Smart infrastructure networks: sensing, big data and mathematical methods – Research Article

International Journal of DistributedSensor Networks2018, Vol. 14(10)� The Author(s) 2018DOI: 10.1177/1550147718803312journals.sagepub.com/home/dsn

Cross-correlation-based algorithm forabsolute stress evaluation in steelmembers using the longitudinalcritically refracted wave

Zuohua Li1, Jingbo He1, Jun Teng1 and Ying Wang2

AbstractThe absolute stress in the in-service steel members is a critical indicator employed for the evaluation of structural per-formance. In the field of structural health monitoring, the stress is usually monitored by the stress monitoring system.However, the monitored stress is the relative value, rather than the absolute value. The longitudinal critically refractedwave has shown potential for use in absolute stress measurement. The accurate measurement of the longitudinal criti-cally refracted wave time-of-flight is the core issue with this method. In this study, a cross-correlation-based algorithm ispresented for stress evaluation using the longitudinal critically refracted wave. Specifically, a cross-correlation theoreticalformula is derived and a five-step framework is proposed for the longitudinal critically refracted wave time-of-flight mea-surement. Four steel members are employed to investigate the parametric calibration using the longitudinal criticallyrefracted wave to measure the stress. On this basis, the proposed cross-correlation-based algorithm is used to evaluatethe stress of a steel member. The results indicate that the cross-correlation-based algorithm can measure the longitudi-nal critically refracted wave time-of-flight without filtering the noise signal, and the stress measurement results are betterthan those of the traditional peak value method. The proposed method provides a potential way to measure the abso-lute stress in practical engineering applications.

Keywordscross-correlation-based algorithm, longitudinal critically refracted wave, absolute stress measurement, steel members,acoustoelasticity

Date received: 8 April 2018; accepted: 8 July 2018

Handling Editor: Lalit Borana

Introduction

Importance of the absolute stress measurement

Structural health monitoring (SHM)1,2 plays an impor-tant role in studying structural performance evolutionand ensuring structural safety. SHM is not only used incivil engineering applications, such as high-rise build-ings,3 million-ton offshore platforms,4 long tunnels,5

and bridges,6 but is also used in aerospace7 and otherfields.8 Stress evaluation is an important part of SHM.9

Traditionally, the structural stress, especially the stressof large steel members, is usually measured by

designing a stress monitoring system.10 The sensors,such as the resistance strain gauge,11 vibrational chordstrain gauge,12 and fiber grating strain sensor13,14 are

1Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen,

People’s Republic of China2Department of Civil and Environmental Engineering, University of

Surrey, Guildford, UK

Corresponding author:

Jun Teng, Shenzhen Graduate School, Harbin Institute of Technology,

Shenzhen 518055, People’s Republic of China.

Email: tengj@hit.edu.cn

Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License

(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

often attached to the steel member surface to collectstress data in real time.15 However, the monitoredstress using SHM technologies is the relative value overa period of time, rather than the absolute value in thecurrent state.12 The absolute stress, which is the appliedmechanical stress at the present moment but not thestress variation in steel members, is a direct indicatorfor the evaluation of structural safety.16 The early eva-luation of absolute stress in steel components can pro-vide valuable information to assist in overallmaintenance and design of civil infrastructures inextreme climatic conditions.

The techniques used for absolute stress measurementcan be destructive methods or non-destructive meth-ods.17 Destructive methods require material removal ofthe tested steel members,18,19 and thus, these methodsare not appropriate for online component evaluation.Non-destructive methods, such as the X-ray diffractionmethod20 and neutron diffraction method,21 measurethe change in waveforms and rays caused by stress. Theimplementation of these methods requires complexequipment, which are usually used at the laboratoryscale. Therefore, these methods are unsuitable for largecomponent evaluation and field applications.

