Research ArticleEffects of Concrete on Propagation Characteristics ofGuided Wave in Steel Bar Embedded in Concrete
Zhupeng Zheng and Ying Lei
Department of Civil Engineering Xiamen University Xiamen 361005 China
Correspondence should be addressed to Ying Lei yleixmueducn
Received 22 June 2014 Revised 22 July 2014 Accepted 28 July 2014 Published 27 August 2014
Academic Editor Ting-Hua Yi
Copyright copy 2014 Z Zheng and Y LeiThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Techniques based on ultrasonic guided waves (UGWs) play important roles in the structural health monitoring (SHM) of large-scale civil infrastructures In this paper dispersion equations of longitudinal wave propagation in reinforced concrete member areinvestigated for the purpose of monitoring steels embedded in concrete For a steel bar embedded in concrete not the velocity butthe attenuation dispersion curves will be affected by the concreteThe effects of steel-to-concrete shear modulus ratio density ratioand Poissonrsquos ratio on propagation characteristics of guided wave in steel bar embedded in concrete were studied by the analysis ofthe real and imaginary parts of the wave numberThe attenuation characteristics of guided waves of steel bar in different conditionsincluding different bar concrete constraint and different diameter of steel bar are also analyzed Studies of the influence of concreteon propagation characteristics of guided wave in steel bars embedded in concrete will increase the accuracy in judging the structureintegrity and promote the level of defect detection for the steel bars embedded in concrete
1 Introduction
In recent years techniques based on ultrasonic guided waves(UGWs) have gained popularities and played important rolesin the structural health monitoring (SHM) of large-scale civilinfrastructures as guided waves have some important advan-tages such as the capability of testing over long range with agreater sensitivity the ability to test multilayered structuresand relatively cheapness due to simplicity and sensor cost[1] Furthermore frequency and mode tuning of UGWs canbe utilized for evaluation of different types of deteriorationor damage because UGWs have many different modes at asingle frequency that are sensitive to different defects Dueto the above various techniques based on guided wave havebeen proposed for damage detection and condition assess-ments of aerospace mechanical civil engineering structuresand other nondestructive test (NDT) areas Raghavan andCesnik [2] presented a review of guided wave for structuralhealth monitoring Sohn et al [3] investigated delamina-tion detection in composites through guided wave imagingSong et al [4 5] developed smart piezoceramic transduc-ers with guided wave for concrete structural health mon-itoring Giurgiutiu [6] studied lamb wave generation with
piezoelectric wafer active sensors for structural health mon-itoring (SHM) Wang et al [7] investigated the effects of adefectrsquos geometric parameters on the two reflection signalsin pipe using guided waves and proposed a new strategyfor accurate and quantitative pipeline defect characterizationCobb et al [8] studied the torsional guided wave attenuationin piping from coating temperature and large-area corrosionand obtained experimental results that wave attenuation is agood indicator of general corrosion level Ahmad and Kundu[9] studied the influence of water flow through pipe networkson damage detection using guided waves Beard et al [10]used guided waves to inspect concrete reinforcing tendonsand evaluated the effect of factors such as leakage and defectgeometry on the inspection Zhu et al [11] used ultrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trusses The finite element method (FEM) isoften carried out to assist structural design [12] and also usedto study the guidedwaves for SHM in the civil infrastructuresYi et al [13] used FEM to predict the location of pittingcorrosion in reinforced concrete based on guided wavesSimulation results show that it is feasible to predict corrosionmonitoring based on ultrasonic guided wave in reinforcedconcreteMoser et al [14] simulated the propagation of elastic
Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 910750 14 pageshttpdxdoiorg1011552014910750
2 Shock and Vibration
wave in the sheet and tubular structure using FEM Theresults are fully consistent with those from experimentwhich further proves the validity of the simulation in wavepropagation using FEM Chen [15] used FEM to simulate thedefect monitoring by longitudinal guided wave and get therelation curves between reflection coefficient and circumfer-ential length or axial length of the defect in pipe He et al[16] studied the propagation of guided waves in bending pipeusing FEM Other recent developments of guided waves forSHMwere also discussed in a review prepared byHuang et al[17]
Among the current available corrosion monitoring tech-niques in reinforced concrete the technique based on ultra-sonic guided wave has gained more and more popularityin the recent years due to its advantages for monitoringcorrosion related damage in reinforcing bars so it has gainedpopularities in the recent years [18] Na et al [19] used bothhigh (1MHz) and low frequencies (150KHz) to study theeffect of various bond levels by surrounding the rebar witha polyvinyl chloride polymer (PVC) pipe in RC beams andthe effect of debonding location on the received waveformsThey in the same way conducted a comparison of steel con-crete interfaces and glass fiber polymer-concrete interfacesusing the guided waves [20 21] Reis et al [22] used thefundamental flexural mode below 250 kHz for estimation ofcorrosion damage in steel reinforced mortar Wu and Chang[23 24] used the piezoelectric discs as sensors and actuatorsto detect debonding in reinforced concrete structures A setof reinforced concrete square beam specimens with variousbond levels were built and tested using guided mechanicalwaves at lower frequencies He et al [25] used frequenciesbetween 1 and 2MHz to relate the effect of debonding onsignals in cylindrical specimens Dongsheng et al [26] usedfive-cycle sinusoidal signals with 120 kHz central frequencyto investigate the UGW energy attenuation on the differentdebonding level between steel bar and concrete in both time-domain and frequency-domain analyses
All the above test results indicated that the receivedwaveform is less attenuated with the increase in debondingfor both low and high frequencies However the lowerfrequencies showed more sensitivity to the change in bondThere was no significant change in the waveform arrivaltime reported The location of debonding is not discerniblethrough pulse transmission as reported by Evin et al [27 28]
These studies were mostly carried out with simulateddebonding In these cases it is in fact the free steel bar withinthe concreteThere is difference in the propagation character-istics of guided waves between the free steel bars and the steelbar surrounded by concrete However the research on theeffects of concrete on propagation characteristics of guidedwave in steel bar embedded in concrete is less reportedbecause there is a difficulty of limitation of monitoring rangeof guided wave in reinforced concrete [18 27] Unlike guidedwave propagation in other multilayered systems such as ametal pipeline in air wave energy in steel bars embeddedin concrete will be reduced (ie attenuated) at high ratesdue to leakage into the surrounding concrete For the defectstest of steel bar embedded in concrete the reflected signalswill be very weak so the general time-frequency methods
are difficult to identify the weak reflection signals of thedefects in the detection signals
In an infinite isotropic solid medium only two types ofindependent wave propagation exist that is compressionand shear waves Both waves propagate with constant veloc-ities and are nondispersive When geometry constraints areintroduced and the dimensions are close to the wavelengththe wave becomes dispersive and is called a guided waveLongitudinal torsional and flexural waves propagate inisotropic cylinders The characteristic equation for solidisotropic cylinders was originally independently derived forthe special case of longitudinal propagation in the late 19thcentury [29 30] Solutions to this equation contain thephase velocity and frequency (assuming no absorption bythe medium) The derivations assume the solid cylinder istraction free and infinite in length and that the wave form hasharmonic motion The characteristic equation for torsionaland flexural propagation was later derived [31 32]
