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Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras 5 6
Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves
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Alireza Farhidzadeh, Salvatore Salamone ⇑
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Smart Structures Research Laboratory, Department of Civil, Structural, and Environmental Engineering, University at Buffalo, Buffalo, NY 14228, United States
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a r t i c l e
1 2 2 5 13 14 15 16 17
i n f o
Article history: Received 3 July 2014 Received in revised form 29 September 2014 Accepted 21 November 2014 Available online xxxx
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Keywords: Corrosion Steel strand Guided ultrasonic wave Wavelet transform Uncertainty analysis
a b s t r a c t This study presents a nondestructive evaluation method based on guided ultrasonic waves (GUW) to quantify corrosion damage of prestressing steel strands. Specifically, a reference-free algorithm is proposed to estimate the strand’s cross-section loss by using dispersion curves, continuous wavelet transform, and wave velocity measurements. Accelerated corrosion tests are carried out to validate the proposed approach. Furthermore, the propagation of Heisenberg uncertainty to diameter measurement is also investigated. The method can reasonably estimate the wires’ diameter without any baseline as a reference. Ó 2014 Published by Elsevier B.V.
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1. Introduction
38
Multiwire steel strands are widely used in civil structures such as cable-stayed bridges, prestressed concrete structures, and re-centering systems. Despite this increase in usage, the corrosion of the multiwire strands has become a concern for designers, owners and regulators. Many of these structures have suffered the failure of strands due to corrosion [1–3]. Extensive inspection and maintenance/repair programs have been established, with attendant direct costs and significant indirect costs due to business interruption [4]. Evaluation of strands is technically challenging. In many structures, inaccessibility of steel strands, eventuate in difficult, expensive and often inconclusive evaluation. Several nondestructive evaluation (NDE) techniques for evaluating the condition of strands have been developed to address these issues in the past few years [5]. Half-Cell potential [6], time domain reflectometry (TDR) [7], linear polarization resistance (LPR) sensors [8], magnetic flux [9], and acoustic emission [10] are some of the most commonly used NDE methods for corrosion diagnosis. Although these techniques have shown promise, very few if any are capable to quantify the cross-sectional loss. A technique that shows potential to quantify the extent of corrosion (e.g., cross-sectional loss) is based on guided ultrasonic waves (GUWs). As opposed to the waves used in traditional
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⇑ Corresponding author. E-mail addresses: [email protected] (A. Farhidzadeh), [email protected] (S. Salamone).
impact-echo (IE), that propagate in 3-D within the structure, GUWs propagate along the strand itself by exploiting its waveguide geometry [11]. Most previous researches attempted to use GUWs to detect and assess corrosion damage on reinforcing bars in reinforced concrete (RC) structures [12–15] and steel strands in prestressed concrete structures [4,16–20]. For instance, to diagnose corrosion in reinforcing bars, energy and attenuation characteristics of longitudinal and flexural GUW modes were used [13,14]. Other methods based on time of flight (ToF) of the first packets were also proposed to investigate various levels of corrosion in reinforcing bars [21]. Longitudinal GUWs were also used to monitor pitting and delamination in steel rebars [15]. To monitor the corrosion process in post-tensioned concrete beams, fractal analysis of GUWs was investigated [4]. In this study, a reference-free algorithm is proposed to quantify the extent of corrosion through estimating the cross-section loss using GUW measurements. An experimental setup was designed to carry out an accelerated corrosion test on a loaded strand. The diameter of the strand’s wires was estimated using the continuous wavelet transform (CWT) [22] of the GUWs. Furthermore, the uncertainty associated with estimated diameter, which originates from the Heisenberg principle in CWT [23], was quantified. In the next section, the behavior of GUW in rods and the effect of crosssection loss on dispersion curves, followed by a short introduction on CWT are presented. The details of experimental setup are then explained. The results of diameter measurement using GUW are discussed and finally the conclusion is given.
http://dx.doi.org/10.1016/j.ultras.2014.11.011 0041-624X/Ó 2014 Published by Elsevier B.V.
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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2. Theory
5000
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2.1. Guided ultrasonic waves in rods
4500
88
106
A guided ultrasonic wave (GUW) is generated whenever an ultrasound propagates into a bounded medium [24]. GUWs are known to be multimode (many vibrating modes can propagate simultaneously) and dispersive (the propagation velocity depends on the wave frequency f ). In cylindrical waveguides, such as rods, three different modes can propagate: longitudinal, flexural, and torsional [24,25]. The dispersive behavior of these modes is represented by the dispersion curves like the ones shown in Fig. 1. These curves describe the relationship between wave velocity and frequency, and can be calculated analytically or they can be computed by approximate solutions derived from numerical methods [24]. In this work a MATLAB open source toolbox, PCdisp (PochhammerChree dispersion) [26,27] was used to generate the dispersion curves (Fig. 1). The longitudinal modes have received significant interest in the past few years, for the nondestructive evaluation of cylindrical waveguides, mostly because the flexural and torsional modes experience high attenuation during the propagation phenomena [16,18,28–30].
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2.2. Effect of corrosion on GUWs
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Corrosion is usually an electrochemical oxidation–reduction process that converts the major component of steel, iron (Fe), to ferrous hydroxide (Fe(OH)2). A consequence of this process is the reduction in member cross sectional area, which eventually may affects its structural performance. To simulate the effect of corrosion and investigate how the reduction in a rod’s cross sectional area affects the guided wave propagation, the dispersion curves for the first longitudinal mode L(0,1), were generated for various diameters, ranging from 5 mm to 2.5 mm. The results are shown in Fig. 2. It can be observed that in a frequency range between 0 and 600 kHz the group velocity increases as the cross-sectional area decreases. Moreover, certain frequencies (e.g., 500 kHz) provide a larger sensitivity to diameter changes than other frequencies (e.g., 300 kHz). It is worth to mention that guided waves thickness measurement methodologies based on group velocity changes have been studied by some researchers for other applications such as plate-like structures [31–35].
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Group velocity [m/sec]
92
4000
3500
3000 d=5.0mm d=4.5mm d=4.0mm d=3.5mm d=3.0mm d=2.5mm
2500
2000
1500
0
100
200
300
400
500
600
700
800
900
1000
Frequency [kHz] Fig. 2. Dispersion curves for the first longitudinal mode L(0,1) in steel rods with various diameters.
Fig. 3 shows the group velocity of the L(0,1) mode as a function of the frequency-diameter product (i.e., v = F(fd)). Therefore, the inverse problem, which is to calculate the diameter given the dispersion curves, can be solved by measuring the velocity of a certain frequency component in the signal. In order to limit the effects of the high order modes the excitation frequency was selected in a frequency range below the first cut-off frequency. In this work, a time–frequency analysis (continuous wavelet transform) was used to measure the group velocity of the L(0,1) mode.
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2.3. Continuous wavelet transform
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Continuous wavelet transform (CWT) for analyzing non-stationary signals has received significant interest in the last few years, due to its ability to extract signal time and frequency information simultaneously [23]. The CWT of a time domain signal f(t) is given by Ref. [23],
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1 tb dt WTðs; bÞ ¼ pffiffiffiffiffi f ðtÞw s jsj 1
140
Z
ð2:1Þ
6000
6000
5000
L(0,1)
5000
L(0,2) 4000 T(0,1) 3000 F(1,3) F(1,2) 2000
F(1,1)
at constant frequency: d↓ → v↑
4000
(470kHz-5.002mm,3440)
3000
2000
1000
1000 0
0 0
100
200
300
400
500
600
700
800
900
1000
0
500
1000
1500
2000
2500
3000
Frequency-diameter [kHz-mm]
Frequency [kHz] Fig. 1. Group velocity dispersion curves for a 5 mm diameter steel rod.
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1
where w⁄(t) denotes the complex conjugate of the mother wavelet w(t), s is the dilation parameter (scale), and b is the translation
Group Velocity [m/sec]
90 91
Group speed [m/s]
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Fig. 3. Dispersion curve for the first longitudinal mode L(0,1) in a steel rod versus product of frequency and diameter.
