Ultrasonics xxx (2014) xxx–xxx

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Ultrasonics

journal homepage: www.elsevier .com/locate /ul t ras

Reference-free corrosion damage diagnosis in steel strands usingguided ultrasonic waves

http://dx.doi.org/10.1016/j.ultras.2014.11.0110041-624X/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: alirezaf@buffalo.edu (A. Farhidzadeh), ssalamon@buffalo.edu

(S. Salamone).

Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonicUltrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011

Alireza Farhidzadeh, Salvatore Salamone ⇑Smart Structures Research Laboratory, Department of Civil, Structural, and Environmental Engineering, University at Buffalo, Buffalo, NY 14228, United States

a r t i c l e i n f o

Article history:Received 3 July 2014Received in revised form 29 September 2014Accepted 21 November 2014Available online xxxx

Keywords:CorrosionSteel strandGuided ultrasonic waveWavelet transformUncertainty analysis

a b s t r a c t

This study presents a nondestructive evaluation method based on guided ultrasonic waves (GUW) toquantify corrosion damage of prestressing steel strands. Specifically, a reference-free algorithm isproposed to estimate the strand’s cross-section loss by using dispersion curves, continuous wavelettransform, and wave velocity measurements. Accelerated corrosion tests are carried out to validate theproposed approach. Furthermore, the propagation of Heisenberg uncertainty to diameter measurementis also investigated. The method can reasonably estimate the wires’ diameter without any baseline asa reference.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Multiwire steel strands are widely used in civil structures suchas cable-stayed bridges, prestressed concrete structures, and re-centering systems. Despite this increase in usage, the corrosionof the multiwire strands has become a concern for designers,owners and regulators. Many of these structures have sufferedthe failure of strands due to corrosion [1–3]. Extensive inspectionand maintenance/repair programs have been established, withattendant direct costs and significant indirect costs due to businessinterruption [4]. Evaluation of strands is technically challenging. Inmany structures, inaccessibility of steel strands, eventuate indifficult, expensive and often inconclusive evaluation. Several non-destructive evaluation (NDE) techniques for evaluating the condi-tion of strands have been developed to address these issues inthe past few years [5]. Half-Cell potential [6], time domain reflec-tometry (TDR) [7], linear polarization resistance (LPR) sensors [8],magnetic flux [9], and acoustic emission [10] are some of the mostcommonly used NDE methods for corrosion diagnosis. Althoughthese techniques have shown promise, very few if any are capableto quantify the cross-sectional loss. A technique that shows poten-tial to quantify the extent of corrosion (e.g., cross-sectional loss) isbased on guided ultrasonic waves (GUWs). As opposed to the

waves used in traditional impact-echo (IE), that propagate in 3-Dwithin the structure, GUWs propagate along the strand itself byexploiting its waveguide geometry [11]. Most previous researchesattempted to use GUWs to detect and assess corrosion damage onreinforcing bars in reinforced concrete (RC) structures [12–15] andsteel strands in prestressed concrete structures [4,16–20]. Forinstance, to diagnose corrosion in reinforcing bars, energy andattenuation characteristics of longitudinal and flexural GUWmodes were used [13,14]. Other methods based on time of flight(ToF) of the first packets were also proposed to investigate variouslevels of corrosion in reinforcing bars [21]. Longitudinal GUWswere also used to monitor pitting and delamination in steel rebars[15]. To monitor the corrosion process in post-tensioned concretebeams, fractal analysis of GUWs was investigated [4].

In this study, a reference-free algorithm is proposed to quantifythe extent of corrosion through estimating the cross-section lossusing GUW measurements. An experimental setup was designedto carry out an accelerated corrosion test on a loaded strand. Thediameter of the strand’s wires was estimated using the continuouswavelet transform (CWT) [22] of the GUWs. Furthermore, theuncertainty associated with estimated diameter, which originatesfrom the Heisenberg principle in CWT [23], was quantified. In thenext section, the behavior of GUW in rods and the effect of cross-section loss on dispersion curves, followed by a short introductionon CWT are presented. The details of experimental setup are thenexplained. The results of diameter measurement using GUW arediscussed and finally the conclusion is given.

waves,

0 100 200 300 400 500 600 700 800 900 10001500

2000

2500

3000

3500

4000

4500

5000

Frequency [kHz]

Gro

up v

eloc

ity [m

/sec

]

d=5.0mmd=4.5mmd=4.0mmd=3.5mmd=3.0mmd=2.5mm

Fig. 2. Dispersion curves for the first longitudinal mode L(0,1) in steel rods withvarious diameters.