In civil infrastructures steel bars are usually embedded inconcrete so the existence of concrete is of strong interferencewith the integrity evaluation of steel bars which lead to thedifficult problem in the detection of steel bars using guidedwave based techniques [27] Steel bar embedded in concretecan be modeled as an isotropic solid cylinder embedded inan infinite isotropic medium The derivation is similar tothe solid cylinder in vacuum however displacement andstress boundary conditions must now be met at the interfacebetween the twomediums In the case of reinforced concretea solution includes phase velocity frequency and attenuationSo there is obviously different propagation characteristics ofUGWs in steel bars embedded in concrete compared withthose in free bars Therefore it is important to study theinfluence of concrete on propagation characteristics of guidedwave in steel bars embedded in concrete which will increasethe accuracy in judging the structure integrity and promotethe level of defect detection for the steel bars embedded inconcrete [33 34]
2 Dispersion Equations of LongitudinalGuided Wave in Reinforced Concrete
For a steel bar embedded in concrete the guided wavespropagate not only along the steel bar but also spreadoutward which leads to the wave energy that diffuses fromsteel bar into the concrete Furthermore the change of waveimpedance of concrete can also cause the reflection of thewaves Thus the wave equations of guided waves in steel bar-concrete system can be established based on the 3D elasticwave theory
The simplified model of a steel bar embedded in concreteis shown in Figure 1 in which the inner layer of mediumis the steel bar with the radius 119886 and peripheral medium isconcrete which is infinite in radial direction Guided wavesare assumed to propagate along the 119885-direction In orderto facilitate the analysis the reinforcement was set as roundsteel bar In the propagation in the steel bar embedded inconcrete the outward propagation and the attenuation due to
Shock and Vibration 3
120579Z
a
r
Steel bar
Concrete
Figure 1 Simplified model of steel bar embedded in concrete
concrete of guided waves can all be expressed by the correla-tion functions in which steel bar is regarded as an isotropicelastic rod and the solution of wave equations can be solvedby separation variables method
Displacement of guided waves propagating in steel barcan be expressed by Bessel function [32] The velocities oflongitudinal and shear waves in steel bar-concrete systemmeet the relations 119862LS gt 119862TS gt 119862LC gt 119862TC where 119862LS and119862TS are the velocity of longitudinal and transverse waves insteel bar respectively and 119862LC and 119862TC are the velocity oflongitudinal and transverse waves in concrete respectivelyOnly Hankel function can meet the required condition whenthewave number 119896 is a complex index [35]Therefore Hankelfunction is chosen to describe the outward propagation ofwaves which attenuates to zero at infinity Based on thedisplacements the stress components in both steel bar andconcrete can be deduced according to the relationships ofstrain-displacement and stress-strain
It is assumed that bonding condition at the interfacebetween steel bar and concrete is good and the displacementand stress at the interface are continuous so
119906119903minus 1199061015840
119903
119906120579minus 1199061015840
120579
119906119911minus 1199061015840
119911
120590119903119903minus 1205901015840
119903119903
120590119903120579minus 1205901015840
119903120579
120590119903119911minus 1205901015840
119903119911
= 0 (119903 = 119886) (1)
where 119906119903 119906120579 119906119911 and 119906
1015840
119903 1199061015840120579 1199061015840119911are the displacement
components of the steel bar and concrete respectively 120590119903119903
120590119903120579 120590119903119911and 1205901015840119903119903 1205901015840119903120579 and 1205901015840
119903119911are the stress components of the
steel bar and concrete respectively
Then substituting displacement and stress into (1) thefollowing matrix can be obtained as follows [23]
11986311
11986312
11986313
11986314
11986315
11986316
11986321
11986322
11986323
11986324
11986325
11986326
11986331
11986332
11986333
11986334
11986335
11986336
11986341
11986342
11986343
11986344
11986345
11986346
11986351
11986352
11986353
11986354
11986355
11986356
11986361
11986362
11986363
11986364
11986365
11986366
1198601
1198604
1198606
1198601015840
1
1198601015840
4
1198601015840
6
= 0 (2)
where detailed expressions of 119863119894119895(119894 119895 = 1 6) are related
to the displacement and stress components of steel barand concrete and 119860
1 1198604 1198606 1198601015840
1 1198601015840
4 and 1198601015840
6are unknown
coefficients [32]If there is a nonzero solution to (2) the coefficient matrix
of the determinant must be zero so the following matrixequation is satisfied to obtain the general dispersion equationfor the steel bar with infinite length embedded in concrete[36]
There are axisymmetric and nonaxisymentric modes ofwaves included in the propagation of guidedwaves in the steelbar [32] The axisymmetric mode is the longitudinal mode119871(0119898) where the first index refers to the circumferentialorder of the mode and all of the longitudinal modes are ofcircumferential order 0 that is 119899 = 0 in (3) and the secondindex 119898 is a counter variable which means the sequentialorder of the mode Then (3) can be further simplified intotwo subdeterminants after exchange of rows and columnsand can get the following equations
When subdeterminant 119863119886= 0 it represents the disper-
sive equation of longitudinal modes in which only nonzerodisplacements 120583
119903and 120583
119911are included
When subdeterminant 119863119887= 0 it represents the dis-
persive equation of torsional modes in which only nonzerodisplacement 120583
120579is included
Further the following definitions are introduced [37]119884 =119896119886 119881 = 120572
119904119886 119882 = 120572
119888119886 119883 = 120573
119904119886 and 119880 = 120573
119888119886 where 119884
is dimensionless wave number and 119881 119882 119883 and 119880 are alldimensionless On the substitution of these definitions intothe equation 119863
119886= 0 we will get the disperse equation of
longitudinal mode of guided waves in reinforced concreteafter simplifying the results into a simple form expressed bydimensionless quantity as shown in the following [37]
Then the following equations can be derived from thetheory of elasticity [22]
119888119879119895
119888119871119895
= [
1 minus 2V119895
2(1 minus V119895)
]
12
119888119879119904
119888119879119888
= (
120583119904
120583119888
times
120588119888
120588119904
)
12
120582119895
120583119895
=
2V119895
1 minus 2V119895
120582119888
120583119904
=
2V119888
1 minus 2V119888
times
120583119888
120583119904
(6)
On the substitution of these variables into (5) all variablesexcept frequency and wave number in the dimensionlessdispersion equation can be determined by the properties ofsteel bar and concrete which include shearmodulus Poissonrsquosratio and density
If the parameters of steel bar and concrete are knownonly two unknown variables (dimensionless frequency and
dimensionless wave number) are involved in (5) In thispaper the related parameters of steel bar and concrete areselected as those in Table 1
In order to solve the relationship between dimensionlessfrequency and dimensionless wave numbers in the dispersionequation (5) the corresponding program is made to solve itThe solving procedures are as follows
(1) Find the cutoff frequency of each order mode from asweep frequency with a specified range which is theinitial root of the equation and also the initial pointof the disperse curve of each order mode Generallythere will be the same number of order modes asthat of initial points which have been found withina specified range of frequency and the correspondingnumber of disperse curves can be obtained
(2) The routine starts from the initial point in the givenstep and direction and converges to the next pointwhich is the second root of the equation
(3) Take the second root as the initial point and repeatstep 2 and so forth The routine will end until thespecified frequency or upper limit wave number isreached
(4) Connect all the found points and a disperse curve willbe achieved after
(5) Repeat the above process from another initial pointand another disperse curve will be achieved in thisway The complete solution to the equation will begotten until all the disperse curves are finished
Based on the solutions the disperse curves betweendimensionless frequency and dimensionless wave numberof the round steel bar embedded in concrete with differentmaterial parameters will be gained by solving (5) as shownin Figures 2ndash4
Shock and Vibration 5
Table 1 Related material parameters of steel bar and concrete
Real partof dimensionless wave numberof dimensionless wave number
0 5 10 15
Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
8
6
4
2
0
10
0
L(0 3)
L(0 3)L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus4 minus3 minus2 minus1
times10minus6
Figure 2 Disperse curves of the first three modes of guided waves in steel bar in the air
From Figures 2ndash4 it can be observed that for free steelbar wave numbers are all real numbers in the solution to thedisperse equation and the corresponding modes of guidedwaves are free modes or modes without attenuation For thesolid round steel bar embedded in concrete wave numbersare all complex in the solution to the disperse equation inwhich the real part of wave number is corresponding tothe propagation of guided waves in the steel bar and theimaginary part is corresponding to the attenuation of guidedwaves