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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Mechanical characteristics 5.2 5.0
Young’s modulus E (GPa) Poisson’s ratio (v)
196 0.29
15.2 230
203 1860
Lay angle b (°)
7.9
Yielding load (kN) Ultimate tensile strength (MPa) Linear weight (kg/m)
1.10
parameter (location). The energy density of a signal is represented by the Power Spectral Density (PSD) as [23],
PSD ¼ 10 logðjWTðs; bÞj2 Þ
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where
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Gaussian [36]. In this study, a complex Morlet was used as mother wavelet [23], defined as:
Core wire diameter dc (mm) Helical wire diameter dh (mm) Strand diameter D (mm) Pitch of helical wire p (mm)
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rt rf
ð2:3Þ
f ¼ psffiffiffiffiffi s fb
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The methods based on wavelet transform have an inherent source of uncertainty that originates from the Heisenberg principle [23]. Based on this principle, the resolution in time and frequency domains is limited according to the following relationship [23]:
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158 160 161 162 163
r2t r2f
1 P 4
ð2:5Þ
where fb is the bandwidth parameter and fc is the wavelet center frequency. Morlet wavelet is a sine wave multiplied by a Gaussian Window. The Morlet wavelet provides a desirable compromise between time and frequency resolution [22]. The complex Morlet parameters were set by trial and error and fixed (fc = 5 Hz, fb = 2 Hz). For a given signal with sampling frequency fs, the complex Morlet transform yields the following equations for rt and rf [23]:
ð2:2Þ
jWTðs; bÞj2 ¼ WTðs; bÞWT ðs; bÞ
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2 1 t wðtÞ ¼ pffiffiffiffiffiffiffiffi expð2ipf c tÞ exp fb pf b
pffiffiffiffiffi s fb ¼ 2f s
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155
3
ð2:6Þ
164 165
166 168 169 170 171 172 173 174 175
176 178 179
ð2:7Þ
in which the dilation parameter (i.e., s) is inversely proportional to the local frequency f according to the following equation:
f f s ¼ s c f
ð2:8Þ
181 182 183
184 186
ð2:4Þ
where r2t and r2f are time and frequency variances, respectively. These quantities represent the local resolution of CWT. This inequality reaches its minimum value of 1/4 when the signal is
3. Experiments
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An experimental setup was designed to validate the proposed approach. Specifically, a loading apparatus was designed and built
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Fig. 4. The loading device designed for the accelerated corrosion test on a steel strand (a) plan view, (b) cross section and side views.
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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Fig. 5. Experimental setup overview.
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to apply a tensile load to a seven-wire strand. Then, accelerated corrosion tests were carried out on the axially loaded strand.
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3.1. Strand characteristics
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In this study, a 15.2 mm Grade 270 steel tendon consisting of six 5.0 mm diameter wires spirally wrapped around a central 5.2 mm diameter wire was considered (see Table 1 for details). The length of each helical wire was assumed equal to the length of the central wire since the lay angle b was very small (the difference was as small as 0.89%) [37,38]. In addition, considering that the pitch length (230 mm) was much longer than the typical wavelength in the signals (smaller than 10 mm [37]), each wire was treated as an individual rod [16]. Beard et al. [16] observed that guided waves in individual wires of a strand behave similar to single wires with minor discrepancies associated to inter-wire contact.
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3.2. Loading apparatus
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Drawings of the loading apparatus are shown in Fig. 4. It consists 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 )) con00 nected with two u 118 all thread steel bars. In the middle of each 00 beam’s web, a 5/8 hole was drilled; the strand passed through the holes and was fixed using wedge anchorages. The load was applied by tightening the nuts, which pushes the rigid beams and stretches the strand; a large wrench with handle extension was used for this purpose. Therefore, the all thread bars are under compressive force. To prevent bucking, two wood beams were connected to the bars using u-shaped hooks (see Fig. 5). Two hollow
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3.3. Accelerated corrosion test
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An accelerated corrosion test [39] was carried out using the impressed current technique. Specifically, the strand was immersed in 3.5% sodium chloride (NaCl) solution. A power supply equipped with built-in ammeter and potentiometer was used to impress a direct current (DC) to the strand to induce significant corrosion in a short period of time. The direction of the current was adjusted so that the steel strand served as the anode, while another metal, superior than steel in electro-chemical series, served as the cathode [40] (see Fig. 4). A constant voltage of 0.16 V was maintained across the solution and the specimen until the 18th day, and then was increased to 0.32 V to failure. Fig. 5 shows an overview of the experimental setup that consists of loading apparatus, water tank, power supply, and the steel strand. To monitor the corrosion process, a video camera with recording rate of 1 frame per minute was mounted on top of the specimen.
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3.4. Mass loss measurement
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The mass loss during the corrosion tests was determined through a Gravimetric analysis in which the iron oxide or rust (Fe2O3) was the analyte. The ferrous ions (Fe2+) precipitate through combination with hydroxyl ions (OH) and form ferrous hydroxide (Fe(OH)2) that are fine solid particles. These particles were collected by filtration, dried, and weighted. When dried, rust (Fe2O3) is produced. The weights of filters were subtracted to obtain the weight of rust. Finally, to calculate the mass loss of iron, the measured weight was multiplied by a correction factor of 0.7. The correction factor is derived from the mass portion of iron in rust (i.e., (2 56)/(2 56 + 3 16) = 0.7). This process is schematically illustrated in Fig. 6.
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3.5. Ultrasonic test equipment
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Ultrasonic tests were performed using broadband piezoelectric transducers in a through-transmission (i.e., pitch-catch) configura-
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Fig. 6. The instruction of mass loss measurement.
Low Frequency transmission
194
load cells passed through the all thread bars and were located between the beam’s web and the nuts’ washers. The tensile load on the strands was continuously measured during the corrosion test as the summation of readings from these two load cells. The strand was initially loaded up to 89 kN (=20 kips 40% of yielding force) which was the designed capacity of the loading apparatus.
PICO sensor
Piezo Linear Amplifier
High Frequency Olympus 5077PR Square pulser
NI PXI-5412 AWG
V1091 transducers
transmission
LabVIEW
PICO sensor
Amplifier
PXI-5105 digizer
Oscilloscope Lecroy wave runner 44Xi-A
Inbuilt amplificaon
V1091 transducers
Fig. 7. Through-transmission testing for low- and high-frequency longitudinal waves.
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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S1 and S2: PICO sensor S3 and S4: 5MHz sensor 3.5 cycles
PICO sensor
S3
S2 S4
S1
Magnet holder 5MHz sensor
(a)
(b)
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.
25 Faraday's law Mass measurment
Mass loss [%]
20 15 10 5 0 0
2
4
6
8
10 12 14 16 18 20 22
Time [day]
transducers (Physical Acoustic Corporation PICO), which are sensitive in a frequency range between 200 kHz and 700 kHz, were used to generate and receive ultrasonic waves. LabVIEW software was used 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 transducers with a central frequency of 5 MHz (Olympus V1091) with a square wave pulse of selectable width. The received signals were recorded using an oscilloscope (Lecroy Wave Runner 44Xi-A). In both experimental setups, transducers were carefully mounted as perpendicular to the wire’s cross section as possible to measure the longitudinal wave motions. The diameter of the transducers was about the same as the diameter of the single wire so that there was no contact between the transducers and the adjacent wires, as shown in Fig. 8.
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4. Results
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Faraday’s law was used to predict the strand mass loss. Faraday’s law provides a relationship between the time of an applied current and the amount of steel weight loss according to the following equation:
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Fig. 9. Mass loss versus time during accelerated corrosion test.
100 90 80
Load [kN]
70
A m¼ ZF
60 50
20
1st wire failure
2nd
and 3rd
wires failures
10 0
1 23 4 5 6
7
10
12
14
18
20
22
Time [day] Fig. 10. Load versus time during accelerated corrosion test.
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IðtÞdt
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 the Faraday’s constant (96,500 A s), t is time in seconds, and I(t) is the current in ampere. Therefore the theoretical percentage of mass loss, Dm, was calculated as:
40 30
Z
tion. A schematic drawing of the experimental setup is shown in Fig. 7. Two through-transmission setup were used to investigate the low frequency modes (below the first cut-off frequency), and the higher frequency modes (above the first cut-off frequency). For the low frequency modes the measurement equipment used for signal generation and data acquisition was mainly constituted by a National Instruments (NI), modular PXI 1042 unit. The unit included an arbitrary waveform generator card (PXI 5412) and one, 20GS/s 12-bit multi-channel digitizers (PXI 5105). A high voltage amplifier was used to amplify the excitation to the ultrasonic transmitters, while a preamplifier (Olympus 5660C) was used to amplify the ultrasonic receivers. Two ultra-mini broadband
m Dm ¼ 100% M
ð4:2Þ
where M is the total mass of the strand. Fig. 9 shows the measured results based on gravimetric mass loss and the corresponding theoretical mass loss based on Faraday’s law. Faraday’s law predicted well the degrees of corrosion (mass loss) with the impressed current technique. Fig. 10 shows the strand axial load measurements through the load cells during the accelerated corrosion test. The failure of three wires was recorded at day 21st. It is noteworthy that load drop after the failure of the 2nd and 3rd wire was almost twice the load drop after the 1st wire breakage. Therefore the load was equally distributed on each wire. The influence of the impressed current on the load drop rate can be observed at the 18th day, when impressed current was increased from 0.5 A to 1.5 A. Fig. 11 illustrates the strand condition at different levels of corrosion. The measured mass loss is also reported for each level of corrosion. The location of the wire breakage is indicated with arrows.
