

Figure 7. Temporal & spectral signal at 5 m

10

5

00 0.005 0.01

At 437 hz(a) (b)

(c) (d)

Wave number

Location-stress history

Wave number

Wave number Wave number

Location (cm) Location (cm)

Location (cm) Location (cm)

Location-stress history

× 108

0.015 0.02

8

6

4

2

00 100 200

× 107

300 400 500

3

2

1

00 0.005 0.0150.01

× 108

0.02 0.025 0.03

15

10

5

00 100 300200

× 106

400 500

10

5

00 0.005 0.01

At frequency of 1288 hz

At frequency 1720 hz At frequency 2156 hz

Location-stress history Location-stress history

× 108

0.015 0.02

8

6

1

00 100 200

× 107

300 400 500

2

1.5

1

0.5

00 0.005 0.0150.01

× 108

0.02 0.025 0.03

10

5

00 100 300200

× 106

400 500

Figure 8. Wave number and modes at the frequency of: (a) 437 Hz; (b) 1,288 Hz; (c) 1,720 Hz; (d) 2,156 Hz

signal containing more peaks. In other words, it hasdemonstrated the presence of multi-modes in highfrequency. Moreover, Figure 10 shows the dispersivecurve for the broadband input frequency of the numericalresults. In terms of the input frequencies, the main

frequency band for this case lies between 400 – 2,600 Hzat which the dispersion curve is drawn. At highfrequency, the velocity of each mode approaches theRayleigh surface wave velocity and the result matcheswith known phenomenon (Kolsky 1963). Analytically,

766 Advances in Structural Engineering Vol. 15 No. 5 2012

Determination of Embedment Depth of Timber Poles and Piles Using Wavelet Transform

Analyzed signal (length = 10181)× 10−6

3

(a)

(b)

(d)(c)

(e)

2

1

0

−1

−2

−3

265

398

784

0.001 0.002 0.003 0.004

Ca,b coefficients − coloration mode: init + by scale + abs

Coefficients line − Ca, b for scale a = 181 (frequency = 441.989)

Coefficients line − Ca, b for scale a = 46 (frequency = 1739.130)

0.005 0.006 0.007 0.008 0.009 0.01

× 10−5

× 10−6

0.0005

Coefficients line − Ca, b for scale a = 62 (frequency = 1290.323)

0.5

1× 10−5

0

−0.5

−1

−1.50.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

5

4

3

2

1

0

−1

−2

−3

−4

−8

−6

−4

−2

0

2

4

6

−50.0005

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

Coefficients line − Ca, b for scale a = 37 (frequency = 2162.162)× 10−6

−5

−4

−3

−2

−1

0

1

2

3

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

Figure 9. (a) Time signal and CWT of the signal; Coefficient plot at frequency of: (b) 437 Hz; (c) 1,288 Hz; (d) 1720 Hz; (e) 2,156 Hz

Rayleigh surface wave velocity should be 2,500 m/s inthis case, whereas Figure-10 shows the value is 2,200m/s. From this curve, the cut-off frequency of secondmode was found at 1,300 Hz. By knowing the cut-offfrequency of the second mode, it is possible to choose thefrequency below that cut-off frequency from CWT andproduce correct velocity from single wave mode. Thevelocity can also be compared from Figure 10.

4. EXPERIMENTAL RESULTSIn the laboratory testing, a 5 meter timber pole with thesame configuration as that of numerical study was testedwith free-end support condition (i.e. the specimen wassuspended in the air by two sling straps) in the structurallaboratory to benchmark test procedures, investigatemethods and accuracy of results. A modally tunedimpact hammer was used to deliver an impact to thespecimens to ensure adequate excitation frequencyrange and quantification of the impulse force. Figure 11shows the experimental set up. After validating thedeveloped sensing system, acquisition system andrequired software, a series of tests were conducted.Figure 12 shows the stress wave velocity pattern ofselected sensors (sensors 2 to 54) located at 3 m, 3.2 mand 3.4 m from the top, respectively. Figure 13 displaysthe coefficient plot by using CWT as well as the originalpattern from time domain for sensor 2, which is located3m from the top under free-end condition.