Absolute stress measurement using the Lcr wavemethod

Recently, the ultrasonic methods, which are based onthe acoustoelastic effect, have received research atten-tion in the field of SHM.22,23 The ultrasonic methodsare not limited by the component size and materialunder study, which makes them potentially useful insteel member absolute stress measurement.24 Among allultrasonic waves, the longitudinal critically refracted(Lcr) wave has been regarded as the best candidate toevaluate the absolute stress in materials.25 The Lcr waveis generated by the longitudinal wave mode conversion,and it can travel parallel to the material surface. Afterbeing refracted into the wedge that is in contact withthe steel member surface, the ultrasonic wave can bereceived by the transducer that is attached to thewedge.26 Because the change in the Lcr wave velocitycaused by stress is hard to be measured, the fixed acous-tic path method is commonly used to measure the Lcrwave time-of-flight (TOF) on a certain path.27 QM Zhuet al.28 studied the effects of stress and microstructureson the Lcr wave TOF and stress coefficient and investi-gated the residual stress evaluation both with and with-out different corrections. Furthermore, the Lcr waveattenuation velocity (LCR-AV) method was presentedto measure the welding residual stresses, and the hole-drilling reference method was used to verify the mea-surement reliability and precision.29 ZH Li et al.22

investigated the relationship between the applied stress

and Lcr wave TOF difference and measured the appliedstress in in-service steel members. Furthermore, theabsolute stress distribution measurement method waspresented to evaluate the stress field in the componentswith variable cross sections.30 Y Javadi et al.31 com-bined the Lcr wave and finite-element (FE) analysis toinvestigate the effects of clamping and post-weld heat-treatment on welding residual stress and deforma-tion.32,33 The combination of FE analysis and the Lcrwave, that is, the FELCR method,34 is an innovativework to evaluate the residual stress distribution throughthickness of the objects, which is an improvement of thetraditional Lcr wave method. The implementation ofthe FELCR wave method provides a reference for theabsolute stress evaluation in steel members. To improvethe Lcr wave TOF measurement accuracy, the laserultrasonic technique35,36 was used to generate a wide-frequency band pulse. The TOF of the ultrasonic pulsescan be measured with an accuracy of approximately0.5 ns using the designed optoacoustic transducer.Generally, the Lcr wave has been widely studied andused in the laboratory.

However, there is a limited number of practical engi-neering applications of the Lcr wave method.17,25 Themain reason for its limited use is that the Lcr waveformis easily affected by environmental factors, whichresults in inaccurate measurements of TOF.17,25 Boudaet al.37 noted that the major drawback of the Lcr wavein field measurements is the microstructure of the mate-rials, and their experiment confirmed that the Lcr waveattenuation and velocity were related to the metal grainsize. ZH Li et al.22 investigated the influence of thematerial properties on the parametric calibration of theLcr wave method and concluded that parametric cali-bration is a necessary step for the absolute stress eva-luation. W Djerir et al.38 explored the effects of crackon the Lcr wave propagation characteristics, and theexperiment showed that the amplitude of the Lcr wavedecreased and its frequency spectrum changed with agiven defect. In addition, temperature39 and couplingconditions40 also have a direct influence on the ultraso-nic wave attenuation and TOF. All these factors mayreduce the signal-to-noise ratio of the Lcr wave. In gen-eral, the goal of improving the stress measurement pre-cision requires accurate measurement of the Lcr waveTOF.

At present, the research on the Lcr wave TOF mea-surement is mainly focused on signal de-noising.17,25

After being filtered, a characteristic point, such as thepeak or valley point, is selected to judge the delay ofthe Lcr wave. In fact, the Lcr wave signal-to-noise ratiodepends on noise reduction parameters, that is, the Lcrwave TOF is different when different de-noising meth-ods and noise reduction parameters are used. In addi-tion, the inhomogeneous material41 and microcrack38

in the steel member may lead to the distortion of the

2 International Journal of Distributed Sensor Networks

Lcr waveform, which may cause the Lcr wave peak orvalley point to fluctuate in a certain range. Therefore,investigating a method to accurately measure the Lcrwave TOF is necessary. In signal processing, the cross-correlation method42 is used to judge the similaritybetween two signals at different times, which has theadvantage of resisting environmental noise. ZW Ling etal.43 employed the cross-correlation algorithm to mea-sure the time delay of the Rayleigh wave and the Lcrwave; and the pressure of the vessel could be evaluated.The cross-correlation method provides a potential wayto measure the Lcr wave TOF.