It can also be drawn from Figures 2ndash4 that the dispersecurves between dimensionless frequency and dimensionlessreal wave number are almost the same whether the steelbar is in the air or embedded in concrete which means thatthe shear and density of surrounded concrete have no effecton the disperse curves that is the velocity of guided waves
propagating in the steel bar is only related to the properties ofsteel bar
As shown in Figures 3 and 4 the disperse curves betweendimensionless frequency and dimensionless imaginary wavenumber change with the shear and density of concreteWhen the concrete has lower shear modulus or density theimaginary wave number is lower the spread of the waveattenuation in steel bar is slow and the transmission distanceis further and vice versa
From the disperse curves in Figure 3 in which steelbar is embedded in ordinary concrete it can be seen thatfor the mode 119871(0 1) when the dimensionless frequencyis less than 18 the imaginary wave number is about 115which is reflecting the energy attenuation of guided wavesAnalogously fromFigure 4 in which steel bar is embedded inconcrete with high strength it can be seen that for the same
6 Shock and Vibration
Real partof dimensionless wave numberof dimensionless wave number
00 5 10 15Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus04 minus03 minus02 minus01
Figure 3 Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)L(0 2)
L(0 2)
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 4 Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete
mode 119871(0 1) when the dimensionless frequency is less than15 the imaginary wave number is even smaller that is about019 This means that energy attenuation of guided waves inordinary strength concrete is slower than that in high strengthconcrete And the imaginary wave number of mode 119871(0 1)increases quickly with the increase of frequency no matter inordinary concrete or in high strength concrete
4 Parametric Analysis
The impact of concrete on the propagation characteristicsof guided wave in steel bar can be evaluated by changingthe material properties of concrete which include the fac-tors such as steel-to-concrete shear modulus ratio densityratio and ratio of Poissonrsquos ratio Furthermore from previous
Shock and Vibration 7
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
10100
1000Free boundary1000
10010
Free boundary
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
wave in the sheet and tubular structure using FEM Theresults are fully consistent with those from experimentwhich further proves the validity of the simulation in wavepropagation using FEM Chen [15] used FEM to simulate thedefect monitoring by longitudinal guided wave and get therelation curves between reflection coefficient and circumfer-ential length or axial length of the defect in pipe He et al[16] studied the propagation of guided waves in bending pipeusing FEM Other recent developments of guided waves forSHMwere also discussed in a review prepared byHuang et al[17]
Among the current available corrosion monitoring tech-niques in reinforced concrete the technique based on ultra-sonic guided wave has gained more and more popularityin the recent years due to its advantages for monitoringcorrosion related damage in reinforcing bars so it has gainedpopularities in the recent years [18] Na et al [19] used bothhigh (1MHz) and low frequencies (150KHz) to study theeffect of various bond levels by surrounding the rebar witha polyvinyl chloride polymer (PVC) pipe in RC beams andthe effect of debonding location on the received waveformsThey in the same way conducted a comparison of steel con-crete interfaces and glass fiber polymer-concrete interfacesusing the guided waves [20 21] Reis et al [22] used thefundamental flexural mode below 250 kHz for estimation ofcorrosion damage in steel reinforced mortar Wu and Chang[23 24] used the piezoelectric discs as sensors and actuatorsto detect debonding in reinforced concrete structures A setof reinforced concrete square beam specimens with variousbond levels were built and tested using guided mechanicalwaves at lower frequencies He et al [25] used frequenciesbetween 1 and 2MHz to relate the effect of debonding onsignals in cylindrical specimens Dongsheng et al [26] usedfive-cycle sinusoidal signals with 120 kHz central frequencyto investigate the UGW energy attenuation on the differentdebonding level between steel bar and concrete in both time-domain and frequency-domain analyses
All the above test results indicated that the receivedwaveform is less attenuated with the increase in debondingfor both low and high frequencies However the lowerfrequencies showed more sensitivity to the change in bondThere was no significant change in the waveform arrivaltime reported The location of debonding is not discerniblethrough pulse transmission as reported by Evin et al [27 28]
These studies were mostly carried out with simulateddebonding In these cases it is in fact the free steel bar withinthe concreteThere is difference in the propagation character-istics of guided waves between the free steel bars and the steelbar surrounded by concrete However the research on theeffects of concrete on propagation characteristics of guidedwave in steel bar embedded in concrete is less reportedbecause there is a difficulty of limitation of monitoring rangeof guided wave in reinforced concrete [18 27] Unlike guidedwave propagation in other multilayered systems such as ametal pipeline in air wave energy in steel bars embeddedin concrete will be reduced (ie attenuated) at high ratesdue to leakage into the surrounding concrete For the defectstest of steel bar embedded in concrete the reflected signalswill be very weak so the general time-frequency methods
are difficult to identify the weak reflection signals of thedefects in the detection signals
In an infinite isotropic solid medium only two types ofindependent wave propagation exist that is compressionand shear waves Both waves propagate with constant veloc-ities and are nondispersive When geometry constraints areintroduced and the dimensions are close to the wavelengththe wave becomes dispersive and is called a guided waveLongitudinal torsional and flexural waves propagate inisotropic cylinders The characteristic equation for solidisotropic cylinders was originally independently derived forthe special case of longitudinal propagation in the late 19thcentury [29 30] Solutions to this equation contain thephase velocity and frequency (assuming no absorption bythe medium) The derivations assume the solid cylinder istraction free and infinite in length and that the wave form hasharmonic motion The characteristic equation for torsionaland flexural propagation was later derived [31 32]
In civil infrastructures steel bars are usually embedded inconcrete so the existence of concrete is of strong interferencewith the integrity evaluation of steel bars which lead to thedifficult problem in the detection of steel bars using guidedwave based techniques [27] Steel bar embedded in concretecan be modeled as an isotropic solid cylinder embedded inan infinite isotropic medium The derivation is similar tothe solid cylinder in vacuum however displacement andstress boundary conditions must now be met at the interfacebetween the twomediums In the case of reinforced concretea solution includes phase velocity frequency and attenuationSo there is obviously different propagation characteristics ofUGWs in steel bars embedded in concrete compared withthose in free bars Therefore it is important to study theinfluence of concrete on propagation characteristics of guidedwave in steel bars embedded in concrete which will increasethe accuracy in judging the structure integrity and promotethe level of defect detection for the steel bars embedded inconcrete [33 34]
2 Dispersion Equations of LongitudinalGuided Wave in Reinforced Concrete
For a steel bar embedded in concrete the guided wavespropagate not only along the steel bar but also spreadoutward which leads to the wave energy that diffuses fromsteel bar into the concrete Furthermore the change of waveimpedance of concrete can also cause the reflection of thewaves Thus the wave equations of guided waves in steel bar-concrete system can be established based on the 3D elasticwave theory
The simplified model of a steel bar embedded in concreteis shown in Figure 1 in which the inner layer of mediumis the steel bar with the radius 119886 and peripheral medium isconcrete which is infinite in radial direction Guided wavesare assumed to propagate along the 119885-direction In orderto facilitate the analysis the reinforcement was set as roundsteel bar In the propagation in the steel bar embedded inconcrete the outward propagation and the attenuation due to
Shock and Vibration 3
120579Z
a
r
Steel bar
Concrete
Figure 1 Simplified model of steel bar embedded in concrete