Figure 12(a) shows the time lag betweenaccelerometers, when the impact is imparted at the topcentre of the cross section of a pole specimen. In thefigure, the time-velocity curve of accelerometer 2, 3 and5 are shown. Figure 12(b) represents the same curve foraccelerometer 3 and 4, which are at the same position buton different sides of the timber pole. Since longitudinalwaves are less dispersive, the time domain graph shows

Advances in Structural Engineering Vol. 15 No. 5 2012 767

Jianchun Li, Mahbube Subhani and Bijan Samali

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

00 500 1000 1500 2000

Frequency

Dispersive curve

Pha

se v

eloc

ity

2500 3000

Phase velocity 1Phase velocity 2Phase velocity 3Phase velocity 4Phase velocity 5

Figure 10. Dispersive curve

1 2 3

4

5 6Flexible constraint(Zero moment)Accelerometer

Impact hammer3 m 1.6 m

0. 2 m 0.2 m

Figure 11. Experimental setup

6× 10−7

4

2

0

0 0.002

Vel

ocity

(m

/s)

−2

0.004

Time (s)

(a) Time velocity curve of sensor/accelerometer 2, 3, 5

Time velocity s: 2, 3, 5

0.006 0.008 0.01−4

Sensor 2Sensor 3Sensor 5

5× 10−7

4

3

2

1

0 0.005

Vel

ocity

(m

/s)

0

−1

0.01

Time (s)

(b) Time velocity curve of sensor 3 and 4

Time velocity sensor 3 & 4

0.015 0.025 0.030.02−2

Sensor 3Sensor 4

Figure 12. Experimental test results (impact at top)

a relatively clear and consistent pattern [Figure 13(a)].However, the first two or three peaks appear to beoutliers because of the presence of other frequencies. Insuch cases, CWT is much better for signal processing[Figure -13(b)] where various contours reflect thepresence of different frequency components.

Figure 13(c) shows the coefficient graph at thefrequency of 672 Hz. The peaks of high coefficient

value represent the presence of contour. This graph canbe used to determine the stress wave velocity (in thiscase it was found equal to 4,367 m/s), which is veryclose to the theoretical result. The small difference intheoretical and experimental results is mainly due touncertainties in modeling the modulus of elasticity anddensity in the theoretical calculation.

Using surface NDT methods usually requires impactat an end of a specimen in order to produce longitudinalwaves only. However, this requirement is difficult tomeet when applying to pole testing in service and it istherefore of great interest and benefit to investigatealternative impact locations and to gain understandingon the influence of these alternatives to the results oftesting and subsequent analysis. A steel bracket wasfabricated and firmly attached to specimens during testsin order to provide an impact point from the middle ofthe specimen. Figure 14 shows the stress wave velocitypattern for three different sensors as aforementionedunder free-end condition and impact from the middle.Figure 16 displays the coefficient plot using wavelet

transform as well as the original pattern from timedomain for sensor 2, which is located 3 m from the topunder free-end condition.

In this case, impact is imparted vertically on thebracket attached to the timber pole. Figure 14 shows theexperimental set up for this case. Figure 15(a) showsclear difference between accelerometers located withinthe distance from the impact point but on the oppositeside of cross sectional area. This observation confirmedthe presence of bending waves in middle-impact case. Inaddition, the longitudinal wave from middle-impact alsoproduces two components (Figure 14) traveling toopposite sides of the timber pole (top and bottom) andreflects back from both ends. As the bending andsurface waves are highly dispersive in nature, the timedomain alone cannot provide the required information[Figure 16(a)]. CWT analysis becomes crucial in thiscase [Figure 16(b)].

Figure 17 shows experimental set up of a damagedpole and Figure 18 shows the testing results from threedifferent sensors under free-end condition. The artificial

768 Advances in Structural Engineering Vol. 15 No. 5 2012

Determination of Embedment Depth of Timber Poles and Piles Using Wavelet Transform

1 2 3

4

5 6Accelerometer

Impact hammer

3 m 1.6 m

0.2 m 0.2 m

Figure 14. Experimental set up

3

2

1

0

−1

−2

−3

2

1.5

1

0.5

0

−0.5

−1

−1.5

× 10−7

0.0020 0.004Time (s)

0.006 0.008 0.01

0.0020 0.004Time (s)

0.006 0.008 0.01

Vel

ocity

(m

/s)

Vel

ocity

(m

/s)

Time velocity s: 2, 3, 5

× 10−7 Time velocity s: 3, 4

(a)

(b)

Sensor 2Sensor 3Sensor 5

Sensor 3 Sensor 4

Figure 15. Time-velocity curve of: (a) sensor 2, 3 and 5; (b) sensor

3 and 4

Figure 13. (a) The original signal of sensor 2; (b) The CWT of the

signal at sensor 2; (c) Coefficient line of sensor 2 at the frequency

of 672.269 Hz

4× 10−7

(a)

(b)

(c)

Analyzed signal (length = 15201)

× 10−6 Coefficients line - Ca, b for scale a = 595 (frequency = 672.269)

Ca, b coefficients - Coloration mode: init + by scale + abs

3210

13467

100133166199232265298331364397430463496529562595628

2000

−1

0

5

−52000 4000 6000 8000 10000 12000 14000

4000 6000 8000 10000 12000 14000

original pattern from time domain for sensor 2, which islocated at 3 m from the top under free-end condition.The stress wave velocity can be calculated from the timelag between the accelerometers (Figure 18).