Goals of this study

In this study, the algorithm of measuring the Lcr waveTOF was developed based on the cross-correlationanalysis by making the best use of the similaritybetween the two Lcr waves received by the two differentreceivers. The cross-correlation-based algorithm has theadvantages of low computational cost and measure-ment of the Lcr wave TOF without signal de-noising.The features of the cross-correlation-based algorithmwere demonstrated in laboratory experimental studies.Specifically, the cross-correlation algorithm is used tomeasure the Lcr wave TOF under different stress condi-tions for four samples, and the correlation between theLcr wave TOF difference and the stress is discussed.The applied stresses on a steel member loaded by thetesting machine are measured using the presentedmethod, and the results are compared with thoseobtained using the traditional peak value method. Inaddition, the influence of the sampling rate on the Lcrwave TOF is discussed.

Theory

Absolute stress measurement using Lcr waves

A typical schematic diagram of the Lcr wave method isshown in Figure 1. The center frequency of the three

transducers (one transmitter and two receivers) is iden-tical. The transmitter transducer transmits a longitudi-nal wave. When the longitudinal wave travels from thepoly(methyl methacrylate) (PMMA) material tothe steel at the first critical angle (approximate 25.7�),the first refracted longitudinal wave transforms into theLcr wave. Then, the Lcr wave is generated and travelsin parallel to the component surface and is finallyreceived by two receiver transducers.

According to the acoustoelastic theory, the relation-ship between the stress and Lcr wave velocity travelingparallel to the stress direction can be written in the fol-lowing form26

r0V 2 = l+ 2m+s

3l+ 2m

l+m

m4l+ 10m+ 4mð Þ+ l+ 2l

� � ð1Þ

where V is the Lcr wave velocity when the steel memberis under free-stress condition; r0 is the material density;l and m are the Lame constants; and l, m, and n are theMurnaghan constants.

By using the fixed acoustic path method, the rela-tionship between stress and the Lcr wave TOF can beobtained. By making further simplifications, the follow-ing formulas can be obtained22

s =B(t0 � t) ð2Þ

B=1

Kt0ð3Þ

r0K =

4l+ 10m+ 4m

m+ 2l�3l�10m�4m

l+ 2m

2(3l+ 2m)ð4Þ

where B is the factor of stress to the acoustic time differ-ence (SATD) and t0 and t are the Lcr wave TOFs in thefree-stressed and stressed steel members, respectively.From equation (2), the accuracy of the stress dependson the Lcr wave TOF measurement.

Figure 1. Schematic diagram of the Lcr wave method.

Li et al. 3

Cross-correlation-based algorithm for Lcr wave TOFmeasurement

Typical waveforms received by the two receiver trans-ducers from Figure 1 are shown in Figure 2. Becausethe Lcr wave received by the second receiver travels alonger distance, its energy is more seriously attenuated,and thus, a lower amplitude in the time-domain signalsis the result. However, the waveforms of the first andsecond Lcr waves are nearly the same, which makes itpossible to measure the Lcr wave TOF between the tworeceivers using the cross-correlation-based algorithm.

Assuming S(n) is a series of points representing thewaveform of the Lcr wave, the waves received by thetwo receivers can be written in the following forms

F(n)= S(n)+N1(n) ð5Þ

G(n)= S(n� m�)+N2(n) ð6Þ

where F(n) and G(n) are two signal series received byreceiver 1 and receiver 2, respectively; m* is the Lcrwave signal delay; N1(n) and N2(n) are noises receivedby receiver 1 and receiver 2, respectively; and n is thenumber of sampling points.

The cross-correlation function between F(n) andG(n) is as follows

RFG(m)=X+‘

n= 0

F(n� m)G(n)½ � ð7Þ

where F(n – m) is the sequence right slided m points byF(n). By substituting equations (5) and (6) into equation(7), a new formula can be obtained

RFG(m)=RSS(m� m�)+RSN2+RN1S(m�m�)+RN1N2

ð8Þ

The cross-correlation between the noise and Lcr waveis zero.42 Therefore, equation (8) can be simplified asfollows

RFG(m)=RSS(m� m�) ð9Þ

According to the nature of the self-correlation function,when m = m*, RSS(m – m*) obtains the maximumvalue. Therefore, the position in time of the maximumamplitude corresponds to the signal delay.

The center frequency of the transducer is approxi-mately 3 MHz. The generated Lcr wave contains thefrequency components from 1 to 5 MHz. Thus, thecross-correlation function, RFG(m), fluctuates over alimited range, which is shown in Figure 3. Because theSavitzky–Golay method can effectively preserve theoriginal features of the data, it is used to smooththe cross-correlation curve. When the peak value of thecross-correlation curve reaches the maximum, the firstLcr wave can be regarded as slided to the position ofthe second Lcr wave, that is, the two Lcr waves coin-cide with each other. Then, the Lcr wave TOF betweenthe two receivers is as follows

Figure 2. Typical waveforms received by the two receivers.