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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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.
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4.1. Attenuation measurements
316
Attenuation coefficients were calculated using the following equation:
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ai ½dB=m ¼
20 A0 log L Ai
ð4:3Þ
where ai is the attenuation coefficient in dB/m, L is the length of strand, A0 is the amplitude of the signal at pristine state, and Ai is the amplitude at ith day. Two PICO sensors in a through-transmission configuration were used to generate and receive GUWs on the same peripheral helical wire. A sweep frequency test between 300 kHz and 700 kHz was performed at different levels of corrosion. Fig. 12a shows the reduction in amplitude of the signals versus mass loss and excitation frequency. Fig. 12b illustrates the attenuation coefficients. At about 20% mass loss, the attenuation coefficient increased to 30 dB/m. It is interesting to note that the attenuation coefficient is quite constant for all frequencies below 15% mass loss. To investigate the attenuation of the higher longitudinal modes during the corrosion process, two piezoelectric broadband transducers with a central frequency of 5 MHz were used in a through-transmission-configuration, as described in Section 3.4. Fig. 13 shows the CWT of the received signals in day 1, 3, 4, and 5. For the first day, the CWT was estimated and compared with numerical results of dispersion curves obtained from the PCdisp toolbox. There is a clear correspondence between theory (solid lines) and experimental results (CWT). After five days (less than 3% mass loss) only the L(0,1) mode was left.
4.2. Reference-free diameter estimation
342
A 3.5-cycle, Hanning-modulated toneburst was used as the actuating signal. For the selection of the excitation frequency a swept of frequencies were performed between 300 kHz and 700 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 frequency of 440 kHz was selected as excitation frequency. A typical received signal and corresponding CWT of the signal before the corrosion test (i.e., pristine condition) is shown in Fig. 15. CWT was compared with the dispersion curves resulting from the numerical simulation described in Section 2.1. It is worth noting a clear correspondence between theory (dashed line) and experimental results (CWT map), which indicate that the first longitudinal mode is the predominant mode of displacement in the received signal. To visualize the Heisenberg’s uncertainty principle discussed in Section 2.3, the time and frequency standard deviations (i.e., rt and rf in Eqs. (2.6) and (2.7)) are also indicated on the Heisenberg box [23]. To estimate the wire diameter at the pristine state (i.e., 5 mm), the maximum value of CWT, called the peak ridge, was identified. The projection of the peak ridge on frequency and time axes corresponds to the dominant frequency f and timeof-flight (ToF) t, respectively. At the pristine state, the peak ridge was at t ¼ 357 ls and f ¼ 470 kHz (see Fig. 15). Given the strand length of 1.23 m, the velocity of 3440 m/s was calculated. Therefore, the dispersion curves shown in Fig. 3 were used to estimate the diameter (i.e., 5.002 mm). There is a clear correspondence between the actual value (5 mm) and the estimated value (5.002 mm). Fig. 16 depicts the time waveforms during the corrosion test at days 3, 6, 10, and 18. As expected, the energy was significantly attenuated due to the corrosion process as well as the inter-wire contacts. However, the wave velocities of the first wave packets, did not shows significantly changes (see the left zoomed-in outset plot) while the time shift was evident on the high energy part of the waveforms (see the right zoomed-in outset plot). This shift is mainly caused by the diameter reduction and its impact on wave velocity at certain frequencies. Fig. 17 shows a typical waveform at day 6th along with its corresponding CWT scalogram. The first packet corresponds to an approximate frequency range of 320– 360 kHz at which the velocity is slightly sensitive to diameter (see Fig. 2). On the contrary, the high energy packet corresponds to a higher frequency component (i.e., 470 kHz). This part of the signal is used for diameter estimation. The reference-free algorithm was repeated every day to estimate the average cross-section loss during the corrosion process. To take into account the fact that, the experimental setup was designed to corrode just a portion of strand, some modifications were needed to find the average velocity in the corroded part. At this aims the strand was divided in three regions: (1) left end side of the salt water tank with length l1, (2) inside the salt water tank with length l2, (3) right end side of the salt water tank with length l3. The middle part (i.e., inside the salt water tank) was the corroding part of the strand, while the other two regions with length l1 and l3 remained pristine during the test. Therefore the arrival time t was estimated as:
343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397
398 3 X t ¼ t i ¼ l1 =v 1 þ l2 =v 2 þ l3 =v 1
ð4:4Þ
i¼1
where t i 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 v 1 considered for the pristine parts was obtained from the dispersion curves (Fig. 3) at d = 5 mm. Since the corrosion was not perfectly uniform, we estimated an average velocity v 2 in the middle part of the strand as:
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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40 30
0.15 0.1 0.05 0
0 800
5 700
Frequency [kHz]
500
15
400 300
20 10 0 -10 300
10
600
Attenuation [db/m]
Amplitude [v]
0.2
20 15
400 500
Mass loss [%]
20
10
600
Frequency [kHz]
(a)
5
700 800
0
Mass loss [%]
(b)
Fig. 12. Effect of corrosion on (a) signal amplitude, and (b) attenuation coefficient.
Fig. 13. Effect of corrosion on higher longitudinal modes.
406
408 409 410 411 412 413
l2 t 2l1 =v 1
v 2 ¼
ð4:5Þ
2 and frequency f , the average diameter of Having the velocity v a wire in the corroding part was estimated from the L(0,1) mode dispersion curve in Fig. 3. This procedure was repeated at different stages of corrosion and the results are presented in Fig. 18. This figure shows a reasonable correspondence between the estimated
and measured diameter. The estimation error increases after day 14th when significant mass loss was observed. The larger error (i.e., 10%) was recorded at the 20th day.
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5. Uncertainty quantification
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The most critical challenge here is to provide a quantitative assessment of how closely our estimates (i.e., diameter) reflect
418
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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0.035 0.03
RMS [V]
0.025
0 3 7 10 14 18 20
0.02 0.015 0.01 0.005 0 300
350
400
450
500
550
600
650
700
Frequency [kHz] Fig. 14. RMS of the received waveforms in sweep frequency test during the accelerated corrosion test.
Fig. 16. Ultrasonic signals at different corrosion states; the velocity does not change considerably for the first packets (bottom left) while it clearly changes at the high energy part of the signal (bottom right).
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Fig. 15. (a) Time-domain signal and (b) spectrogram of signal taken from a 1.23 length rod (5 mm diameter) through-transmission. The dispersion curve for L(0,1), time and frequency of the ridge as well as Heisenberg uncertainties are superimposed. Fig. 17. Guided wave signal during corrosion at day 6 (top) and the corresponding CWT scalogram (bottom).
420 421 422 423 424 425 426 427 428 429 430 431 432 433
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 unpredictable changes in measurements (e.g., ToF), including instrumentation noise, temperature changes, etc. [41]. Systematic errors are mostly caused by the digital signal processing technique (e.g., CWT) used for analyzing the time waveforms [41]. This section aims at quantifying the systematic uncertainty, caused by the time–frequency analysis (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 r2t and r2f defined according to Eqs. (2.6) and (2.7), respectively. Also the group velocity (v) was treated as a Gaussian random variable with and variance r2v defined as (see Appendix A): mean v
434
v ¼ tl 2
r2v ¼ tl4 r2t
ð5:1Þ
The two random variables f and v are related by the Pochhammer-Chree equation v ¼ FðfdÞ (i.e., dispersion curves). Therefore, given the pdf of f and v, the systematic uncertainty associated to the estimated diameter was evaluated by determining its pdf. Several techniques exist in the literature to approximate the pdf of 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, require extensive computational resources and effort, and become increasingly
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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Measured
Histogram Monte Carlo Unscented transform Delta method
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PDF
Diameter [mm]
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Time [day]
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Fig. 18. Diameter of a helical wire estimated using the reference-free approach and measured through visual inspection during accelerated corrosion test.
0
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10
Diameter [mm] Fig. 20. Probability density functions of diameter at the 10th day with different approaches.
Measured Estimated
5.5
Diameter [mm]
5
4.5
4
Fig. 19. Heisenberg uncertainty propagation through the nonlinear dispersion function of L(0,1) analyzed by Monte Carlo (MC), Unscented Transform (UT) and Delta methods.