concrete of guided waves can all be expressed by the correla-tion functions in which steel bar is regarded as an isotropicelastic rod and the solution of wave equations can be solvedby separation variables method
Displacement of guided waves propagating in steel barcan be expressed by Bessel function [32] The velocities oflongitudinal and shear waves in steel bar-concrete systemmeet the relations 119862LS gt 119862TS gt 119862LC gt 119862TC where 119862LS and119862TS are the velocity of longitudinal and transverse waves insteel bar respectively and 119862LC and 119862TC are the velocity oflongitudinal and transverse waves in concrete respectivelyOnly Hankel function can meet the required condition whenthewave number 119896 is a complex index [35]Therefore Hankelfunction is chosen to describe the outward propagation ofwaves which attenuates to zero at infinity Based on thedisplacements the stress components in both steel bar andconcrete can be deduced according to the relationships ofstrain-displacement and stress-strain
It is assumed that bonding condition at the interfacebetween steel bar and concrete is good and the displacementand stress at the interface are continuous so
119906119903minus 1199061015840
119903
119906120579minus 1199061015840
120579
119906119911minus 1199061015840
119911
120590119903119903minus 1205901015840
119903119903
120590119903120579minus 1205901015840
119903120579
120590119903119911minus 1205901015840
119903119911
= 0 (119903 = 119886) (1)
where 119906119903 119906120579 119906119911 and 119906
1015840
119903 1199061015840120579 1199061015840119911are the displacement
components of the steel bar and concrete respectively 120590119903119903
120590119903120579 120590119903119911and 1205901015840119903119903 1205901015840119903120579 and 1205901015840
119903119911are the stress components of the
steel bar and concrete respectively
Then substituting displacement and stress into (1) thefollowing matrix can be obtained as follows [23]
11986311
11986312
11986313
11986314
11986315
11986316
11986321
11986322
11986323
11986324
11986325
11986326
11986331
11986332
11986333
11986334
11986335
11986336
11986341
11986342
11986343
11986344
11986345
11986346
11986351
11986352
11986353
11986354
11986355
11986356
11986361
11986362
11986363
11986364
11986365
11986366
1198601
1198604
1198606
1198601015840
1
1198601015840
4
1198601015840
6
= 0 (2)
where detailed expressions of 119863119894119895(119894 119895 = 1 6) are related
to the displacement and stress components of steel barand concrete and 119860
1 1198604 1198606 1198601015840
1 1198601015840
4 and 1198601015840
6are unknown
coefficients [32]If there is a nonzero solution to (2) the coefficient matrix
of the determinant must be zero so the following matrixequation is satisfied to obtain the general dispersion equationfor the steel bar with infinite length embedded in concrete[36]
There are axisymmetric and nonaxisymentric modes ofwaves included in the propagation of guidedwaves in the steelbar [32] The axisymmetric mode is the longitudinal mode119871(0119898) where the first index refers to the circumferentialorder of the mode and all of the longitudinal modes are ofcircumferential order 0 that is 119899 = 0 in (3) and the secondindex 119898 is a counter variable which means the sequentialorder of the mode Then (3) can be further simplified intotwo subdeterminants after exchange of rows and columnsand can get the following equations
When subdeterminant 119863119886= 0 it represents the disper-
sive equation of longitudinal modes in which only nonzerodisplacements 120583
119903and 120583
119911are included
When subdeterminant 119863119887= 0 it represents the dis-
persive equation of torsional modes in which only nonzerodisplacement 120583
120579is included
Further the following definitions are introduced [37]119884 =119896119886 119881 = 120572
119904119886 119882 = 120572
119888119886 119883 = 120573
119904119886 and 119880 = 120573
119888119886 where 119884
is dimensionless wave number and 119881 119882 119883 and 119880 are alldimensionless On the substitution of these definitions intothe equation 119863
119886= 0 we will get the disperse equation of
longitudinal mode of guided waves in reinforced concreteafter simplifying the results into a simple form expressed bydimensionless quantity as shown in the following [37]
Then the following equations can be derived from thetheory of elasticity [22]
119888119879119895
119888119871119895
= [
1 minus 2V119895
2(1 minus V119895)
]
12
119888119879119904
119888119879119888
= (
120583119904
120583119888
times
120588119888
120588119904
)
12
120582119895
120583119895
=
2V119895
1 minus 2V119895
120582119888
120583119904
=
2V119888
1 minus 2V119888
times
120583119888
120583119904
(6)
On the substitution of these variables into (5) all variablesexcept frequency and wave number in the dimensionlessdispersion equation can be determined by the properties ofsteel bar and concrete which include shearmodulus Poissonrsquosratio and density
If the parameters of steel bar and concrete are knownonly two unknown variables (dimensionless frequency and
dimensionless wave number) are involved in (5) In thispaper the related parameters of steel bar and concrete areselected as those in Table 1
In order to solve the relationship between dimensionlessfrequency and dimensionless wave numbers in the dispersionequation (5) the corresponding program is made to solve itThe solving procedures are as follows
(1) Find the cutoff frequency of each order mode from asweep frequency with a specified range which is theinitial root of the equation and also the initial pointof the disperse curve of each order mode Generallythere will be the same number of order modes asthat of initial points which have been found withina specified range of frequency and the correspondingnumber of disperse curves can be obtained
(2) The routine starts from the initial point in the givenstep and direction and converges to the next pointwhich is the second root of the equation
(3) Take the second root as the initial point and repeatstep 2 and so forth The routine will end until thespecified frequency or upper limit wave number isreached
(4) Connect all the found points and a disperse curve willbe achieved after
(5) Repeat the above process from another initial pointand another disperse curve will be achieved in thisway The complete solution to the equation will begotten until all the disperse curves are finished
Based on the solutions the disperse curves betweendimensionless frequency and dimensionless wave numberof the round steel bar embedded in concrete with differentmaterial parameters will be gained by solving (5) as shownin Figures 2ndash4
Shock and Vibration 5
Table 1 Related material parameters of steel bar and concrete
Real partof dimensionless wave numberof dimensionless wave number
0 5 10 15
Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
8
6
4
2
0
10
0
L(0 3)
L(0 3)L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus4 minus3 minus2 minus1
times10minus6
Figure 2 Disperse curves of the first three modes of guided waves in steel bar in the air
From Figures 2ndash4 it can be observed that for free steelbar wave numbers are all real numbers in the solution to thedisperse equation and the corresponding modes of guidedwaves are free modes or modes without attenuation For thesolid round steel bar embedded in concrete wave numbersare all complex in the solution to the disperse equation inwhich the real part of wave number is corresponding tothe propagation of guided waves in the steel bar and theimaginary part is corresponding to the attenuation of guidedwaves
It can also be drawn from Figures 2ndash4 that the dispersecurves between dimensionless frequency and dimensionlessreal wave number are almost the same whether the steelbar is in the air or embedded in concrete which means thatthe shear and density of surrounded concrete have no effecton the disperse curves that is the velocity of guided waves
propagating in the steel bar is only related to the properties ofsteel bar
As shown in Figures 3 and 4 the disperse curves betweendimensionless frequency and dimensionless imaginary wavenumber change with the shear and density of concreteWhen the concrete has lower shear modulus or density theimaginary wave number is lower the spread of the waveattenuation in steel bar is slow and the transmission distanceis further and vice versa
From the disperse curves in Figure 3 in which steelbar is embedded in ordinary concrete it can be seen thatfor the mode 119871(0 1) when the dimensionless frequencyis less than 18 the imaginary wave number is about 115which is reflecting the energy attenuation of guided wavesAnalogously fromFigure 4 in which steel bar is embedded inconcrete with high strength it can be seen that for the same
6 Shock and Vibration
Real partof dimensionless wave numberof dimensionless wave number
00 5 10 15Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus04 minus03 minus02 minus01
Figure 3 Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)L(0 2)
L(0 2)