4 International Journal of Distributed Sensor Networks

t=Dt � m� ð10Þ

where Dt is the interval time between two points andm* is the m*th point when the cross-correlation curvereaches the maximum value.

Methodology

According to the Lcr wave stress measurement theoryand cross-correlation-based algorithm, the measure-ment devices are selected and a measurement system isdesigned in this study, which is shown in Figure 4. Asensor group, which contains one transmitter and tworeceivers, is designed to transmit and receive Lcr waves.The sensor group can accurately characterize the Lcrwave propagation path and the Lcr wave TOF.30 Inaddition, the environmental effects, such as ambienttemperature, on the Lcr wave velocity can bedecreased.31,32,34,44

The ultrasonic generator transmits a beam of pulsevoltage signals and shunts them into the transmitter,which contains a chip made of piezoelectric ceramics.Because of the inverse piezoelectric effect, a longitudi-nal wave is produced and travels from the PMMA tothe steel at the angle of 25.7�. The Lcr wave is gener-ated and travels in the steel member. Then, the Lcrwave is received by two receivers with the same centerfrequency. The two Lcr waves are amplified by twoultrasonic preamplifiers and then collected by the oscil-loscope. The Lcr wave TOF between the two receiverscan be obtained by measuring the time delay of the sec-ond Lcr wave. To make a constant coupling film thick-ness between the transducers and steel member, twohigh-strength magnets are embedded in each probe toensure a constant pressure.

In this work, the critical angle is calculated from theSnell’s law. Then, the piezoelectric chip angle knob onthe transducer is fine-tuned to ensure that the ampli-tude of Lcr wave reaches maximum. Therefore, the crit-ical angle of the transmitter and receivers is 25.7�. Notethat the critical angle is different from some previousworks.31,32,34,44 This difference may be caused by thedifferent steel materials under study. The dimension ofthe magnet is 10 mm 3 5 mm 3 2 mm. The magnetis embedded in the edge of the PMMA to enable closecontact and effective coupling between the PMMA andthe steel member. The Lcr wave signals were collectedby oscilloscope with a sampling rate of 2.5 GSa/s, andthe wavelet transform method was used for signal de-noising. The experiment results show that the Lcr waveTOFs with and without the magnet are identical.Therefore, the influence of the magnets on the Lcr

Figure 4. Experimental setup for the measurement system.

Figure 3. Illustration of the cross-correlation sequence RFG(m).

Li et al. 5

wave propagation is not taken into account during theexperiments.

A five-step framework for measuring the Lcr waveTOF and absolute stress is proposed:

1. A replication member, with the same material asthe tested object, is employed for parametriccalibration. The transducers are attached to thereplicated member surface when it is under freestress. The signal sequence received by receiver 1when the steel member is under a free-stress con-dition is defined as F(n). The signal sequencesreceived by receiver 1 and receiver 2 when thesteel member is under the stress condition aredefined as G1(n) and G2(n), respectively.

2. Calculating the cross-correlation function of thetwo sequences. For the sequence F(n), the start-ing and ending points are on the left and rightsides of the Lcr wave, respectively, such that thewhole Lcr wave signal is selected as an analysisobject. For the sequences G1(n) and G2(n), thestarting points are the same position as F(n),while the ending point is on the right side of thesecond Lcr wave. By right sliding the sequenceF(n), the cross-correlation function in the differ-ent sliding positions can be calculated, and thecross-correlation curve can be obtained. Aftersmoothing the cross-correlation curve using theSavitzky–Golay method, the maximum value inthe cross-correlation curve can be collected.When the sample is under the free-stress condi-tion, the signals received from the receivers 1and 2 can be used to calculate the Lcr waveTOF, t0, using equation (10).

3. Measurement of the Lcr wave TOF under dif-ferent stress conditions. The testing machine isused to increase the tensile stress in the repli-cated member. The Lcr wave TOF between thetwo receivers under the ith stress condition is asfollows

ti =Dt m2i� � m1i

�ð Þ ð11Þ

where ti is the Lcr wave TOF between the two receiversunder the ith stress condition, m2i

� is the maximumpoint in the cross-correlation curve between F(n) andG2i(n), m1i

� is the maximum point in the cross-correlation curve between F(n) and G1i(n), and Dt is the

interval time between two points. The correspondingstress data, si, under the ith stress condition is thencollected.