3.5
0
2
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14
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22
Time [day] 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466
infeasible for portable NDT devices with small processing units. The UT and Delta method, are both used to estimate the statistics of a random variable which undergoes a nonlinear transformation as in F 1 ðv Þ [44]. In the UT method, instead of randomly selecting multiple data points from a pdf, a few data points (called sigma points) are intelligently selected. The Delta method is motivated by the assumption that in this study the dispersion curve can be approximated by a linear function in the interested velocity 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 methods for uncertainty propagation analysis. It focuses the dispersion . The curve (shown by dashed-dot line) at the velocity mean v velocity pdf is shown by a thick solid line. The sampled points from the velocity pdf are shown by narrow gray lines, representing the MC simulation. These points are projected to fd axis through the dispersion curve (i.e., ðfdÞi ¼ F 1 ðv i Þ). The UT sigma points are also Q4 superimposed using three red1 dotted lines, signifying the significant reduction in required sampling points. The Delta method is represented 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 of this article.
Fig. 21. Estimated diameters and their associated uncertainty.
(i.e., fd Nðfd; r2fd Þ) that is shown by a black narrow line. Given the parameters of fd pdf, the diameter’s mean and variance are calculated via the following equations:
467 468 469
470
¼ fd=f d
r2d ¼
r2fd d2 r2f f 2 þr2 f
ð5:2Þ
Fig. 20 shows the estimated pdf of diameter using the aforementioned methods at the 10th day as an example. The histogram and pdf obtained using MC method are shown by vertical bars and a dashed line. The solid line represents the pdf of the diameter using UT, and the dotted line is that of Delta method. There is a clear correspondence between the pdf obtained with all three methods. However, UT and Delta method provides significant computational cost reduction compared to Monte Carlo method. Finally Fig. 21 illustrates the estimated diameter’s mean and standard deviation as well as the measured diameter during the corrosion test. It is remarkable to note that the difference between measured and estimated diameter is always less than one standard deviation. The standard deviations are approximately 0.7 ± 0.1 mm. The total diameter reduction is less than 1.5 mm and this evidence the high resolution of the proposed velocity-based approach to estimate the diameter.
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6. Conclusions
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This study aims at designing a reference-free nondestructive evaluation technique for corrosion damage estimation in prestressed seven wires steel strands using guided ultrasonic waves. A small-scale experiment was designed to investigate corrosion process and behavior of ultrasonic waves in under accelerated corrosion. Axial load and mass loss were measured during the test. Attenuation coefficients were monitored and reached to 35 dB at critical stage of corrosion. A new method based on the velocity of certain velocity components was proposed and validated to estimate the cross-section loss. Continuous wavelet transform was used to localize the highest energy content in time–frequency domain. Having the frequency and ToF (that yields the velocity), the diameter was estimated using the numerically derived dispersion function for the first longitudinal mode. This function correlates velocity to multiplication of frequency and diameter. In addition, an uncertainty analysis was carried out to study propagation of time and frequency uncertainties from Heisenberg principle to the diameter calculation. The proposed algorithm does not need a baseline as a reference and directly measure the diameter via through-transmission configuration in ultrasonic testing. The algorithm was validated experimentally and the results conformed very well to the diameter measurements during visual inspection.
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Acknowledgments
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Funding provided by the Research Foundation of SUNY through the research collaboration fund. Any opinions, findings, conclusions or recommendations expressed are those of the author(s) and do not necessarily reflect the views of the RF/SUNY.
515 516
517
Appendix A
518
A.1. Velocity pdf
519
In general, the mean and variance of the reciprocal of a random variable can be obtained using its equivalent Taylor series at mean value as follows: denoting the expected value (mean) of a random variable t as E½t ¼ t and defining a random variable v as a function of t as v ðtÞ ¼ tl , we have:
520 521 522 523 524
l Taylor series : v ðt ¼ tÞ ¼ t t¼t
2 1 t t ðt tÞ ¼ l 2 þ 3 þ . . . t t t
526 527
528 530
531 532 533
534
Thus the expected value of
! ðA:1Þ
ðA:2Þ
In our case, since the variance in time, r is very small in comparison to the measured time t, we can eliminate the second term in Eq. (A.2) and simply define the velocity mean as: 2 t;
l t
537
and the velocity variance as,
ðA:3Þ
542 543
The Unscented Transform is a method for estimating the statistics of a random variable which undergoes a nonlinear transformation as in F 1 ðv Þ [44]. Assuming the velocity as a random variable v and variance r2v ; only three weighted sigma with the mean v points v0, v1, and v2 are needed to calculate the statistics of v. These points are given by Ref. [44]:
560
v0 ¼ v ; W0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 v1 ¼ v þ ð1 þ jÞrv ; W 1 ¼ 2ð1þ1 jÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ¼ v ð1 þ jÞr2v ; W 1 ¼ 2ð1þ1 jÞ
545 546 547 548 549 550 551 552 553 554 555 556 557 558
561 562 563 564 565
566
ðA:5Þ 568
where Wi is the weight associated with the ith sigma point and j is an arbitrary number providing 1 + j – 0; j = 0 is chosen in this work. Given the set of sigma points calculated by Eq. (A.5), the transformation of velocity pdf to the approximated fd pdf is summarized in the following steps [44]:
569
1. Calculate the fd corresponding to each sigma point using dispersion function:
574
ðfdÞi ¼ F 1 ðvi Þ
ðA:6Þ
ðA:7Þ
i¼0
r2fd ¼
ðA:4Þ
A.2. Monte Carlo simulation The Monte Carlo simulation samples a large number of data from the underlying probability space to generate a family of test
2 W i ðfdÞi fd
571 572 573
575
576 578 581
2 X fd ¼ W i ðfdÞi
2 X
570
579 580
2. The mean of fd is calculated from:
583 584 585
586
ðA:8Þ
i¼0
588 589
Since f and d are statistically independent, the following equation holds for estimating fd
590
fd ¼ E½fd ¼ E½f E½d ¼ f d
594
ðA:9Þ
is calculated as: and thus d
¼ fd=f d
2
l r2v ¼ E½ðv v Þ ¼ 4 r2t t 2
541
559
v can be found as:
1 r2 E½v ¼ l þ 3t þ . . . t t
v ¼ E½v ¼
540
A.3. Unscented Transformation
3. The variance of random variable fd is determined by:
536 538
544
j 1þj
512
514
points random, feeds forward the samples individually through the exact nonlinear function to find the output data deterministically, and finally evaluates the pdf of outputs [43]. In this study, a large number of data points (>10,000) were sampled from the ; r2v Þ; v i ; i ¼ 1; . . . ; n), and were fed into the velocity pdf (v Nðv inverse of dispersion function to find (fd)i deterministically (i.e., ðfdÞi ¼ F 1 ðv i Þ). Next, another set of n data points are randomly sampled from the frequency pdf (i.e., f Nðf ; r2f Þ; f i ; i ¼ 1; . . . ; n). Assuming statistical independency between frequency and diameter, the diameter di is found (di ¼ ðfdÞi =f i ). Therefore, the diameter and r2 ; are computed having the set of d ’s. It is pdf parameters, d i d worth to mention that the outliers are eliminated from calculation > 3r and recalcuof mean and variance (i.e., remove di if jdi dj d and r2 ) since the number of sampled points were limited late d d and outliers could cause error.
591
592
595
596
ðA:10Þ
598
one can calculate r2 using the variance relationship Given r2fd and d, d for the product of independent variables (i.e., var½xy ¼ ðE½xÞ2 var½y þðE½yÞ2 var½x þ var½xvar½y) as follows:
599
r2 d2 r2f r2d ¼ fd2 f þ r2f
600 601
602
ðA:11Þ
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A.4. Delta method
606
609
Delta method uses the second-order Taylor expansions to find the variance of a function of a random variable. Given a random variable x and a function g(x), the variance of this function is obtained as follows [46],
612
var½gðxÞ ðg 0 ðE½xÞÞ var½x
613 615
where g 0 is the derivative of the function g with respect to x. In case of this study, velocity v is the variable and F 1 ðv Þ is the function. Therefore, we have:
618
r2fd ¼ var½F 1 ðv Þ F 10 ðv Þ r2v
607 608
610
614
616
2
ðA:12Þ
2
ðA:13Þ
620
Given r2fd , the variance of diameter r2d is calculated via Eq. (A.11) is given through Eq. (A.10). and d
621
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Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras 5 6
Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves
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Alireza Farhidzadeh, Salvatore Salamone ⇑
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Smart Structures Research Laboratory, Department of Civil, Structural, and Environmental Engineering, University at Buffalo, Buffalo, NY 14228, United States
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a r t i c l e
1 2 2 5 13 14 15 16 17
i n f o
Article history: Received 3 July 2014 Received in revised form 29 September 2014 Accepted 21 November 2014 Available online xxxx
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Keywords: Corrosion Steel strand Guided ultrasonic wave Wavelet transform Uncertainty analysis
a b s t r a c t This study presents a nondestructive evaluation method based on guided ultrasonic waves (GUW) to quantify corrosion damage of prestressing steel strands. Specifically, a reference-free algorithm is proposed to estimate the strand’s cross-section loss by using dispersion curves, continuous wavelet transform, and wave velocity measurements. Accelerated corrosion tests are carried out to validate the proposed approach. Furthermore, the propagation of Heisenberg uncertainty to diameter measurement is also investigated. The method can reasonably estimate the wires’ diameter without any baseline as a reference. Ó 2014 Published by Elsevier B.V.