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 4 Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete
mode 119871(0 1) when the dimensionless frequency is less than15 the imaginary wave number is even smaller that is about019 This means that energy attenuation of guided waves inordinary strength concrete is slower than that in high strengthconcrete And the imaginary wave number of mode 119871(0 1)increases quickly with the increase of frequency no matter inordinary concrete or in high strength concrete
4 Parametric Analysis
The impact of concrete on the propagation characteristicsof guided wave in steel bar can be evaluated by changingthe material properties of concrete which include the fac-tors such as steel-to-concrete shear modulus ratio densityratio and ratio of Poissonrsquos ratio Furthermore from previous
Shock and Vibration 7
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
10100
1000Free boundary1000
10010
Free boundary
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Figure 1 Simplified model of steel bar embedded in concrete
concrete of guided waves can all be expressed by the correla-tion functions in which steel bar is regarded as an isotropicelastic rod and the solution of wave equations can be solvedby separation variables method
Displacement of guided waves propagating in steel barcan be expressed by Bessel function [32] The velocities oflongitudinal and shear waves in steel bar-concrete systemmeet the relations 119862LS gt 119862TS gt 119862LC gt 119862TC where 119862LS and119862TS are the velocity of longitudinal and transverse waves insteel bar respectively and 119862LC and 119862TC are the velocity oflongitudinal and transverse waves in concrete respectivelyOnly Hankel function can meet the required condition whenthewave number 119896 is a complex index [35]Therefore Hankelfunction is chosen to describe the outward propagation ofwaves which attenuates to zero at infinity Based on thedisplacements the stress components in both steel bar andconcrete can be deduced according to the relationships ofstrain-displacement and stress-strain
It is assumed that bonding condition at the interfacebetween steel bar and concrete is good and the displacementand stress at the interface are continuous so
119906119903minus 1199061015840
119903
119906120579minus 1199061015840
120579
119906119911minus 1199061015840
119911
120590119903119903minus 1205901015840
119903119903
120590119903120579minus 1205901015840
119903120579
120590119903119911minus 1205901015840
119903119911
= 0 (119903 = 119886) (1)
where 119906119903 119906120579 119906119911 and 119906
1015840
119903 1199061015840120579 1199061015840119911are the displacement
components of the steel bar and concrete respectively 120590119903119903
120590119903120579 120590119903119911and 1205901015840119903119903 1205901015840119903120579 and 1205901015840
119903119911are the stress components of the
steel bar and concrete respectively
Then substituting displacement and stress into (1) thefollowing matrix can be obtained as follows [23]
11986311
11986312
11986313
11986314
11986315
11986316
11986321
11986322
11986323
11986324
11986325
11986326
11986331
11986332
11986333
11986334
11986335
11986336
11986341
11986342
11986343
11986344
11986345
11986346
11986351
11986352
11986353
11986354
11986355
11986356
11986361
11986362
11986363
11986364
11986365
11986366
1198601
1198604
1198606
1198601015840
1
1198601015840
4
1198601015840
6
= 0 (2)
where detailed expressions of 119863119894119895(119894 119895 = 1 6) are related
to the displacement and stress components of steel barand concrete and 119860
1 1198604 1198606 1198601015840
1 1198601015840
4 and 1198601015840
6are unknown
coefficients [32]If there is a nonzero solution to (2) the coefficient matrix
of the determinant must be zero so the following matrixequation is satisfied to obtain the general dispersion equationfor the steel bar with infinite length embedded in concrete[36]
There are axisymmetric and nonaxisymentric modes ofwaves included in the propagation of guidedwaves in the steelbar [32] The axisymmetric mode is the longitudinal mode119871(0119898) where the first index refers to the circumferentialorder of the mode and all of the longitudinal modes are ofcircumferential order 0 that is 119899 = 0 in (3) and the secondindex 119898 is a counter variable which means the sequentialorder of the mode Then (3) can be further simplified intotwo subdeterminants after exchange of rows and columnsand can get the following equations
When subdeterminant 119863119886= 0 it represents the disper-
sive equation of longitudinal modes in which only nonzerodisplacements 120583
119903and 120583
119911are included
When subdeterminant 119863119887= 0 it represents the dis-
persive equation of torsional modes in which only nonzerodisplacement 120583
120579is included
Further the following definitions are introduced [37]119884 =119896119886 119881 = 120572
119904119886 119882 = 120572
119888119886 119883 = 120573
119904119886 and 119880 = 120573
119888119886 where 119884
is dimensionless wave number and 119881 119882 119883 and 119880 are alldimensionless On the substitution of these definitions intothe equation 119863
119886= 0 we will get the disperse equation of
longitudinal mode of guided waves in reinforced concreteafter simplifying the results into a simple form expressed bydimensionless quantity as shown in the following [37]
Then the following equations can be derived from thetheory of elasticity [22]
119888119879119895
119888119871119895
= [
1 minus 2V119895
2(1 minus V119895)
]
12
119888119879119904
119888119879119888
= (
120583119904
120583119888
times
120588119888
120588119904
)
12
120582119895
120583119895
=
2V119895
1 minus 2V119895
120582119888
120583119904
=
2V119888
1 minus 2V119888
times
120583119888
120583119904
(6)
On the substitution of these variables into (5) all variablesexcept frequency and wave number in the dimensionlessdispersion equation can be determined by the properties ofsteel bar and concrete which include shearmodulus Poissonrsquosratio and density
If the parameters of steel bar and concrete are knownonly two unknown variables (dimensionless frequency and
dimensionless wave number) are involved in (5) In thispaper the related parameters of steel bar and concrete areselected as those in Table 1
In order to solve the relationship between dimensionlessfrequency and dimensionless wave numbers in the dispersionequation (5) the corresponding program is made to solve itThe solving procedures are as follows
(1) Find the cutoff frequency of each order mode from asweep frequency with a specified range which is theinitial root of the equation and also the initial pointof the disperse curve of each order mode Generallythere will be the same number of order modes asthat of initial points which have been found withina specified range of frequency and the correspondingnumber of disperse curves can be obtained
(2) The routine starts from the initial point in the givenstep and direction and converges to the next pointwhich is the second root of the equation
(3) Take the second root as the initial point and repeatstep 2 and so forth The routine will end until thespecified frequency or upper limit wave number isreached
(4) Connect all the found points and a disperse curve willbe achieved after
(5) Repeat the above process from another initial pointand another disperse curve will be achieved in thisway The complete solution to the equation will begotten until all the disperse curves are finished
Based on the solutions the disperse curves betweendimensionless frequency and dimensionless wave numberof the round steel bar embedded in concrete with differentmaterial parameters will be gained by solving (5) as shownin Figures 2ndash4
Shock and Vibration 5
Table 1 Related material parameters of steel bar and concrete
Real partof dimensionless wave numberof dimensionless wave number
0 5 10 15
Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
8
6
4
2
0
10
0
L(0 3)
L(0 3)L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus4 minus3 minus2 minus1
times10minus6
Figure 2 Disperse curves of the first three modes of guided waves in steel bar in the air
From Figures 2ndash4 it can be observed that for free steelbar wave numbers are all real numbers in the solution to thedisperse equation and the corresponding modes of guidedwaves are free modes or modes without attenuation For thesolid round steel bar embedded in concrete wave numbersare all complex in the solution to the disperse equation inwhich the real part of wave number is corresponding tothe propagation of guided waves in the steel bar and theimaginary part is corresponding to the attenuation of guidedwaves
It can also be drawn from Figures 2ndash4 that the dispersecurves between dimensionless frequency and dimensionlessreal wave number are almost the same whether the steelbar is in the air or embedded in concrete which means thatthe shear and density of surrounded concrete have no effecton the disperse curves that is the velocity of guided waves
propagating in the steel bar is only related to the properties ofsteel bar