4. Linear fitting of the factor, B, in equation (2).The Lcr wave TOF difference between free-stressed and stressed condition can be calcu-lated. Thus, a set of data ((t1 – t0, s1), (t2 – t0,s2), ., (tn – t0, sn)) can be obtained. Using theleast square method, the slope of the trend linecan be calibrated.

5. The absolute stress measurement for the testedobject. The Lcr wave TOF in the tested steelmember can be measured as that in step 3.Using the fitted SATD factor B and equation(2), the absolute stress in the tested object canbe calculated.

Experimental procedures

Sample description

In civil engineering, Q235 steel is widely used in load-bearing components and in connecting members. Inthis work, four Q235 steel members (samples A, B, Cand D), with the same dimensions of 370 mm 3 40mm 3 10 mm, are employed as research objects. Thefour specimens are cut and machined from the samesteel plate. To reduce the influence of the specimen sur-face roughness on the coupling film thickness, the steelmember surface is polished until the surface roughnessis less than 10 mm. The mechanical properties andchemical composition of the four steel members areshown in Table 1. The calibration of the parametersand the stress measurement for the steel members arecompleted at room temperature.

Parametric calibration

The four samples were used for parametric calibration.The transducers, which are connected by a Vernier cali-per, were attached to the center of the four steel mem-ber surfaces. The distance between receiver 1 andreceiver 2 is approximately 60 mm. The central fre-quency of the three transducers is approximately3 MHz, and the frequency spectrum is shown inFigure 5.

Table 1. Mechanical properties and chemical composition of Q235 steel.

Mechanical properties Chemical composition

Ultimate strength Yield strength Young’s modulus C Si Mn P S

420 MPa 235 MPa 210 GPa 0.20% 0.35% 1.40% 0.045% 0.045%

6 International Journal of Distributed Sensor Networks

The steel members were loaded with an increase instress of approximately 20 MPa using the electronicuniversal testing machine. The Lcr waves received bythe two receivers were recorded by the oscilloscope at asampling rate of 2.5 GSa/s under each stress condition.Using equation (7), the cross-correlation sequenceswere calculated, and the corresponding cross-correlation curves were obtained. After beingsmoothed, the maximum peak points in the cross-correlation curves were collected and the Lcr waveTOFs under each stress condition were calculated usingequation (11). Thus, the SATD factors in equation (2)were fitted for the four specimens using the least-squares method. To investigate the influence of thesampling rate on the Lcr wave TOF measurement andparameter calibration, the above process was repeatedat sampling rates of 2.0, 1.0, and 0.5 GSa/s.

The peak value method was also employed to pro-cess the above signals, and the measurement methodcan be found in previous papers.22,30 Because the high-frequency noises hinder the accurate capture of thepeak point in the Lcr wave, the Lcr wave signals werefiltered using the wavelet transform method. TheMATLAB software (MATLAB R2014b) was used towrite a program. Specifically, the function ‘‘ddencmp’’returned default values for denoising the Lcr wave sig-nal; then, the function ‘‘wavedec’’ returned the waveletdecomposition of the Lcr wave signal at level 4, using‘‘db16’’; finally, the function ‘‘wdencmp’’ performed ade-noising process of the Lcr wave signal. The SATDfactors were fitted for the four samples with samplingrates of 2.5, 2.0, 1.0, and 0.5 GSa/s.

Absolute stress measurement in a steel member

A set of loads were applied to sample A using the test-ing machine. The transducers were attached to the

sample A surface, and the distance of the two receiverswas the same as that of the parametric calibration. Thecross-correlation-based algorithm and the peak valuemethod were used to measure the Lcr wave TOF. Withthe calibrated coefficients under the sampling rate of2.5 GSa/s, the applied stresses in sample A can be cal-culated using equation (2). The strain gauge methodwas used to verify the results measured by the cross-cor-relation-based algorithm and the peak value method.

Results and discussions

Stress coefficient under different sampling rates

Change in the cross-correlation curves affected by thestresses in sample A (sampling rate: 2.5 GSa/s) is illu-strated in Figure 6, and the fitting lines of the SATDfactors for the four specimens are shown in Figure 7.For the peak value method, the Lcr waves (received by

Figure 5. The frequency spectrum of the transducers.