26 27 28 29 30 31 32 33 34
35 36 37
1. Introduction
38
Multiwire steel strands are widely used in civil structures such as cable-stayed bridges, prestressed concrete structures, and re-centering systems. Despite this increase in usage, the corrosion of the multiwire strands has become a concern for designers, owners and regulators. Many of these structures have suffered the failure of strands due to corrosion [1–3]. Extensive inspection and maintenance/repair programs have been established, with attendant direct costs and significant indirect costs due to business interruption [4]. Evaluation of strands is technically challenging. In many structures, inaccessibility of steel strands, eventuate in difficult, expensive and often inconclusive evaluation. Several nondestructive evaluation (NDE) techniques for evaluating the condition of strands have been developed to address these issues in the past few years [5]. Half-Cell potential [6], time domain reflectometry (TDR) [7], linear polarization resistance (LPR) sensors [8], magnetic flux [9], and acoustic emission [10] are some of the most commonly used NDE methods for corrosion diagnosis. Although these techniques have shown promise, very few if any are capable to quantify the cross-sectional loss. A technique that shows potential to quantify the extent of corrosion (e.g., cross-sectional loss) is based on guided ultrasonic waves (GUWs). As opposed to the waves used in traditional
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Q3
⇑ Corresponding author. E-mail addresses: [email protected] (A. Farhidzadeh), [email protected] (S. Salamone).
impact-echo (IE), that propagate in 3-D within the structure, GUWs propagate along the strand itself by exploiting its waveguide geometry [11]. Most previous researches attempted to use GUWs to detect and assess corrosion damage on reinforcing bars in reinforced concrete (RC) structures [12–15] and steel strands in prestressed concrete structures [4,16–20]. For instance, to diagnose corrosion in reinforcing bars, energy and attenuation characteristics of longitudinal and flexural GUW modes were used [13,14]. Other methods based on time of flight (ToF) of the first packets were also proposed to investigate various levels of corrosion in reinforcing bars [21]. Longitudinal GUWs were also used to monitor pitting and delamination in steel rebars [15]. To monitor the corrosion process in post-tensioned concrete beams, fractal analysis of GUWs was investigated [4]. In this study, a reference-free algorithm is proposed to quantify the extent of corrosion through estimating the cross-section loss using GUW measurements. An experimental setup was designed to carry out an accelerated corrosion test on a loaded strand. The diameter of the strand’s wires was estimated using the continuous wavelet transform (CWT) [22] of the GUWs. Furthermore, the uncertainty associated with estimated diameter, which originates from the Heisenberg principle in CWT [23], was quantified. In the next section, the behavior of GUW in rods and the effect of crosssection loss on dispersion curves, followed by a short introduction on CWT are presented. The details of experimental setup are then explained. The results of diameter measurement using GUW are discussed and finally the conclusion is given.
http://dx.doi.org/10.1016/j.ultras.2014.11.011 0041-624X/Ó 2014 Published by Elsevier B.V.
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2. Theory
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2.1. Guided ultrasonic waves in rods
4500
88
106
A guided ultrasonic wave (GUW) is generated whenever an ultrasound propagates into a bounded medium [24]. GUWs are known to be multimode (many vibrating modes can propagate simultaneously) and dispersive (the propagation velocity depends on the wave frequency f ). In cylindrical waveguides, such as rods, three different modes can propagate: longitudinal, flexural, and torsional [24,25]. The dispersive behavior of these modes is represented by the dispersion curves like the ones shown in Fig. 1. These curves describe the relationship between wave velocity and frequency, and can be calculated analytically or they can be computed by approximate solutions derived from numerical methods [24]. In this work a MATLAB open source toolbox, PCdisp (PochhammerChree dispersion) [26,27] was used to generate the dispersion curves (Fig. 1). The longitudinal modes have received significant interest in the past few years, for the nondestructive evaluation of cylindrical waveguides, mostly because the flexural and torsional modes experience high attenuation during the propagation phenomena [16,18,28–30].
107
2.2. Effect of corrosion on GUWs
108
Corrosion is usually an electrochemical oxidation–reduction process that converts the major component of steel, iron (Fe), to ferrous hydroxide (Fe(OH)2). A consequence of this process is the reduction in member cross sectional area, which eventually may affects its structural performance. To simulate the effect of corrosion and investigate how the reduction in a rod’s cross sectional area affects the guided wave propagation, the dispersion curves for the first longitudinal mode L(0,1), were generated for various diameters, ranging from 5 mm to 2.5 mm. The results are shown in Fig. 2. It can be observed that in a frequency range between 0 and 600 kHz the group velocity increases as the cross-sectional area decreases. Moreover, certain frequencies (e.g., 500 kHz) provide a larger sensitivity to diameter changes than other frequencies (e.g., 300 kHz). It is worth to mention that guided waves thickness measurement methodologies based on group velocity changes have been studied by some researchers for other applications such as plate-like structures [31–35].
93 94 95 96 97 98 99 100 101 102 103 104 105
109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
Group velocity [m/sec]
92
4000
3500
3000 d=5.0mm d=4.5mm d=4.0mm d=3.5mm d=3.0mm d=2.5mm
2500
2000
1500
0
100
200
300
400
500
600
700
800
900
1000
Frequency [kHz] Fig. 2. Dispersion curves for the first longitudinal mode L(0,1) in steel rods with various diameters.
Fig. 3 shows the group velocity of the L(0,1) mode as a function of the frequency-diameter product (i.e., v = F(fd)). Therefore, the inverse problem, which is to calculate the diameter given the dispersion curves, can be solved by measuring the velocity of a certain frequency component in the signal. In order to limit the effects of the high order modes the excitation frequency was selected in a frequency range below the first cut-off frequency. In this work, a time–frequency analysis (continuous wavelet transform) was used to measure the group velocity of the L(0,1) mode.
125
2.3. Continuous wavelet transform
134
Continuous wavelet transform (CWT) for analyzing non-stationary signals has received significant interest in the last few years, due to its ability to extract signal time and frequency information simultaneously [23]. The CWT of a time domain signal f(t) is given by Ref. [23],
135
1 tb dt WTðs; bÞ ¼ pffiffiffiffiffi f ðtÞw s jsj 1
140
Z
ð2:1Þ
6000
6000
5000
L(0,1)
5000
L(0,2) 4000 T(0,1) 3000 F(1,3) F(1,2) 2000
F(1,1)
at constant frequency: d↓ → v↑
4000
(470kHz-5.002mm,3440)
3000
2000
1000
1000 0
0 0
100
200
300
400
500
600
700
800
900
1000
0
500
1000
1500
2000
2500
3000
Frequency-diameter [kHz-mm]
Frequency [kHz] Fig. 1. Group velocity dispersion curves for a 5 mm diameter steel rod.
126 127 128 129 130 131 132 133
136 137 138 139
1
where w⁄(t) denotes the complex conjugate of the mother wavelet w(t), s is the dilation parameter (scale), and b is the translation
Group Velocity [m/sec]
90 91
Group speed [m/s]
89
Fig. 3. Dispersion curve for the first longitudinal mode L(0,1) in a steel rod versus product of frequency and diameter.