2 A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx

2. Theory

2.1. Guided ultrasonic waves in rods

A guided ultrasonic wave (GUW) is generated whenever anultrasound propagates into a bounded medium [24]. GUWs areknown to be multimode (many vibrating modes can propagatesimultaneously) and dispersive (the propagation velocity dependson the wave frequency f). In cylindrical waveguides, such as rods,three different modes can propagate: longitudinal, flexural, andtorsional [24,25]. The dispersive behavior of these modes is repre-sented by the dispersion curves like the ones shown in Fig. 1. Thesecurves describe the relationship between wave velocity and fre-quency, and can be calculated analytically or they can be computedby approximate solutions derived from numerical methods [24]. Inthis work a MATLAB open source toolbox, PCdisp (Pochhammer-Chree dispersion) [26,27] was used to generate the dispersioncurves (Fig. 1).

The longitudinal modes have received significant interest in thepast few years, for the nondestructive evaluation of cylindricalwaveguides, mostly because the flexural and torsional modesexperience high attenuation during the propagation phenomena[16,18,28–30].

2.2. Effect of corrosion on GUWs

Corrosion is usually an electrochemical oxidation–reductionprocess that converts the major component of steel, iron (Fe), toferrous hydroxide (Fe(OH)2). A consequence of this process is thereduction in member cross sectional area, which eventually mayaffects its structural performance. To simulate the effect of corro-sion and investigate how the reduction in a rod’s cross sectionalarea affects the guided wave propagation, the dispersion curvesfor the first longitudinal mode L(0,1), were generated for variousdiameters, ranging from 5 mm to 2.5 mm. The results are shownin Fig. 2. It can be observed that in a frequency range between 0and 600 kHz the group velocity increases as the cross-sectionalarea decreases. Moreover, certain frequencies (e.g., 500 kHz) pro-vide a larger sensitivity to diameter changes than other frequencies(e.g., 300 kHz). It is worth to mention that guided waves thicknessmeasurement methodologies based on group velocity changeshave been studied by some researchers for other applications suchas plate-like structures [31–35].

0 100 200 300 400 500 600 700 800 900 10000

1000

2000

3000

4000

5000

6000

Frequency [kHz]

Gro

up sp

eed

[m/s

]

T(0,1)

L(0,1)

L(0,2)

F(1,1)

F(1,2)F(1,3)

Fig. 1. Group velocity dispersion curves for a 5 mm diameter steel rod.

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Fig. 3 shows the group velocity of the L(0,1) mode as a functionof the frequency-diameter product (i.e., v = F(fd)). Therefore, theinverse problem, which is to calculate the diameter given the dis-persion curves, can be solved by measuring the velocity of a certainfrequency component in the signal. In order to limit the effects ofthe high order modes the excitation frequency was selected in afrequency range below the first cut-off frequency. In this work, atime–frequency analysis (continuous wavelet transform) was usedto measure the group velocity of the L(0,1) mode.

2.3. Continuous wavelet transform

Continuous wavelet transform (CWT) for analyzing non-sta-tionary signals has received significant interest in the last fewyears, due to its ability to extract signal time and frequency infor-mation simultaneously [23]. The CWT of a time domain signal f(t)is given by Ref. [23],

WTðs; bÞ ¼ 1ffiffiffiffiffijsj

p Z 1

�1f ðtÞw � t � b

s

� �dt ð2:1Þ

where w⁄(t) denotes the complex conjugate of the mother waveletw(t), s is the dilation parameter (scale), and b is the translation

0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

5000

6000

Gro

up V

eloc

ity [m

/sec

]

Frequency-diameter [kHz-mm]

(470kHz-5.002mm,3440)

at constant frequency:d↓ → v↑

Fig. 3. Dispersion curve for the first longitudinal mode L(0,1) in a steel rod versusproduct of frequency and diameter.

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

Table 1Geometrical and mechanical characteristics of the tested strand.

Geometrical characteristics Mechanical characteristics

Core wire diameter dc (mm) 5.2 Young’s modulus E (GPa) 196Helical wire diameter dh

(mm)5.0 Poisson’s ratio (v) 0.29

Strand diameter D (mm) 15.2 Yielding load (kN) 203Pitch of helical wire p (mm) 230 Ultimate tensile strength

(MPa)1860

Lay angle b (�) 7.9 Linear weight (kg/m) 1.10

A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx 3

parameter (location). The energy density of a signal is representedby the Power Spectral Density (PSD) as [23],

PSD ¼ 10� logðjWTðs; bÞj2Þ ð2:2Þ

where

jWTðs; bÞj2 ¼WTðs; bÞWT � ðs; bÞ ð2:3Þ

The methods based on wavelet transform have an inherentsource of uncertainty that originates from the Heisenberg principle[23]. Based on this principle, the resolution in time and frequencydomains is limited according to the following relationship [23]:

r2t r

2f P

14

ð2:4Þ

where r2t and r2

f are time and frequency variances, respectively.These quantities represent the local resolution of CWT. Thisinequality reaches its minimum value of 1/4 when the signal is