As shown in Figures 3 and 4 the disperse curves betweendimensionless frequency and dimensionless imaginary wavenumber change with the shear and density of concreteWhen the concrete has lower shear modulus or density theimaginary wave number is lower the spread of the waveattenuation in steel bar is slow and the transmission distanceis further and vice versa
From the disperse curves in Figure 3 in which steelbar is embedded in ordinary concrete it can be seen thatfor the mode 119871(0 1) when the dimensionless frequencyis less than 18 the imaginary wave number is about 115which is reflecting the energy attenuation of guided wavesAnalogously fromFigure 4 in which steel bar is embedded inconcrete with high strength it can be seen that for the same
6 Shock and Vibration
Real partof dimensionless wave numberof dimensionless wave number
00 5 10 15Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus04 minus03 minus02 minus01
Figure 3 Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)L(0 2)
L(0 2)
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 4 Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete
mode 119871(0 1) when the dimensionless frequency is less than15 the imaginary wave number is even smaller that is about019 This means that energy attenuation of guided waves inordinary strength concrete is slower than that in high strengthconcrete And the imaginary wave number of mode 119871(0 1)increases quickly with the increase of frequency no matter inordinary concrete or in high strength concrete
4 Parametric Analysis
The impact of concrete on the propagation characteristicsof guided wave in steel bar can be evaluated by changingthe material properties of concrete which include the fac-tors such as steel-to-concrete shear modulus ratio densityratio and ratio of Poissonrsquos ratio Furthermore from previous
Shock and Vibration 7
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
10100
1000Free boundary1000
10010
Free boundary
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
When subdeterminant 119863119886= 0 it represents the disper-
sive equation of longitudinal modes in which only nonzerodisplacements 120583
119903and 120583
119911are included
When subdeterminant 119863119887= 0 it represents the dis-
persive equation of torsional modes in which only nonzerodisplacement 120583
120579is included
Further the following definitions are introduced [37]119884 =119896119886 119881 = 120572
119904119886 119882 = 120572
119888119886 119883 = 120573
119904119886 and 119880 = 120573
119888119886 where 119884
is dimensionless wave number and 119881 119882 119883 and 119880 are alldimensionless On the substitution of these definitions intothe equation 119863
119886= 0 we will get the disperse equation of
longitudinal mode of guided waves in reinforced concreteafter simplifying the results into a simple form expressed bydimensionless quantity as shown in the following [37]
Then the following equations can be derived from thetheory of elasticity [22]
119888119879119895
119888119871119895
= [
1 minus 2V119895
2(1 minus V119895)
]
12
119888119879119904
119888119879119888
= (
120583119904
120583119888
times
120588119888
120588119904
)
12
120582119895
120583119895
=
2V119895
1 minus 2V119895
120582119888
120583119904
=
2V119888
1 minus 2V119888
times
120583119888
120583119904
(6)
On the substitution of these variables into (5) all variablesexcept frequency and wave number in the dimensionlessdispersion equation can be determined by the properties ofsteel bar and concrete which include shearmodulus Poissonrsquosratio and density
If the parameters of steel bar and concrete are knownonly two unknown variables (dimensionless frequency and
dimensionless wave number) are involved in (5) In thispaper the related parameters of steel bar and concrete areselected as those in Table 1
In order to solve the relationship between dimensionlessfrequency and dimensionless wave numbers in the dispersionequation (5) the corresponding program is made to solve itThe solving procedures are as follows
(1) Find the cutoff frequency of each order mode from asweep frequency with a specified range which is theinitial root of the equation and also the initial pointof the disperse curve of each order mode Generallythere will be the same number of order modes asthat of initial points which have been found withina specified range of frequency and the correspondingnumber of disperse curves can be obtained
(2) The routine starts from the initial point in the givenstep and direction and converges to the next pointwhich is the second root of the equation
(3) Take the second root as the initial point and repeatstep 2 and so forth The routine will end until thespecified frequency or upper limit wave number isreached
(4) Connect all the found points and a disperse curve willbe achieved after
(5) Repeat the above process from another initial pointand another disperse curve will be achieved in thisway The complete solution to the equation will begotten until all the disperse curves are finished
Based on the solutions the disperse curves betweendimensionless frequency and dimensionless wave numberof the round steel bar embedded in concrete with differentmaterial parameters will be gained by solving (5) as shownin Figures 2ndash4
Shock and Vibration 5
Table 1 Related material parameters of steel bar and concrete
Real partof dimensionless wave numberof dimensionless wave number
0 5 10 15
Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
8
6
4
2
0
10
0
L(0 3)
L(0 3)L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus4 minus3 minus2 minus1
times10minus6
Figure 2 Disperse curves of the first three modes of guided waves in steel bar in the air
From Figures 2ndash4 it can be observed that for free steelbar wave numbers are all real numbers in the solution to thedisperse equation and the corresponding modes of guidedwaves are free modes or modes without attenuation For thesolid round steel bar embedded in concrete wave numbersare all complex in the solution to the disperse equation inwhich the real part of wave number is corresponding tothe propagation of guided waves in the steel bar and theimaginary part is corresponding to the attenuation of guidedwaves
It can also be drawn from Figures 2ndash4 that the dispersecurves between dimensionless frequency and dimensionlessreal wave number are almost the same whether the steelbar is in the air or embedded in concrete which means thatthe shear and density of surrounded concrete have no effecton the disperse curves that is the velocity of guided waves
propagating in the steel bar is only related to the properties ofsteel bar
As shown in Figures 3 and 4 the disperse curves betweendimensionless frequency and dimensionless imaginary wavenumber change with the shear and density of concreteWhen the concrete has lower shear modulus or density theimaginary wave number is lower the spread of the waveattenuation in steel bar is slow and the transmission distanceis further and vice versa
From the disperse curves in Figure 3 in which steelbar is embedded in ordinary concrete it can be seen thatfor the mode 119871(0 1) when the dimensionless frequencyis less than 18 the imaginary wave number is about 115which is reflecting the energy attenuation of guided wavesAnalogously fromFigure 4 in which steel bar is embedded inconcrete with high strength it can be seen that for the same
6 Shock and Vibration
Real partof dimensionless wave numberof dimensionless wave number
00 5 10 15Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus04 minus03 minus02 minus01
Figure 3 Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)L(0 2)
L(0 2)
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 4 Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete
mode 119871(0 1) when the dimensionless frequency is less than15 the imaginary wave number is even smaller that is about019 This means that energy attenuation of guided waves inordinary strength concrete is slower than that in high strengthconcrete And the imaginary wave number of mode 119871(0 1)increases quickly with the increase of frequency no matter inordinary concrete or in high strength concrete
4 Parametric Analysis
The impact of concrete on the propagation characteristicsof guided wave in steel bar can be evaluated by changingthe material properties of concrete which include the fac-tors such as steel-to-concrete shear modulus ratio densityratio and ratio of Poissonrsquos ratio Furthermore from previous
Shock and Vibration 7
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
10100
1000Free boundary1000
10010
Free boundary
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Real partof dimensionless wave numberof dimensionless wave number
0 5 10 15
Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
8
6
4
2
0
10
0
L(0 3)
L(0 3)L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus4 minus3 minus2 minus1
times10minus6
Figure 2 Disperse curves of the first three modes of guided waves in steel bar in the air