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145 146
147
Mechanical characteristics 5.2 5.0
Young’s modulus E (GPa) Poisson’s ratio (v)
196 0.29
15.2 230
203 1860
Lay angle b (°)
7.9
Yielding load (kN) Ultimate tensile strength (MPa) Linear weight (kg/m)
1.10
parameter (location). The energy density of a signal is represented by the Power Spectral Density (PSD) as [23],
PSD ¼ 10 logðjWTðs; bÞj2 Þ
150
where
151
Gaussian [36]. In this study, a complex Morlet was used as mother wavelet [23], defined as:
Core wire diameter dc (mm) Helical wire diameter dh (mm) Strand diameter D (mm) Pitch of helical wire p (mm)
149
rt rf
ð2:3Þ
f ¼ psffiffiffiffiffi s fb
154
The methods based on wavelet transform have an inherent source of uncertainty that originates from the Heisenberg principle [23]. Based on this principle, the resolution in time and frequency domains is limited according to the following relationship [23]:
157
158 160 161 162 163
r2t r2f
1 P 4
ð2:5Þ
where fb is the bandwidth parameter and fc is the wavelet center frequency. Morlet wavelet is a sine wave multiplied by a Gaussian Window. The Morlet wavelet provides a desirable compromise between time and frequency resolution [22]. The complex Morlet parameters were set by trial and error and fixed (fc = 5 Hz, fb = 2 Hz). For a given signal with sampling frequency fs, the complex Morlet transform yields the following equations for rt and rf [23]:
ð2:2Þ
jWTðs; bÞj2 ¼ WTðs; bÞWT ðs; bÞ
156
2 1 t wðtÞ ¼ pffiffiffiffiffiffiffiffi expð2ipf c tÞ exp fb pf b
pffiffiffiffiffi s fb ¼ 2f s
153
155
3
ð2:6Þ
164 165
166 168 169 170 171 172 173 174 175
176 178 179
ð2:7Þ
in which the dilation parameter (i.e., s) is inversely proportional to the local frequency f according to the following equation:
f f s ¼ s c f
ð2:8Þ
181 182 183
184 186
ð2:4Þ
where r2t and r2f are time and frequency variances, respectively. These quantities represent the local resolution of CWT. This inequality reaches its minimum value of 1/4 when the signal is
3. Experiments
187
An experimental setup was designed to validate the proposed approach. Specifically, a loading apparatus was designed and built
188
Fig. 4. The loading device designed for the accelerated corrosion test on a steel strand (a) plan view, (b) cross section and side views.
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Fig. 5. Experimental setup overview.
190 191
to apply a tensile load to a seven-wire strand. Then, accelerated corrosion tests were carried out on the axially loaded strand.
192
3.1. Strand characteristics
193
204
In this study, a 15.2 mm Grade 270 steel tendon consisting of six 5.0 mm diameter wires spirally wrapped around a central 5.2 mm diameter wire was considered (see Table 1 for details). The length of each helical wire was assumed equal to the length of the central wire since the lay angle b was very small (the difference was as small as 0.89%) [37,38]. In addition, considering that the pitch length (230 mm) was much longer than the typical wavelength in the signals (smaller than 10 mm [37]), each wire was treated as an individual rod [16]. Beard et al. [16] observed that guided waves in individual wires of a strand behave similar to single wires with minor discrepancies associated to inter-wire contact.
205
3.2. Loading apparatus
206
Drawings of the loading apparatus are shown in Fig. 4. It consists 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 )) con00 nected with two u 118 all thread steel bars. In the middle of each 00 beam’s web, a 5/8 hole was drilled; the strand passed through the holes and was fixed using wedge anchorages. The load was applied by tightening the nuts, which pushes the rigid beams and stretches the strand; a large wrench with handle extension was used for this purpose. Therefore, the all thread bars are under compressive force. To prevent bucking, two wood beams were connected to the bars using u-shaped hooks (see Fig. 5). Two hollow
195 196 197 198 199 200 201 202 203
207 208 209 210 211 212 213 214 215 216
217
3.3. Accelerated corrosion test
223
An accelerated corrosion test [39] was carried out using the impressed current technique. Specifically, the strand was immersed in 3.5% sodium chloride (NaCl) solution. A power supply equipped with built-in ammeter and potentiometer was used to impress a direct current (DC) to the strand to induce significant corrosion in a short period of time. The direction of the current was adjusted so that the steel strand served as the anode, while another metal, superior than steel in electro-chemical series, served as the cathode [40] (see Fig. 4). A constant voltage of 0.16 V was maintained across the solution and the specimen until the 18th day, and then was increased to 0.32 V to failure. Fig. 5 shows an overview of the experimental setup that consists of loading apparatus, water tank, power supply, and the steel strand. To monitor the corrosion process, a video camera with recording rate of 1 frame per minute was mounted on top of the specimen.
224
3.4. Mass loss measurement
239
The mass loss during the corrosion tests was determined through a Gravimetric analysis in which the iron oxide or rust (Fe2O3) was the analyte. The ferrous ions (Fe2+) precipitate through combination with hydroxyl ions (OH) and form ferrous hydroxide (Fe(OH)2) that are fine solid particles. These particles were collected by filtration, dried, and weighted. When dried, rust (Fe2O3) is produced. The weights of filters were subtracted to obtain the weight of rust. Finally, to calculate the mass loss of iron, the measured weight was multiplied by a correction factor of 0.7. The correction factor is derived from the mass portion of iron in rust (i.e., (2 56)/(2 56 + 3 16) = 0.7). This process is schematically illustrated in Fig. 6.
240
3.5. Ultrasonic test equipment
252
Ultrasonic tests were performed using broadband piezoelectric transducers in a through-transmission (i.e., pitch-catch) configura-
253
Fig. 6. The instruction of mass loss measurement.
Low Frequency transmission
194
load cells passed through the all thread bars and were located between the beam’s web and the nuts’ washers. The tensile load on the strands was continuously measured during the corrosion test as the summation of readings from these two load cells. The strand was initially loaded up to 89 kN (=20 kips 40% of yielding force) which was the designed capacity of the loading apparatus.
PICO sensor
Piezo Linear Amplifier
High Frequency Olympus 5077PR Square pulser
NI PXI-5412 AWG
V1091 transducers
transmission
LabVIEW
PICO sensor
Amplifier
PXI-5105 digizer
Oscilloscope Lecroy wave runner 44Xi-A
Inbuilt amplificaon
V1091 transducers
Fig. 7. Through-transmission testing for low- and high-frequency longitudinal waves.
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S1 and S2: PICO sensor S3 and S4: 5MHz sensor 3.5 cycles
PICO sensor
S3
S2 S4
S1
Magnet holder 5MHz sensor
(a)
(b)
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.
25 Faraday's law Mass measurment
Mass loss [%]
20 15 10 5 0 0
2
4
6
8
10 12 14 16 18 20 22
Time [day]
transducers (Physical Acoustic Corporation PICO), which are sensitive in a frequency range between 200 kHz and 700 kHz, were used to generate and receive ultrasonic waves. LabVIEW software was used 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 transducers with a central frequency of 5 MHz (Olympus V1091) with a square wave pulse of selectable width. The received signals were recorded using an oscilloscope (Lecroy Wave Runner 44Xi-A). In both experimental setups, transducers were carefully mounted as perpendicular to the wire’s cross section as possible to measure the longitudinal wave motions. The diameter of the transducers was about the same as the diameter of the single wire so that there was no contact between the transducers and the adjacent wires, as shown in Fig. 8.
267
4. Results
282
Faraday’s law was used to predict the strand mass loss. Faraday’s law provides a relationship between the time of an applied current and the amount of steel weight loss according to the following equation:
283
268 269 270 271 272 273 274 275 276 277 278 279 280 281
Fig. 9. Mass loss versus time during accelerated corrosion test.
100 90 80
Load [kN]
70
A m¼ ZF
60 50
20
1st wire failure
2nd
and 3rd
wires failures
10 0
1 23 4 5 6
7
10
12
14
18
20
22
Time [day] Fig. 10. Load versus time during accelerated corrosion test.
255 256 257 258 259 260 261 262 263 264 265 266
IðtÞdt
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 the Faraday’s constant (96,500 A s), t is time in seconds, and I(t) is the current in ampere. Therefore the theoretical percentage of mass loss, Dm, was calculated as:
40 30
Z
tion. A schematic drawing of the experimental setup is shown in Fig. 7. Two through-transmission setup were used to investigate the low frequency modes (below the first cut-off frequency), and the higher frequency modes (above the first cut-off frequency). For the low frequency modes the measurement equipment used for signal generation and data acquisition was mainly constituted by a National Instruments (NI), modular PXI 1042 unit. The unit included an arbitrary waveform generator card (PXI 5412) and one, 20GS/s 12-bit multi-channel digitizers (PXI 5105). A high voltage amplifier was used to amplify the excitation to the ultrasonic transmitters, while a preamplifier (Olympus 5660C) was used to amplify the ultrasonic receivers. Two ultra-mini broadband
m Dm ¼ 100% M
ð4:2Þ
where M is the total mass of the strand. Fig. 9 shows the measured results based on gravimetric mass loss and the corresponding theoretical mass loss based on Faraday’s law. Faraday’s law predicted well the degrees of corrosion (mass loss) with the impressed current technique. Fig. 10 shows the strand axial load measurements through the load cells during the accelerated corrosion test. The failure of three wires was recorded at day 21st. It is noteworthy that load drop after the failure of the 2nd and 3rd wire was almost twice the load drop after the 1st wire breakage. Therefore the load was equally distributed on each wire. The influence of the impressed current on the load drop rate can be observed at the 18th day, when impressed current was increased from 0.5 A to 1.5 A. Fig. 11 illustrates the strand condition at different levels of corrosion. The measured mass loss is also reported for each level of corrosion. The location of the wire breakage is indicated with arrows.