Fig. 4. The loading device designed for the accelerated corrosion test

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Gaussian [36]. In this study, a complex Morlet was used as motherwavelet [23], defined as:

wðtÞ ¼ 1ffiffiffiffiffiffiffiffipf b

p expð2ipf ctÞ expt2

f b

� �ð2:5Þ

where fb is the bandwidth parameter and fc is the wavelet centerfrequency. Morlet wavelet is a sine wave multiplied by a GaussianWindow. The Morlet wavelet provides a desirable compromisebetween time and frequency resolution [22]. The complex Morletparameters were set by trial and error and fixed (fc = 5 Hz, fb = 2 Hz).For a given signal with sampling frequency fs, the complex Morlettransform yields the following equations for rt and rf [23]:

rt ¼sffiffiffiffiffif b

p2f s

ð2:6Þ

rf ¼f s

sffiffiffiffiffif b

p ð2:7Þ

in which the dilation parameter (i.e., s) is inversely proportional tothe local frequency �f according to the following equation:

s ¼ f sf c�f

ð2:8Þ

3. Experiments

An experimental setup was designed to validate the proposedapproach. Specifically, a loading apparatus was designed and built

on a steel strand (a) plan view, (b) cross section and side views.

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

Fig. 5. Experimental setup overview.

4 A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx

to apply a tensile load to a seven-wire strand. Then, acceleratedcorrosion tests were carried out on the axially loaded strand.

3.1. Strand characteristics

In this study, a 15.2 mm Grade 270 steel tendon consisting ofsix 5.0 mm diameter wires spirally wrapped around a central5.2 mm diameter wire was considered (see Table 1 for details).The length of each helical wire was assumed equal to the lengthof the central wire since the lay angle b was very small (the differ-ence was as small as 0.89%) [37,38]. In addition, considering thatthe pitch length (230 mm) was much longer than the typical wave-length in the signals (smaller than 10 mm [37]), each wire wastreated as an individual rod [16]. Beard et al. [16] observed thatguided waves in individual wires of a strand behave similar to sin-gle wires with minor discrepancies associated to inter-wirecontact.

3.2. Loading apparatus

Drawings of the loading apparatus are shown in Fig. 4. It con-sists of two I-shape rigid beams (web: 76.2 cm � 7.6 cm � 1.9 cm(3000 � 300 � 3/400), flanges: 50.8 cm � 1.3 cm (2000 � 400 � 1/200)) con-nected with two u11

800 all thread steel bars. In the middle of each

beam’s web, a 5/800 hole was drilled; the strand passed throughthe holes and was fixed using wedge anchorages. The load wasapplied by tightening the nuts, which pushes the rigid beamsand stretches the strand; a large wrench with handle extensionwas used for this purpose. Therefore, the all thread bars are undercompressive force. To prevent bucking, two wood beams were con-nected to the bars using u-shaped hooks (see Fig. 5). Two hollow

Fig. 6. The instruction of m

Low Frequency

NI PXI-5412 AWG

Piezo Linear Amplifier

PICO sensor Amplifier

PICO sensor

PXI-5105 digi�zer

tran

smis

sion

LabVIEW

Fig. 7. Through-transmission testing for low-

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load cells passed through the all thread bars and were locatedbetween the beam’s web and the nuts’ washers. The tensile loadon the strands was continuously measured during the corrosiontest as the summation of readings from these two load cells. Thestrand was initially loaded up to 89 kN (=20 kips � 40% of yieldingforce) which was the designed capacity of the loading apparatus.

3.3. Accelerated corrosion test

An accelerated corrosion test [39] was carried out using theimpressed current technique. Specifically, the strand wasimmersed in 3.5% sodium chloride (NaCl) solution. A power supplyequipped with built-in ammeter and potentiometer was used toimpress a direct current (DC) to the strand to induce significantcorrosion in a short period of time. The direction of the currentwas adjusted so that the steel strand served as the anode, whileanother metal, superior than steel in electro-chemical series,served as the cathode [40] (see Fig. 4). A constant voltage of0.16 V was maintained across the solution and the specimen untilthe 18th day, and then was increased to 0.32 V to failure. Fig. 5shows an overview of the experimental setup that consists of load-ing apparatus, water tank, power supply, and the steel strand. Tomonitor the corrosion process, a video camera with recording rateof 1 frame per minute was mounted on top of the specimen.