From Figures 2ndash4 it can be observed that for free steelbar wave numbers are all real numbers in the solution to thedisperse equation and the corresponding modes of guidedwaves are free modes or modes without attenuation For thesolid round steel bar embedded in concrete wave numbersare all complex in the solution to the disperse equation inwhich the real part of wave number is corresponding tothe propagation of guided waves in the steel bar and theimaginary part is corresponding to the attenuation of guidedwaves
It can also be drawn from Figures 2ndash4 that the dispersecurves between dimensionless frequency and dimensionlessreal wave number are almost the same whether the steelbar is in the air or embedded in concrete which means thatthe shear and density of surrounded concrete have no effecton the disperse curves that is the velocity of guided waves
propagating in the steel bar is only related to the properties ofsteel bar
As shown in Figures 3 and 4 the disperse curves betweendimensionless frequency and dimensionless imaginary wavenumber change with the shear and density of concreteWhen the concrete has lower shear modulus or density theimaginary wave number is lower the spread of the waveattenuation in steel bar is slow and the transmission distanceis further and vice versa
From the disperse curves in Figure 3 in which steelbar is embedded in ordinary concrete it can be seen thatfor the mode 119871(0 1) when the dimensionless frequencyis less than 18 the imaginary wave number is about 115which is reflecting the energy attenuation of guided wavesAnalogously fromFigure 4 in which steel bar is embedded inconcrete with high strength it can be seen that for the same
6 Shock and Vibration
Real partof dimensionless wave numberof dimensionless wave number
00 5 10 15Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus04 minus03 minus02 minus01
Figure 3 Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)L(0 2)
L(0 2)
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 4 Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete
mode 119871(0 1) when the dimensionless frequency is less than15 the imaginary wave number is even smaller that is about019 This means that energy attenuation of guided waves inordinary strength concrete is slower than that in high strengthconcrete And the imaginary wave number of mode 119871(0 1)increases quickly with the increase of frequency no matter inordinary concrete or in high strength concrete
4 Parametric Analysis
The impact of concrete on the propagation characteristicsof guided wave in steel bar can be evaluated by changingthe material properties of concrete which include the fac-tors such as steel-to-concrete shear modulus ratio densityratio and ratio of Poissonrsquos ratio Furthermore from previous
Shock and Vibration 7
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
10100
1000Free boundary1000
10010
Free boundary
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Real partof dimensionless wave numberof dimensionless wave number
00 5 10 15Imaginary part
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)
L(0 2)
L(0 2)
minus04 minus03 minus02 minus01
Figure 3 Disperse curves of the first three modes of guided waves in steel bar embedded in ordinary concrete
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
L(0 3)
L(0 3)
L(0 1)
L(0 1)L(0 2)
L(0 2)
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 4 Disperse curves of the first three modes of guided waves in steel bar embedded in high strength concrete
mode 119871(0 1) when the dimensionless frequency is less than15 the imaginary wave number is even smaller that is about019 This means that energy attenuation of guided waves inordinary strength concrete is slower than that in high strengthconcrete And the imaginary wave number of mode 119871(0 1)increases quickly with the increase of frequency no matter inordinary concrete or in high strength concrete
4 Parametric Analysis
The impact of concrete on the propagation characteristicsof guided wave in steel bar can be evaluated by changingthe material properties of concrete which include the fac-tors such as steel-to-concrete shear modulus ratio densityratio and ratio of Poissonrsquos ratio Furthermore from previous
Shock and Vibration 7
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
10100
1000Free boundary1000
10010
Free boundary
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Figure 5 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete shear modulus ratio
studies by the authors [38] it is known that the cutoff freq-uency of the lowest order mode 119871(0 1) in steel bar is zeroso those modes propagating along the steel bar will alwaysinclude it Thus the effects of related factors on the lowestorder mode 119871(0 1) are mainly taken into considerationwhich are studied respectively in the following part
41 Effect of Shear Modulus Ratio When the steel bar isembedded in ordinary concrete the shear modulus ratio isabout 10 which will increase when the surrounding concretebecomes softer Until it is up to a certain extent the boundaryof steel bar can be regarded as a free boundary By changingthe ratio the relationship between dimensionless wave num-ber and dimensionless frequency can be gained as shown inFigure 5
From Figure 5 it can be seen that the change of shearmodulus ratio has little effect on the real part of dimensionlesswave number However with the increase of shear modu-lus ratio (that is the surrounding concrete becomes moreand more soft) the imaginary part of dimensionless wavenumber which representing the attenuation of guided wavesbecomes smaller and smallerThis shows that the attenuationof energy due to leakage into the surrounding concrete isgetting smaller and farther distance of propagation can beachieved It is considered that no energy leak into concretein the condition of free boundary so the dimensionless wavenumber will be real number
42 Effect of Density Ratio When the density of concrete ischanged and other parameters remain constant the disperse
relation will surely be affected While steel bar is embeddedin high strength concrete the steel-to-concrete density ratiois 36 Then decreasing gradually the density of concretethe steel-to-concrete density ratio will be increased and thechange of the disperse curve between dimensionless wavenumber and dimensionless frequency will be gotten as shownin Figure 6When the ratio is close to infinity it indicates thatthe effect of surrounding concrete can be neglected or it canbe regarded as free boundary
From Figure 6 it can be seen that the change of densityratio has little effect on the real part of dimensionless wavenumber However with the decrease of density ratio (ie thesurrounding concrete becomes more and more dense) theimaginary part of dimensionless wave number representingthe attenuation of guided waves becomes more and morelarge which shows that the attenuation of energy due toleakage into the surrounding concrete is getting larger andshorter distance of propagation can be achieved It is alsoconsidered that no energy leak into concrete in the conditionof free boundary so the dimensionless wave number is realnumber
43 Effect of Ratio of Poissonrsquos Ratio As Poissonrsquos ratio of steelbar and concrete can be changed and large changes in thePoissonrsquos ratio of concrete from low grade to high grade takeplace the effect on the disperse curve by Poissonrsquos ratio can begotten by assuming Poissonrsquos ratio of steel bar as a constant of02865 and changing that of concrete as shown in Figure 7where V is Poissonrsquos ratio of concrete
From Figure 7 it can be found that the variations both inthe real and imaginary parts of wave number are small with
8 Shock and Vibration
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
360363636
36360Free boundary Free boundary
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Figure 6 Influence on the real and imaginary parts of 119871(0 1) by the steel-bar-to-concrete density ratio
= 012
= 027
= 020
= 012
= 027
= 020
Real partof dimensionless wave number
0 5 10 15
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
Dim
ensio
nles
s fre
quen
cy
8
6
4
2
0
10
of dimensionless wave number
0Imaginary part
minus04 minus03 minus02 minus01
Figure 7 Influence on the real and imaginary parts of 119871(0 1) by Poissonrsquos ratio of concrete
the change of Poissonrsquos ratio of concrete in low frequencyWhen the dimensionless frequency is greater than 18 theattenuation will be obviously affected with the increase offrequency and the smaller Poissonrsquos ratio of concrete thegreater the attenuation
5 Attenuation Characteristics ofGuided Wave in Reinforced Concrete
Through the previous definition and analysis it usuallyregarding the disperse curve between the imaginary part
Shock and Vibration 9
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