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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.
315
4.1. Attenuation measurements
316
Attenuation coefficients were calculated using the following equation:
317
318 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341
ai ½dB=m ¼
20 A0 log L Ai
ð4:3Þ
where ai is the attenuation coefficient in dB/m, L is the length of strand, A0 is the amplitude of the signal at pristine state, and Ai is the amplitude at ith day. Two PICO sensors in a through-transmission configuration were used to generate and receive GUWs on the same peripheral helical wire. A sweep frequency test between 300 kHz and 700 kHz was performed at different levels of corrosion. Fig. 12a shows the reduction in amplitude of the signals versus mass loss and excitation frequency. Fig. 12b illustrates the attenuation coefficients. At about 20% mass loss, the attenuation coefficient increased to 30 dB/m. It is interesting to note that the attenuation coefficient is quite constant for all frequencies below 15% mass loss. To investigate the attenuation of the higher longitudinal modes during the corrosion process, two piezoelectric broadband transducers with a central frequency of 5 MHz were used in a through-transmission-configuration, as described in Section 3.4. Fig. 13 shows the CWT of the received signals in day 1, 3, 4, and 5. For the first day, the CWT was estimated and compared with numerical results of dispersion curves obtained from the PCdisp toolbox. There is a clear correspondence between theory (solid lines) and experimental results (CWT). After five days (less than 3% mass loss) only the L(0,1) mode was left.
4.2. Reference-free diameter estimation
342
A 3.5-cycle, Hanning-modulated toneburst was used as the actuating signal. For the selection of the excitation frequency a swept of frequencies were performed between 300 kHz and 700 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 frequency of 440 kHz was selected as excitation frequency. A typical received signal and corresponding CWT of the signal before the corrosion test (i.e., pristine condition) is shown in Fig. 15. CWT was compared with the dispersion curves resulting from the numerical simulation described in Section 2.1. It is worth noting a clear correspondence between theory (dashed line) and experimental results (CWT map), which indicate that the first longitudinal mode is the predominant mode of displacement in the received signal. To visualize the Heisenberg’s uncertainty principle discussed in Section 2.3, the time and frequency standard deviations (i.e., rt and rf in Eqs. (2.6) and (2.7)) are also indicated on the Heisenberg box [23]. To estimate the wire diameter at the pristine state (i.e., 5 mm), the maximum value of CWT, called the peak ridge, was identified. The projection of the peak ridge on frequency and time axes corresponds to the dominant frequency f and timeof-flight (ToF) t, respectively. At the pristine state, the peak ridge was at t ¼ 357 ls and f ¼ 470 kHz (see Fig. 15). Given the strand length of 1.23 m, the velocity of 3440 m/s was calculated. Therefore, the dispersion curves shown in Fig. 3 were used to estimate the diameter (i.e., 5.002 mm). There is a clear correspondence between the actual value (5 mm) and the estimated value (5.002 mm). Fig. 16 depicts the time waveforms during the corrosion test at days 3, 6, 10, and 18. As expected, the energy was significantly attenuated due to the corrosion process as well as the inter-wire contacts. However, the wave velocities of the first wave packets, did not shows significantly changes (see the left zoomed-in outset plot) while the time shift was evident on the high energy part of the waveforms (see the right zoomed-in outset plot). This shift is mainly caused by the diameter reduction and its impact on wave velocity at certain frequencies. Fig. 17 shows a typical waveform at day 6th along with its corresponding CWT scalogram. The first packet corresponds to an approximate frequency range of 320– 360 kHz at which the velocity is slightly sensitive to diameter (see Fig. 2). On the contrary, the high energy packet corresponds to a higher frequency component (i.e., 470 kHz). This part of the signal is used for diameter estimation. The reference-free algorithm was repeated every day to estimate the average cross-section loss during the corrosion process. To take into account the fact that, the experimental setup was designed to corrode just a portion of strand, some modifications were needed to find the average velocity in the corroded part. At this aims the strand was divided in three regions: (1) left end side of the salt water tank with length l1, (2) inside the salt water tank with length l2, (3) right end side of the salt water tank with length l3. The middle part (i.e., inside the salt water tank) was the corroding part of the strand, while the other two regions with length l1 and l3 remained pristine during the test. Therefore the arrival time t was estimated as:
343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397
398 3 X t ¼ t i ¼ l1 =v 1 þ l2 =v 2 þ l3 =v 1
ð4:4Þ
i¼1
where t i 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 v 1 considered for the pristine parts was obtained from the dispersion curves (Fig. 3) at d = 5 mm. Since the corrosion was not perfectly uniform, we estimated an average velocity v 2 in the middle part of the strand as:
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40 30
0.15 0.1 0.05 0
0 800
5 700
Frequency [kHz]
500
15
400 300
20 10 0 -10 300
10
600
Attenuation [db/m]
Amplitude [v]
0.2
20 15
400 500
Mass loss [%]
20
10
600
Frequency [kHz]
(a)
5
700 800
0
Mass loss [%]
(b)
Fig. 12. Effect of corrosion on (a) signal amplitude, and (b) attenuation coefficient.
Fig. 13. Effect of corrosion on higher longitudinal modes.
406
408 409 410 411 412 413
l2 t 2l1 =v 1
v 2 ¼
ð4:5Þ
2 and frequency f , the average diameter of Having the velocity v a wire in the corroding part was estimated from the L(0,1) mode dispersion curve in Fig. 3. This procedure was repeated at different stages of corrosion and the results are presented in Fig. 18. This figure shows a reasonable correspondence between the estimated
and measured diameter. The estimation error increases after day 14th when significant mass loss was observed. The larger error (i.e., 10%) was recorded at the 20th day.
414
5. Uncertainty quantification
417
The most critical challenge here is to provide a quantitative assessment of how closely our estimates (i.e., diameter) reflect
418
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0.035 0.03
RMS [V]
0.025
0 3 7 10 14 18 20
0.02 0.015 0.01 0.005 0 300
350
400
450
500
550
600
650
700
Frequency [kHz] Fig. 14. RMS of the received waveforms in sweep frequency test during the accelerated corrosion test.
Fig. 16. Ultrasonic signals at different corrosion states; the velocity does not change considerably for the first packets (bottom left) while it clearly changes at the high energy part of the signal (bottom right).
Q8
Fig. 15. (a) Time-domain signal and (b) spectrogram of signal taken from a 1.23 length rod (5 mm diameter) through-transmission. The dispersion curve for L(0,1), time and frequency of the ridge as well as Heisenberg uncertainties are superimposed. Fig. 17. Guided wave signal during corrosion at day 6 (top) and the corresponding CWT scalogram (bottom).
420 421 422 423 424 425 426 427 428 429 430 431 432 433
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 unpredictable changes in measurements (e.g., ToF), including instrumentation noise, temperature changes, etc. [41]. Systematic errors are mostly caused by the digital signal processing technique (e.g., CWT) used for analyzing the time waveforms [41]. This section aims at quantifying the systematic uncertainty, caused by the time–frequency analysis (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 r2t and r2f defined according to Eqs. (2.6) and (2.7), respectively. Also the group velocity (v) was treated as a Gaussian random variable with and variance r2v defined as (see Appendix A): mean v
434
v ¼ tl 2
r2v ¼ tl4 r2t
ð5:1Þ
The two random variables f and v are related by the Pochhammer-Chree equation v ¼ FðfdÞ (i.e., dispersion curves). Therefore, given the pdf of f and v, the systematic uncertainty associated to the estimated diameter was evaluated by determining its pdf. Several techniques exist in the literature to approximate the pdf of 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, require extensive computational resources and effort, and become increasingly
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5.2
Estimated 5
Measured
Histogram Monte Carlo Unscented transform Delta method
0.6
0.5 4.6 0.4
4.4
Diameter [mm]
4.8
4.2
0.3 4 0.2 3.8 0
2
4
6
8
10
12
14
16
18
20
Time [day]
0.1
Fig. 18. Diameter of a helical wire estimated using the reference-free approach and measured through visual inspection during accelerated corrosion test.
0
2
3
4
5
6
7
8
9
10
Diameter [mm] Fig. 20. Probability density functions of diameter at the 10th day with different approaches.