3.4. Mass loss measurement

The mass loss during the corrosion tests was determinedthrough a Gravimetric analysis in which the iron oxide or rust(Fe2O3) was the analyte. The ferrous ions (Fe2+) precipitate throughcombination with hydroxyl ions (OH�) and form ferrous hydroxide(Fe(OH)2) that are fine solid particles. These particles were col-lected by filtration, dried, and weighted. When dried, rust (Fe2O3)is produced. The weights of filters were subtracted to obtain theweight of rust. Finally, to calculate the mass loss of iron, themeasured weight was multiplied by a correction factor of 0.7.The correction factor is derived from the mass portion of ironin rust (i.e., (2 � 56)/(2 � 56 + 3 � 16) = 0.7). This process isschematically illustrated in Fig. 6.

3.5. Ultrasonic test equipment

Ultrasonic tests were performed using broadband piezoelectrictransducers in a through-transmission (i.e., pitch-catch) configura-

ass loss measurement.

High Frequency

OscilloscopeLecroy wave

runner 44Xi-A

Olympus 5077PR Square pulser

V1091 transducers

V1091 transducers

Inbuilt amplifica�on

transmission

and high-frequency longitudinal waves.

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

(b)(a)

S1S2

S4S3

S1 and S2: PICO sensorS3 and S4: 5MHz sensor

Magnet holder

5MHz sensor

PICO sensor3.5 cycles

Fig. 8. Instrumenting a multi wire steel strand with transducers attached to the cross-section of helical wires (a) sensor layout, and (b) sensor coupling.

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20 22

Mas

s los

s [%

]

Time [day]

Faraday's lawMass measurment

Fig. 9. Mass loss versus time during accelerated corrosion test.

0

10

20

30

40

50

60

70

80

90

100

Time [day]

Load

[kN

]

1st wire failure

2nd and 3rd wires failures

2220181412107654321

Fig. 10. Load versus time during accelerated corrosion test.

A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx 5

tion. A schematic drawing of the experimental setup is shown inFig. 7. Two through-transmission setup were used to investigatethe low frequency modes (below the first cut-off frequency), andthe higher frequency modes (above the first cut-off frequency).For the low frequency modes the measurement equipment usedfor signal generation and data acquisition was mainly constitutedby a National Instruments (NI), modular PXI 1042 unit. The unitincluded an arbitrary waveform generator card (PXI 5412) andone, 20GS/s 12-bit multi-channel digitizers (PXI 5105). A high volt-age amplifier was used to amplify the excitation to the ultrasonictransmitters, while a preamplifier (Olympus 5660C) was used toamplify the ultrasonic receivers. Two ultra-mini broadband

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transducers (Physical Acoustic Corporation PICO), which are sensi-tive in a frequency range between 200 kHz and 700 kHz, were usedto generate and receive ultrasonic waves. LabVIEW software wasused to control the sensors, and acquire the data.

For the high frequency modes a commercial pulser–receiver(Olympus 5077PR) was used to drive two piezoelectric transducerswith a central frequency of 5 MHz (Olympus V1091) with a squarewave pulse of selectable width. The received signals were recordedusing an oscilloscope (Lecroy Wave Runner 44Xi-A).

In both experimental setups, transducers were carefullymounted as perpendicular to the wire’s cross section as possibleto measure the longitudinal wave motions. The diameter of thetransducers was about the same as the diameter of the single wireso that there was no contact between the transducers and the adja-cent wires, as shown in Fig. 8.

4. Results

Faraday’s law was used to predict the strand mass loss. Faraday’slaw provides a relationship between the time of an applied currentand the amount of steel weight loss according to the followingequation [47]:

m ¼ AZ � F

ZIðtÞdt ð4:1Þ

where m is the mass in grams, A is the atomic mass of iron (56 g),Z is the valence of the reacting electrode (2 for iron), F is theFaraday’s constant (96,500 A s), t is time in seconds, and I(t) is thecurrent in ampere. Therefore the theoretical percentage of massloss, Dm, was calculated as:

Dm ¼ 100%�mM

ð4:2Þ

where M is the total mass of the strand. Fig. 9 shows the measuredresults based on gravimetric mass loss and the corresponding theo-retical mass loss based on Faraday’s law. Faraday’s law predictedwell the degrees of corrosion (mass loss) with the impressed cur-rent technique.

Fig. 10 shows the strand axial load measurements through theload cells during the accelerated corrosion test. The failure of threewires was recorded at day 21st. It is noteworthy that load dropafter the failure of the 2nd and 3rd wire was almost twice the loaddrop after the 1st wire breakage. Therefore the load was equallydistributed on each wire. The influence of the impressed currenton the load drop rate can be observed at the 18th day, whenimpressed current was increased from 0.5 A to 1.5 A.