The wave passing through the timber pole is actuallydivided in two parts when it propagates to the damagelocation that is a reflective wave and a continuallypropagated wave through the undamaged part seen fromthe graph. As the time difference between the receivedsignal at accelerometer 2 and damage is known fromthis graph; and the velocity is also known, the distancecan be found. The distance is found to be 1.13 m fromaccelerometer 2 which is 1 m in real case, an error ofmere 13%.

5. CONCLUSIONIn this paper, the method of FFT and DFT is used to findout various modes in a longitudinally dominated wavesignal. Dispersive curve is also generated numericallyfor impact load to find out velocities at different modespresent in the signal. Continuous Wavelet Transform(CWT) technique has been studied for the determinationof wave mode of timber poles through numericalinvestigations. The results have demonstrated that CWTis an effective tool for processing stress wave signals foridentifying reflective waves, especially under morecomplicated situations such as impact at middle ofspecimens and existence of damage. For these cases,traditional time domain analyses cannot providesatisfactory results due to multiple wave modes andbroadband frequency excitation. Further investigation is

Advances in Structural Engineering Vol. 15 No. 5 2012 769

Jianchun Li, Mahbube Subhani and Bijan Samali

× 10−7(a)

(b)

(c)

Analyzed signal (length = 10741)

Coefficients line - Ca, b for scale a = 56 (frequency = 7142.857)

Ca, b coefficients - Coloration mode: init + by scale + abs

13467

100133166199232265298331364397430463496529562595628

−2

0

2

500 1000 1500 2000 2500 3000 3500 4000 4500

500 1000 1500 2000 2500 3000 3500 4000 4500

0.5

0

0.5

−1

1× 10−6

Figure 16. (a) The original signal of sensor 2; (b) The CWT of the

signal at sensor 2; (c) Coefficient line of sensor 2 at the frequency

of 7,142 Hz

1 2 3

4

5 6Flexible constraint(Zero moment)Accelerometer

Impact hammer3 m 1.6 m

0.2 m 0.2 m

Figure 17. Experimental set up

0.0020 0.004Time (s)

0.006 0.008 0.01

× 10−7 Time velocity s: 2, 3, 5

4

3

2

1

0

−1

−2

5

−3

Vel

ocity

(m

/s)

Sensor 2Sensor 3Sensor 5

Figure 18. Time velocity graph of sensor 2, 3 and 5

× 10−7 Analyzed signal (length = 15301)

Coefficients line - Ca, b for scale a = 639 (frequency = 625.978)× 10−6

2000 4000 6000 8000 10000 12000 14000

2000

Damage

Receive

4000 6000 8000 10000 12000 14000

0

2

4

341

67100133166199232265298331364397430463496529562595628

0

5

−5

Ca, b coefficients - Coloration mode: init + by scale + abs

(a)

(b)

(c)

damage was introduced by removing the half section of1 m length of the specimen from the bottom. This is tosimulate termite damage, which is often found in thetimber poles in-service. Figure 19 displays thecoefficient plot using wavelet transform as well as the

Figure 19. (a) The original signal of sensor 2; (b) The CWT of the

signal at sensor 2; (c) Coefficient line of sensor 2 at the frequency

of 625.978 Hz

needed to gain full understanding on effects of thegeotechnical conditions and uncertainties of fieldtesting.

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NOTATIONψ(u,s)(t) ψ(t) is translated by the translation parameter

u and dilated by the scaling parameter s* complex conjugateψ~(ω) Fourier transform of ψ(t)ψ(t) mother wavelet of wavelet transform in time

domaina accelerationC1 velocity of dilatational wave in infinite mediaC2 velocity of distortional wave in infinite mediaCL traditional bar velocityCp phase velocityCs velocity of Rayleigh surface waveE modulus of elasticityF forcef(t) any square integral functionGs Gabor wavelet transformJ0 Bessel function of the first kind of order zeroJ1 Bessel function of the first kind of order onek wavenumberm massr radius of the cylinders time scaling factor or dilationt timeu the time shift or translationWf (u,s) continuous wavelet transform w.r to u and sα (ω2/C1

2)-k2 in Pocchammer-Chree equationα1 ratio between the velocity of dilatational

wave and wave of distortion.β (ω2/C2

2)-k2 in Pocchammer-Chree equationε strainη centre frequency of ψ~(ω)κ1 ratio between the velocity of surface wave

and wave of distortion.λ Lame’s constantµ shear modulus of elasticityρ density of the materialσ stressσt time spread of ψ(t)σw frequency spread of ψ(t)ω frequency

770 Advances in Structural Engineering Vol. 15 No. 5 2012

Determination of Embedment Depth of Timber Poles and Piles Using Wavelet Transform