Measured Estimated
5.5
Diameter [mm]
5
4.5
4
Fig. 19. Heisenberg uncertainty propagation through the nonlinear dispersion function of L(0,1) analyzed by Monte Carlo (MC), Unscented Transform (UT) and Delta methods.
3.5
0
2
4
6
8
10
12
14
16
18
20
22
Time [day] 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466
infeasible for portable NDT devices with small processing units. The UT and Delta method, are both used to estimate the statistics of a random variable which undergoes a nonlinear transformation as in F 1 ðv Þ [44]. In the UT method, instead of randomly selecting multiple data points from a pdf, a few data points (called sigma points) are intelligently selected. The Delta method is motivated by the assumption that in this study the dispersion curve can be approximated by a linear function in the interested velocity 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 methods for uncertainty propagation analysis. It focuses the dispersion . The curve (shown by dashed-dot line) at the velocity mean v velocity pdf is shown by a thick solid line. The sampled points from the velocity pdf are shown by narrow gray lines, representing the MC simulation. These points are projected to fd axis through the dispersion curve (i.e., ðfdÞi ¼ F 1 ðv i Þ). The UT sigma points are also Q4 superimposed using three red1 dotted lines, signifying the significant reduction in required sampling points. The Delta method is represented 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 of this article.
Fig. 21. Estimated diameters and their associated uncertainty.
(i.e., fd Nðfd; r2fd Þ) that is shown by a black narrow line. Given the parameters of fd pdf, the diameter’s mean and variance are calculated via the following equations:
467 468 469
470
¼ fd=f d
r2d ¼
r2fd d2 r2f f 2 þr2 f
ð5:2Þ
Fig. 20 shows the estimated pdf of diameter using the aforementioned methods at the 10th day as an example. The histogram and pdf obtained using MC method are shown by vertical bars and a dashed line. The solid line represents the pdf of the diameter using UT, and the dotted line is that of Delta method. There is a clear correspondence between the pdf obtained with all three methods. However, UT and Delta method provides significant computational cost reduction compared to Monte Carlo method. Finally Fig. 21 illustrates the estimated diameter’s mean and standard deviation as well as the measured diameter during the corrosion test. It is remarkable to note that the difference between measured and estimated diameter is always less than one standard deviation. The standard deviations are approximately 0.7 ± 0.1 mm. The total diameter reduction is less than 1.5 mm and this evidence the high resolution of the proposed velocity-based approach to estimate the diameter.
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6. Conclusions
490
This study aims at designing a reference-free nondestructive evaluation technique for corrosion damage estimation in prestressed seven wires steel strands using guided ultrasonic waves. A small-scale experiment was designed to investigate corrosion process and behavior of ultrasonic waves in under accelerated corrosion. Axial load and mass loss were measured during the test. Attenuation coefficients were monitored and reached to 35 dB at critical stage of corrosion. A new method based on the velocity of certain velocity components was proposed and validated to estimate the cross-section loss. Continuous wavelet transform was used to localize the highest energy content in time–frequency domain. Having the frequency and ToF (that yields the velocity), the diameter was estimated using the numerically derived dispersion function for the first longitudinal mode. This function correlates velocity to multiplication of frequency and diameter. In addition, an uncertainty analysis was carried out to study propagation of time and frequency uncertainties from Heisenberg principle to the diameter calculation. The proposed algorithm does not need a baseline as a reference and directly measure the diameter via through-transmission configuration in ultrasonic testing. The algorithm was validated experimentally and the results conformed very well to the diameter measurements during visual inspection.
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Acknowledgments
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Funding provided by the Research Foundation of SUNY through the research collaboration fund. Any opinions, findings, conclusions or recommendations expressed are those of the author(s) and do not necessarily reflect the views of the RF/SUNY.
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Appendix A
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A.1. Velocity pdf
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In general, the mean and variance of the reciprocal of a random variable can be obtained using its equivalent Taylor series at mean value as follows: denoting the expected value (mean) of a random variable t as E½t ¼ t and defining a random variable v as a function of t as v ðtÞ ¼ tl , we have:
520 521 522 523 524
l Taylor series : v ðt ¼ tÞ ¼ t t¼t
2 1 t t ðt tÞ ¼ l 2 þ 3 þ . . . t t t
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528 530
531 532 533
534
Thus the expected value of
! ðA:1Þ
ðA:2Þ
In our case, since the variance in time, r is very small in comparison to the measured time t, we can eliminate the second term in Eq. (A.2) and simply define the velocity mean as: 2 t;
l t
537
and the velocity variance as,
ðA:3Þ
542 543
The Unscented Transform is a method for estimating the statistics of a random variable which undergoes a nonlinear transformation as in F 1 ðv Þ [44]. Assuming the velocity as a random variable v and variance r2v ; only three weighted sigma with the mean v points v0, v1, and v2 are needed to calculate the statistics of v. These points are given by Ref. [44]:
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v0 ¼ v ; W0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 v1 ¼ v þ ð1 þ jÞrv ; W 1 ¼ 2ð1þ1 jÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ¼ v ð1 þ jÞr2v ; W 1 ¼ 2ð1þ1 jÞ
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561 562 563 564 565
566
ðA:5Þ 568
where Wi is the weight associated with the ith sigma point and j is an arbitrary number providing 1 + j – 0; j = 0 is chosen in this work. Given the set of sigma points calculated by Eq. (A.5), the transformation of velocity pdf to the approximated fd pdf is summarized in the following steps [44]:
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1. Calculate the fd corresponding to each sigma point using dispersion function:
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ðfdÞi ¼ F 1 ðvi Þ
ðA:6Þ
ðA:7Þ
i¼0
r2fd ¼
ðA:4Þ
A.2. Monte Carlo simulation The Monte Carlo simulation samples a large number of data from the underlying probability space to generate a family of test
2 W i ðfdÞi fd
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2 X fd ¼ W i ðfdÞi
2 X
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2. The mean of fd is calculated from:
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ðA:8Þ
i¼0
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Since f and d are statistically independent, the following equation holds for estimating fd
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fd ¼ E½fd ¼ E½f E½d ¼ f d
594
ðA:9Þ
is calculated as: and thus d
¼ fd=f d
2
l r2v ¼ E½ðv v Þ ¼ 4 r2t t 2
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v can be found as:
1 r2 E½v ¼ l þ 3t þ . . . t t
v ¼ E½v ¼
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A.3. Unscented Transformation
3. The variance of random variable fd is determined by:
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544
j 1þj
512
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points random, feeds forward the samples individually through the exact nonlinear function to find the output data deterministically, and finally evaluates the pdf of outputs [43]. In this study, a large number of data points (>10,000) were sampled from the ; r2v Þ; v i ; i ¼ 1; . . . ; n), and were fed into the velocity pdf (v Nðv inverse of dispersion function to find (fd)i deterministically (i.e., ðfdÞi ¼ F 1 ðv i Þ). Next, another set of n data points are randomly sampled from the frequency pdf (i.e., f Nðf ; r2f Þ; f i ; i ¼ 1; . . . ; n). Assuming statistical independency between frequency and diameter, the diameter di is found (di ¼ ðfdÞi =f i ). Therefore, the diameter and r2 ; are computed having the set of d ’s. It is pdf parameters, d i d worth to mention that the outliers are eliminated from calculation > 3r and recalcuof mean and variance (i.e., remove di if jdi dj d and r2 ) since the number of sampled points were limited late d d and outliers could cause error.
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ðA:10Þ
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one can calculate r2 using the variance relationship Given r2fd and d, d for the product of independent variables (i.e., var½xy ¼ ðE½xÞ2 var½y þðE½yÞ2 var½x þ var½xvar½y) as follows:
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r2 d2 r2f r2d ¼ fd2 f þ r2f
600 601
602
ðA:11Þ
Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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No. of Pages 11, Model 5G
2 December 2014 A. Farhidzadeh, S. Salamone / Ultrasonics xxx (2014) xxx–xxx 605
A.4. Delta method
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Delta method uses the second-order Taylor expansions to find the variance of a function of a random variable. Given a random variable x and a function g(x), the variance of this function is obtained as follows [46],
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var½gðxÞ ðg 0 ðE½xÞÞ var½x
613 615
where g 0 is the derivative of the function g with respect to x. In case of this study, velocity v is the variable and F 1 ðv Þ is the function. Therefore, we have:
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r2fd ¼ var½F 1 ðv Þ F 10 ðv Þ r2v
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610
614
616
2
ðA:12Þ
2
ðA:13Þ
620
Given r2fd , the variance of diameter r2d is calculated via Eq. (A.11) is given through Eq. (A.10). and d
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Please cite this article in press as: A. Farhidzadeh, S. Salamone, Reference-free corrosion damage diagnosis in steel strands using guided ultrasonic waves, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.11.011
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