Figure 6. Change in the cross-correlation curves affected bythe stresses in sample A (sampling rate: 2.5 GSa/s).

Figure 7. Fitting lines of the SATD factors for the fourspecimens using the cross-correlation-based algorithm (samplingrate: 2.5 GSa/s).

Li et al. 7

the second receiver) affected by the stresses and the fit-ting lines are shown in Figures 8 and 9, respectively.Figure 6 shows that the cross-correlation curve has atendency to move right with the increase in the tensilestress in sample A. This indicates that the Lcr waveTOF between the two receivers increases with the ten-sile stress, which is in accordance with the phenomenashown in Figure 8. Both Figures 7 and 9 illustrate thatthe Lcr wave TOF difference exhibits a perfect linearrelationship with the stress in all four samples.

All the fitted SATD factors for the four specimensunder different sampling rate and the correspondingcorrelation coefficient (R2) using the two methods areshown in Figures 10 and 11, respectively. The compari-son of the coefficients is listed in Table 2. In compari-son with the peak value method, the correlationcoefficients of the SATD factors using the cross-

correlation-based algorithm are better at each samplingrate, which is shown in Figure 11. In addition, thestress errors caused by the Lcr wave TOF measurementare smaller. This indicates that the cross-correlation-based algorithm can obtain a more accurate Lcr waveTOF than the peak value method. The main reason isthat the cross-correlation-based algorithm calculatesthe similarity of two Lcr waves at different times, whilethe peak value method evaluates the time delay of apeak point in the Lcr wave. In fact, when the Lcr wavepropagates in steel members, the waveform is easilyaffected by factors such as inhomogeneous materialand microcracks in steel, which may lead to the distor-tion of the Lcr waveform. Thus, the peak and valleypoints may fluctuate over a small range, which resultsin an inaccurate measurement of the Lcr wave TOF. Inthis study, the aim of using the cross-correlation algo-rithm is to analyze the whole Lcr wave signal, but not asingle peak point in the Lcr wave. The fluctuation ofthe distortion points exerts a relatively small impact onthe Lcr wave TOF measurement. Therefore, the cross-

Figure 8. Change in the Lcr waves (received by the secondreceiver) affected by the stresses in sample A (sampling rate:2.5 GSa/s).

Figure 9. Fitting lines of the SATD factors for the fourspecimens using the peak value method (sampling rate: 2.5 GSa/s).

Figure 10. The SATD factors of the four samples withdifferent sampling rates.

Figure 11. The correlation coefficients (R2) of the foursamples with different sampling rates.

8 International Journal of Distributed Sensor Networks

correlation-based algorithm obtains a better correlationcoefficient than the peak value method.

The coefficients in Figure 11 with red color are mea-sured using the cross-correlation-based algorithm. Thefour samples show the same tendency, in which the cor-relation coefficient (R2) is reduced with a decrease inthe sampling rate. When the sampling rates are2.5 GSa/n and 0.5 GSa/s, the correlation coefficientsare 0.9994 and 0.9533, respectively, which shows a cer-tain difference. In fact, the sampling rate of the signalis related to the time resolution. When the samplingrates are 2.5 GSa/n and 0.5 GSa/s, the time resolutionis 0.4 and 2.0 ns, respectively. This illustrates that thelow sampling rate leads to a decrease in the Lcr waveTOF recognition accuracy. Although the cross-correla-tion-based algorithm can offer better Lcr wave TOFresults than the peak value method, it is difficult toimprove the Lcr wave TOF resolution. The effectiveway to obtain a better Lcr wave TOF resolution is toincrease the signal sampling rate.

When the sampling rate is 2.5 GSa/s, Figure 11shows some discrepancies of the fitted SATD factorsfor the four samples, which may be caused by the tex-ture of the materials. Although the four samples are cutfrom the same steel plate, the machine process may leadto uneven changes in the surface and internal materialof the four samples. However, these differences aresmall and can usually be ignored.

Absolute stress measurement results

The measurement results of the applied stresses in sam-ple A using the three methods are shown in Figure 12.The error comparison between the cross-correlation-

based algorithm and peak value method is shown inFigure 13. Figure 12 shows that the applied stress insample A can be measured using the cross-correla-tion-based algorithm and the peak value method. Themaximum difference for the cross-correlation-basedalgorithm is 6.7 MPa, while the peak value is approxi-mately 16.0 MPa. Overall, the cross-correlation-basedalgorithm is more consistent with the strain gaugemethod.