Fig. 11 illustrates the strand condition at different levels of cor-rosion. The measured mass loss is also reported for each level ofcorrosion. The location of the wire breakage is indicated witharrows.

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

No corrosion (Mass loss = 0.0%)

Light corrosion (Mass loss = 0.19%)

Pitting (Mass loss = 1.26%)

Heavy Pitting (Mass loss = 2.72%)

Cross section loss (Mass loss = 8.38%)

Fracture (Mass loss = 21.34%)

Fig. 11. Visual inspection of corrosion progress in a steel strand.

6 A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx

4.1. Attenuation measurements

Attenuation coefficients were calculated using the followingequation [30]:

ai ½dB=m� ¼ 20L

logA0

Aið4:3Þ

where ai is the attenuation coefficient in dB/m, L is the length ofstrand, A0 is the amplitude of the signal at pristine state, and Ai isthe amplitude at ith day. Two PICO sensors in a through-transmis-sion configuration were used to generate and receive GUWs on thesame peripheral helical wire. A sweep frequency test between300 kHz and 700 kHz was performed at different levels of corrosion.Fig. 12a shows the reduction in amplitude of the signals versusmass loss and excitation frequency. Fig. 12b illustrates the attenua-tion coefficients. At about 20% mass loss, the attenuation coefficientincreased to 30 dB/m. It is interesting to note that the attenuationcoefficient is quite constant for all frequencies below 15% mass loss.

To investigate the attenuation of the higher longitudinal modesduring the corrosion process, two piezoelectric broadband trans-ducers with a central frequency of 5 MHz were used in athrough-transmission-configuration, as described in Section 3.4.Fig. 13 shows the CWT of the received signals in day 1, 3, 4, and5. For the first day, the CWT was estimated and compared withnumerical results of dispersion curves obtained from the PCdisptoolbox. There is a clear correspondence between theory (solidlines) and experimental results (CWT). After five days (less than3% mass loss) only the L(0,1) mode was left.

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4.2. Reference-free diameter estimation

A 3.5-cycle, Hanning-modulated toneburst was used as theactuating signal. For the selection of the excitation frequency aswept of frequencies were performed between 300 kHz and700 kHz. The dominant L(0,1) mode was received and analyzed.Fig. 14 shows the signal strengths (RMS) during the corrosion test.The highest RMS was obtained at 440 kHz. Therefore, the fre-quency of 440 kHz was selected as excitation frequency.

A typical received signal and corresponding CWT of the signalbefore the corrosion test (i.e., pristine condition) is shown inFig. 15. CWT was compared with the dispersion curves resultingfrom the numerical simulation described in Section 2.1. It is worthnoting a clear correspondence between theory (dashed line) andexperimental results (CWT map), which indicate that the first lon-gitudinal mode is the predominant mode of displacement in thereceived signal. To visualize the Heisenberg’s uncertainty principlediscussed in Section 2.3, the time and frequency standard devia-tions (i.e., rt and rf in Eqs. (2.6) and (2.7)) are also indicated onthe Heisenberg box [23]. To estimate the wire diameter at the pris-tine state (i.e., 5 mm), the maximum value of CWT, called the peakridge, was identified. The projection of the peak ridge on frequencyand time axes corresponds to the dominant frequency �f and time-of-flight (ToF) �t, respectively. At the pristine state, the peak ridgewas at �t ¼ 357 ls and �f ¼ 470 kHz (see Fig. 15). Given the strandlength of 1.23 m, the velocity of 3440 m/s was calculated.Therefore, the dispersion curves shown in Fig. 3 were used to esti-mate the diameter (i.e., 5.002 mm). There is a clear correspondencebetween the actual value (5 mm) and the estimated value(5.002 mm).

Fig. 16 depicts the time waveforms during the corrosion test atdays 3, 6, 10, and 18. As expected, the energy was significantlyattenuated due to the corrosion process as well as the inter-wirecontacts. However, the wave velocities of the first wave packets,did not shows significantly changes (see the left zoomed-in outsetplot) while the time shift was evident on the high energy part ofthe waveforms (see the right zoomed-in outset plot). This shift ismainly caused by the diameter reduction and its impact on wavevelocity at certain frequencies. Fig. 17 shows a typical waveformat day 6th along with its corresponding CWT scalogram. The firstpacket corresponds to an approximate frequency range of320–360 kHz at which the velocity is slightly sensitive to diameter(see Fig. 2). On the contrary, the high energy packet corresponds toa higher frequency component (i.e., 470 kHz). This part of the sig-nal is used for diameter estimation.

The reference-free algorithm was repeated every day to esti-mate the average cross-section loss during the corrosion process.To take into account the fact that, the experimental setup wasdesigned to corrode just a portion of strand, some modificationswere needed to find the average velocity in the corroded part. Atthis aims the strand was divided in three regions: (1) left end sideof the salt water tank with length l1, (2) inside the salt water tankwith length l2, (3) right end side of the salt water tank with lengthl3. The middle part (i.e., inside the salt water tank) was the corrod-ing part of the strand, while the other two regions with length l1and l3 remained pristine during the test. Therefore the arrival time�t was estimated as:

�t ¼X3

i¼1

ti ¼ l1=v1 þ l2=�v2 þ l3=v1 ð4:4Þ

where ti is the ToF at each region, and l1, l2, and l3, are constant (i.e.,l1 = l3 = 31 cm, l2 = 61 cm). The wave velocity v1 considered for thepristine parts was obtained from the dispersion curves (Fig. 3) atd = 5 mm. Since the corrosion was not perfectly uniform, we esti-mated an average velocity �v2 in the middle part of the strand as:

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

(a) (b)

300400

500600

700800

0

5

10

15

20

0

0.05

0.1

0.15

0.2

Mass loss [%]Frequency [kHz]

Am

plitu

de [v

]

300400

500600

700800 0

510

1520-10

0

10

20

30

40

Mass loss [%]Frequency [kHz]

Atte

nuat

ion

[db/

m]

Fig. 12. Effect of corrosion on (a) signal amplitude, and (b) attenuation coefficient.

Fig. 13. Effect of corrosion on higher longitudinal modes.

A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx 7

�v2 ¼l2

�t � 2l1=v1ð4:5Þ

Having the velocity �v2 and frequency �f , the average diameter ofa wire in the corroding part was estimated from the L(0,1) modedispersion curve in Fig. 3. This procedure was repeated at differentstages of corrosion and the results are presented in Fig. 18. This fig-ure shows a reasonable correspondence between the estimated

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and measured diameter. The estimation error increases after day14th when significant mass loss was observed. The larger error(i.e., 10%) was recorded at the 20th day.

5. Uncertainty quantification

The most critical challenge here is to provide a quantitativeassessment of how closely our estimates (i.e., diameter) reflect

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

300 350 400 450 500 550 600 650 7000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Frequency [kHz]

RM

S [V

]

Day 0Day 3Day 7Day 10Day 14Day 18Day 20

Fig. 14. RMS of the received waveforms in sweep frequency test during theaccelerated corrosion test.

Fig. 15. (a) Time-domain signal and (b) spectrogram of signal taken from a 1.23length rod (5 mm diameter) through-transmission. The dispersion curve for L(0,1),time and frequency of the ridge as well as Heisenberg uncertainties aresuperimposed.

Fig. 16. Ultrasonic signals at different corrosion states; the velocity does not changeconsiderably for the first packets (bottom left) while it clearly changes at the highenergy part of the signal (bottom right).

Fig. 17. Guided wave signal during corrosion at day 6 (top) and the correspondingCWT scalogram (bottom).

8 A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx

reality in the presence of measurement uncertainty. In general,uncertainties can be caused by random and systematic errors.The random errors are caused by unknown and unpredictablechanges in measurements (e.g., ToF), including instrumentationnoise, temperature changes, etc. [41]. Systematic errors are mostlycaused by the digital signal processing technique (e.g., CWT) usedfor analyzing the time waveforms [41]. This section aims at quan-tifying the systematic uncertainty, caused by the time–frequencyanalysis (i.e., CWT), which arises from the Heisenberg principle,as described in Section 2.3. Thus, both ToF (t) and frequency (f)were treated as Gaussian random variables with variance r2

t andr2

f defined according to Eqs. (2.6) and (2.7), respectively. Also thegroup velocity (v) was treated as a Gaussian random variable withmean �v and variance r2

v defined as (see Appendix A):

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�v ¼ l�t

r2v ¼ l2

�t4 r2t

ð5:1Þ

The two random variables f and v are related by the Pochham-mer-Chree equation v ¼ FðfdÞ (i.e., dispersion curves). Therefore,given the pdf of f and v, the systematic uncertainty associated tothe estimated diameter was evaluated by determining its pdf.Several techniques exist in the literature to approximate the pdfof random variables [42], including Monte Carlo (MC) methods[43], Unscented Transformation (UT) [44], and Delta method[45]. Monte Carlo method, which is the most popular, requireextensive computational resources and effort, and become

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

0 2 4 6 8 10 12 14 16 18 203.8

4

4.2

4.4

4.6

4.8

5

5.2

Time [day]

Dia

met

er [m

m]

EstimatedMeasured

Fig. 18. Diameter of a helical wire estimated using the reference-free approach andmeasured through visual inspection during accelerated corrosion test.