Note that the original Lcr waves are not de-noisedwhen the cross-correlation-based algorithm is used tomeasure TOF. The essence of the cross-correlation-based algorithm is the amplification of the Lcr waveenergy and the suppression of the noise signal.Theoretically, the effects of the noise signal on thecross-correlation-based curve are ignored, which is

Table 2. Coefficients of the fitting lines with four sampling rates using the two methods.

Sampling rate The cross-correlation-based algorithm The peak value method

Averagea Standarddeviation/average

Stresserrorb (MPa)

Average Standarddeviation/average

Stress errorb

(MPa)

2.5 GSa/s B (MPa/ns) 5.7022 2.69% 15.9 5.7020 2.68% 31.0t0 (ns) 14,691.1 0.19& 14,691.3 0.37&R2 0.9994 0.04% 0.9978 0.18%

2.0 GSa/s B (MPa/ns) 5.7058 2.59% 18.4 5.7517 3.01% 36.3t0 (ns) 14,690.6 0.22& 14,691.1 0.43%R2 0.9987 0.04% 0.9943 0.50&

1.0 GSa/s B (MPa/ns) 5.7974 3.92% 25.5 5.8468 5.22% 49.0t0 (ns) 14,690.2 0.30& 14,691.2 0.57&R2 0.9840 0.58% 0.9723 0.65%

0.5 GSa/s B (MPa/ns) 5.5637 2.68% 31.1 5.6873 9.52% 60.2t0 (ns) 14,690.1 0.38& 14,691.2 0.72&R2 0.9533 1.09% 0.9286 1.51%

aAverage is the mean value of a specific coefficient for the four samples.bThe stress error is the product of the average value of the SATD factor (B) and the standard deviation of the Lcr wave TOF (t0).

Figure 12. The measurement results of the three methods.

Li et al. 9

illustrated in equations (8) and (9). The noise signal canlead to small fluctuations in the cross-correlationsequences. Thus, the Savitzky–Golay method is used tosmooth the cross-correlation curve. Figure 3 indicatesthat there is little difference between the two sequencesbefore and after smoothing, but the smoothed curve ismore convenient for the maximum peak value collec-tion. The cross-correlation algorithm in this study pro-vides a potential way to measure the Lcr wave TOF forsteel member absolute stress evaluation in practicalengineering applications.

Conclusion

In this article, a cross-correlation-based algorithm ispresented to measure the Lcr wave TOF and absolutestress in steel members. According to the achievedresults, the conclusions are summarized as follows:

1. The cross-correlation-based algorithm makesuse of the cross-correlation to study the similar-ity of two Lcr waves, which are received by twodifferent receivers. The original Lcr waves areused directly without being filtered to calculatethe cross-correlation sequence. The maximumpeak point in the cross-correlation curve corre-sponds to the Lcr wave TOF.

2. The cross-correlation-based algorithm and thepeak value method are used to measure the Lcrwave TOF under different stress conditions.The stress and corresponding Lcr TOF differ-ence is linearly fitted. The correlation coeffi-cients of the cross-correlation-based algorithmare better than the peak value method.

3. With the decrease in the sampling rate, the cor-relation coefficient is reduced and the stress

measurement error increases. The presentedcross-correlation-based algorithm cannot com-pensate for the shortage of high sampling ratesof the Lcr wave method.

4. A set of random stresses are measured using thecross-correlation-based algorithm and the peakvalue method, and the results of the former aremore accurate. As the cross-correlation-basedalgorithm has the advantage of resisting envi-ronmental noise, it provides a potential way toaccurately measure the Lcr wave TOF and eval-uate the absolute stress in practical engineeringapplications.

Acknowledgements

The co-first authors (Z.L. and J.H.) contributed equally tothis work.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This work was financially supported by the NationalKey Research and Development Program of China underGrant 2016YFC0701102, the National Natural ScienceFoundation of China under Grant 51538003, the NationalMajor Scientific Research Instrument Development Programof China under Grant 51827811, and Shenzhen TechnologyInnovation Program under Grant JCYJ20170811160003571.The authors would like to thank these financial supports.

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