Fig. 19. Heisenberg uncertainty propagation through the nonlinear dispersionfunction of L(0,1) analyzed by Monte Carlo (MC), Unscented Transform (UT) andDelta methods.

2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Diameter [mm]

PDF

HistogramMonte CarloUnscented transformDelta method

Fig. 20. Probability density functions of diameter at the 10th day with differentapproaches.

0 2 4 6 8 10 12 14 16 18 20 223.5

4

4.5

5

5.5

Time [day]

Dia

met

er [m

m]

MeasuredEstimated

Fig. 21. Estimated diameters and their associated uncertainty.

A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx 9

increasingly infeasible for portable NDT devices with small pro-cessing units. The UT and Delta method, are both used to estimatethe statistics of a random variable which undergoes a nonlineartransformation as in F�1ðvÞ [44]. In the UT method, instead of ran-domly selecting multiple data points from a pdf, a few data points(called sigma points) are intelligently selected. The Delta method ismotivated by the assumption that in this study the dispersioncurve can be approximated by a linear function in the interestedvelocity range. More details on the application of these techniques(MC, UT and Delta method) are provided in Appendix A.

Fig. 19 illustrates the application of the aforementioned meth-ods for uncertainty propagation analysis. It focuses the dispersioncurve (shown by dashed-dot line) at the velocity mean �v . Thevelocity pdf is shown by a thick solid line. The sampled points fromthe velocity pdf are shown by narrow gray lines, representing theMC simulation. These points are projected to fd axis through thedispersion curve (i.e., ðfdÞi ¼ F�1ðv iÞ). The UT sigma points are alsosuperimposed using three red1 dotted lines, signifying the signifi-cant reduction in required sampling points. The Delta method is rep-resented by a dashed tangent line on the dispersion curve at v ¼ �v .These methods yield the pdf for product of frequency and diameter

1 For interpretation of color in Fig. 19, the reader is referred to the web version ofthis article.

Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free cUltrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011

(i.e., fd � Nðfd; r2fdÞ) that is shown by a black narrow line. Given the

parameters of fd pdf, the diameter’s mean and variance are calcu-lated via the following equations:

�d ¼ fd=�f

r2d ¼

r2fd��d2r2

f�f 2þr2

f

ð5:2Þ

Fig. 20 shows the estimated pdf of diameter using the afore-mentioned methods at the 10th day as an example. The histogramand pdf obtained using MC method are shown by vertical bars anda dashed line. The solid line represents the pdf of the diameterusing UT, and the dotted line is that of Delta method. There is aclear correspondence between the pdf obtained with all threemethods. However, UT and Delta method provides significant com-putational cost reduction compared to Monte Carlo method.

Finally Fig. 21 illustrates the estimated diameter’s mean andstandard deviation as well as the measured diameter during thecorrosion test. It is remarkable to note that the difference betweenmeasured and estimated diameter is always less than one standarddeviation. The standard deviations are approximately0.7 ± 0.1 mm. The total diameter reduction is less than 1.5 mmand this evidence the high resolution of the proposed velocity-based approach to estimate the diameter.

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

10 A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx

6. Conclusions

This study aims at designing a reference-free nondestructiveevaluation technique for corrosion damage estimation in pre-stressed seven wires steel strands using guided ultrasonic waves.A small-scale experiment was designed to investigate corrosionprocess and behavior of ultrasonic waves in under accelerated cor-rosion. Axial load and mass loss were measured during the test.Attenuation coefficients were monitored and reached to 35 dB atcritical stage of corrosion. A new method based on the velocity ofcertain velocity components was proposed and validated to esti-mate the cross-section loss. Continuous wavelet transform wasused to localize the highest energy content in time–frequencydomain. Having the frequency and ToF (that yields the velocity),the diameter was estimated using the numerically derived disper-sion function for the first longitudinal mode. This function corre-lates velocity to multiplication of frequency and diameter. Inaddition, an uncertainty analysis was carried out to study propaga-tion of time and frequency uncertainties from Heisenberg principleto the diameter calculation. The proposed algorithm does not needa baseline as a reference and directly measure the diameter viathrough-transmission configuration in ultrasonic testing. The algo-rithm was validated experimentally and the results conformedvery well to the diameter measurements during visual inspection.

Acknowledgments

Funding provided by the Research Foundation of SUNY throughthe research collaboration fund. Any opinions, findings, conclu-sions or recommendations expressed are those of the author(s)and do not necessarily reflect the views of the RF/SUNY.

Appendix A

A.1. Velocity pdf

In general, the mean and variance of the reciprocal of a randomvariable can be obtained using its equivalent Taylor series at meanvalue as follows: denoting the expected value (mean) of a randomvariable t as E½t� ¼ �t and defining a random variable v as a functionof t as vðtÞ ¼ l

t, we have:

Taylor series : vðt ¼ �tÞ ¼ lt

����t¼�t

¼ l1�t� t � �t

�t2þ ðt �

�tÞ2�t3

þ . . .

!ðA:1Þ

Thus the expected value of v can be found as:

E½v � ¼ l1�tþ r2

t�t3þ . . .

� �ðA:2Þ

In our case, since the variance in time, r2t ; is very small in com-

parison to the measured time �t, we can eliminate the second termin Eq. (A.2) and simply define the velocity mean as:

�v ¼ E½v � ¼ l�t

ðA:3Þ

and the velocity variance as,

r2v ¼ E½ðv � �vÞ2� ¼ l2

�t4r2

t ðA:4Þ

A.2. Monte Carlo simulation

The Monte Carlo simulation samples a large number of datafrom the underlying probability space to generate a family of test

Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free cUltrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011

points random, feeds forward the samples individually throughthe exact nonlinear function to find the output data deterministi-cally, and finally evaluates the pdf of outputs [43]. In this study,a large number of data points (>10,000) were sampled from thevelocity pdf (v � Nð�v;r2

v Þ; v i; i ¼ 1; . . . ;n), and were fed into theinverse of dispersion function to find (fd)i deterministically (i.e.,ðfdÞi ¼ F�1ðv iÞ). Next, another set of n data points are randomlysampled from the frequency pdf (i.e., f � Nð�f ;r2

f Þ; f i; i ¼ 1; . . . ;n).Assuming statistical independency between frequency and diame-ter, the diameter di is found (di ¼ ðfdÞi=f i). Therefore, the diameterpdf parameters, �d and r2

d ; are computed having the set of di’s. It isworth to mention that the outliers are eliminated from calculationof mean and variance (i.e., remove di if jdi � �dj > 3rd and recalcu-late �d and r2

d) since the number of sampled points were limitedand outliers could cause error.

A.3. Unscented Transformation

The Unscented Transform is a method for estimating the statis-tics of a random variable which undergoes a nonlinear transforma-tion as in F�1ðvÞ [44]. Assuming the velocity as a random variable vwith the mean �v and variance r2

v ; only three weighted sigmapoints v0, v1, and v2 are needed to calculate the statistics of v.These points are given by Ref. [44]:

v0 ¼ �v; W0 ¼ j1þj

v1 ¼ �v þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ jÞr2

vp

; W1 ¼ 12ð1þjÞ

v2 ¼ �v �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ jÞr2

vp

; W1 ¼ 12ð1þjÞ

ðA:5Þ

where Wi is the weight associated with the ith sigma point and j isan arbitrary number providing 1 + j – 0; j = 0 is chosen in thiswork. Given the set of sigma points calculated by Eq. (A.5), thetransformation of velocity pdf to the approximated fd pdf is sum-marized in the following steps [44]:

1. Calculate the fd corresponding to each sigma point usingdispersion function:

ðfdÞi ¼ F�1ðviÞ ðA:6Þ

2. The mean of fd is calculated from:

fd ¼X2

i¼0

WiðfdÞi ðA:7Þ

3. The variance of random variable fd is determined by:

r2fd ¼

X2

i¼0

Wi ðfdÞi � fd� 2

ðA:8Þ

Since f and d are statistically independent, the following equa-tion holds for estimating fd

fd ¼ E½fd� ¼ E½f �E½d� ¼ �f � �d ðA:9Þ

and thus �d is calculated as:

�d ¼ fd=�f ðA:10Þ

Given r2fd and �d, one can calculate r2

d using the variance relationshipfor the product of independent variables (i.e.,var½xy� ¼ ðE½x�Þ2var½y� þ ðE½y�Þ2var½x� þ var½x�var½y�) as follows:

r2d ¼

r2fd � �d2r2

f

�f 2 þ r2f

ðA:11Þ

orrosion damage diagnosis in steel strands using guided ultrasonic waves,

A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx 11

A.4. Delta method

Delta method uses the second-order Taylor expansions to findthe variance of a function of a random variable. Given a randomvariable x and a function g(x), the variance of this function isobtained as follows [46],

var½gðxÞ� � ðg0ðE½x�ÞÞ2var½x� ðA:12Þ

where g0 is the derivative of the function g with respect to x. In caseof this study, velocity v is the variable and F�1ðvÞ is the function.Therefore, we have:

r2fd ¼ var½F�1ðvÞ� � F�10 ð�vÞ

� 2r2

v ðA:13Þ

Given r2fd, the variance of diameter r2

d is calculated via Eq. (A.11)and �d is given through Eq. (A.10).

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