of dimensionless wave number and dimensionless frequencyas the attenuation curve In fact the forward propagationof guided waves in steel bar is a process in which energyattenuates gradually due to the effects of constrains ofconcrete impact excitation energy material damping of steeland defects on steel bar If there are much residues or defectsin steel bar or its characteristics is close to the surroundingconcrete (close density or shear modulus) the waves willvanish due to attenuation after a short distance of propagationand then it will be very difficult to detect the defect of steelbar using the traditional echo method Based on the researchof the attenuation curve the energy dissipation of guidedwaves propagating along the steel bar and the effects on theattenuation bymaterial properties propagation distance andfrequency can be known more clearly which will be a benefitfor the defect detection in the reinforced concrete
When the steel bar is in the air (free steel bar) the solvedimaginary wave number is very small and even close to zerobecause there is great difference in the physical propertiesbetween steel bar and the airTherefore its propagationmodeis approximated free or no attenuation mode
For the steel bar embedded in concrete the imaginarywave number depends on the shearmodulus ratio anddensityratio between steel bar and concrete The energy loss of wavecan be quantified by the displacement 119906 which is expressedin the steel bar as follows [39]
inwhich 119896119903and 119896119894are the real and imaginary parts of thewave
number respectively 119886 is the radius of steel bar and 119911 is theaxial direction of steel bar
Thus the attenuation of guided wave along the steel barcan be measured using the amplitude variation of the signalalong the 119911-direction In (7) the first part119880(119886) is the functionof radius which is a constant taking a fixed point along theradiusThe second part shows that the wave propagates alongthe +119885 directionThe third part is corresponding to the expo-nential increase or decrease of wave energy which dependson the symbol of 119896
119894 The solution to the disperse equation in
the steel bar surrounded by concrete shows that 119896119894is negative
and its displacement attenuates exponentially along the +119885directionThe attenuating value of displacement along the +119885direction depends on the amplitude of imaginary part of wavenumberThe attenuation coefficient can be signified byNeperabbreviated as119873
119901
120574 = log119890(
1199062
1199061
)119873119901 (8)
where 120574 is the attenuation coefficient and 1199061and 119906
2are the
amplitudes of signals at the initial point and last point ofmeasuring The ratio of amplitudes 119860 is defined as
119860 =
1199062
1199061
= 119890120574
(9)
In (9) the first two parts have no effect on the amplitudeof displacement in 119911-direction and they can be substituted bya constant 119864 Thus the attenuation coefficient is expressed as
where 1199111and 1199112are the initial and final positions of measure-
ment The attenuation coefficient can be turned into decibelsas shown in the following
120574 = 119896119894nepersm = 8686 119896
119894dBm (11)
51 Steel Bar Embedded in Ordinary Concrete The dispersecurves for steel bar with the diameter of 22 mm embedded inordinary concrete are shown in Figure 8 In the figure thegreen and red dotscircles are the cutoff frequency of eachmode and the roots of the dispersive equation respectivelyThe involved material parameters are tabulated in Table 1
By comparing the energy velocity curve to phase velocitycurve in Figure 8 it can be seen that when the local mini-mum value of attenuation is achieved the correspondingenergy velocity at the frequency is the maximum While themaximum value attenuation is reached the correspondingenergy velocity at the frequency is the minimum
From the phase velocity disperse curves in Figure 8(a) itcan be observed that the fundamental 119871(0 1)mode is startingat zero frequency while each higher-order mode is startingfrom a higher cutoff frequency Each of the higher modesshows a plateau region around the steel longitudinal bulkvelocity line first and then decreases eventually approachingthe transverse wave velocity The plateau regions correspondto the points of the maximum energy velocity (Figure 8(b))and minimum attenuation (Figure 8(c)) But 119871(0 8) modeshows a different pattern Instead of each plateau regionbelonging to a single mode 119871(0 8) breaks from this patternand links the subsequent plateau regions together to form asingle mode that propagates close to the longitudinal bulkvelocity of steel This 119871(0 8) mode is chosen for study At afrequency of 172MHz the mode exhibits global attenuationminima of 18 dBm and it is the fastest propagating modeThe phase velocity obtained from dispersion curve at thisfrequency is 6 Kms
By (11) the corresponding amplitude ratio of signals atdifferent propagating distance of guided waves in steel barembedded in ordinary concrete can be calculated as tabulatedin Table 2
As can be seen from Table 2 for the steel bar embeddedin ordinary concrete after more and more energy leaksinto the concrete with the increase of propagation distancethe internal energy becomes less and its particle vibrationbecomes more and more weak and then the amplitude ratioof signals is becoming smaller and smaller
Figure 9 shows the axial displacement 119880119911and strain
energy density (SED) distribution for 119871(0 8) mode studiedin the present investigationThe energy is concentrated in thecentral core portion of the bar and has relatively less surface
10 Shock and Vibration
05 10 15 20 25 3000
20
40
60
80
100
Frequency (MHz)
Vph
(mm
s)
L(0 8)
L(0 1)
(a) Phase velocity disperse curves
05 10 15 20 25 3000
20
40
60
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms)
L(0 8)
L(0 1)
(b) Energy velocity disperse curves
05 10 15 20 25 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 8)
L(0 1)
(c) Attenuation disperse curves
Figure 8 Disperse curves of guided waves in steel bar embedded in ordinary concrete
Table 2 Attenuation coefficient and the signal amplitude ratio ofmode 119871(0 8)
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Figure 9 Mode shape of 119871(0 8) at 119891 = 172MHz
component Hence it should be more sensitive to local bartopography or loss of material changes and not the surfaceprofile changes
52 Steel Bar Embedded in Concrete with High Strength Thedisperse curves for steel bar embedded in high strengthconcrete with the diameter of 22mm are shown in Figure 10In the figure the green and red dotscircles are the cutofffrequency of each mode and the roots of the dispersiveequation respectively The involved material parameters arealso tabulated in Table 1
The phase velocity disperse curves in Figure 10(a) showthat similar to those of ordinary concrete the fundamental119871(0 1)mode is starting at zero frequency while each higher-order mode is starting from a higher cutoff frequency Eachof the higher modes shows a plateau region around thesteel longitudinal bulk velocity line first and then decreaseseventually approaching the transverse wave velocity Theplateau regions correspond to the points of the maximumenergy velocity (Figure 10(b)) and minimum attenuation(Figure 10(c)) But 119871(0 14) mode shows a different patternInstead of each plateau region belonging to a single mode119871(0 14) breaks from this pattern and links the subsequentplateau regions together to form a single mode that propa-gates close to the longitudinal bulk velocity of steel At the fre-quency band corresponding to the plateau regions of modes119871(0 6) and 119871(0 14) they are nearly the fastest propagatingmodes and exhibit global attenuation minima as shown inFigures 10(b) and 10(c) At a frequency of 102MHz themode119871(0 6) has the attenuation minima of 368 dBm and at afrequency of 295MHz the mode 119871(0 14) has the attenua-tion minima of 366 dBm The phase velocities of the two
Shock and Vibration 11
10 20 3000
20
40
60
80
100
120
Frequency (MHz)
L(0 6)
L(0 14)
Vph
(mm
s)
(a) Phase velocity disperse curves
10 20 3000
10
20
30
40
50
Frequency (MHz)
Ener
gy v
eloci
ty (z
) (m
ms) L(0 6) L(0 14)
(b) Energy velocity disperse curves
10 20 300
100
200
300
400
Frequency (MHz)
Att
(dB
m)
L(0 6) L(0 14)
(c) Attenuation disperse curves
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Figure 10 Disperse curves of guided waves in steel bar embedded in high strength concrete
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 11 Mode shape of 119871(0 6) at 119891 = 102MHz
modes as obtained from dispersion curves at the frequencycorresponding to the minimum attenuation are both about6Kms The attenuation value of 368 dBm is taken andthen the corresponding amplitude ratio of signals at differentpropagating distance of guided waves in steel bar embeddedin high strength concrete is calculated as tabulated in Table 3
Figures 11 and 12 show the axial displacement 119880119911and
strain energy density (SED) distribution for modes 119871(0 6)and 119871(0 14) respectively It can be seen that the energy ofmode 119871(0 6) is concentrated in the central core portion of thebar and has relatively less surface component And the energy
00
20
40
60
80
100
Pos
(mm
)
SED
Uz
Figure 12 Mode shape of 119871(0 14) at 119891 = 294MHz
of mode 119871(0 14) is more concentrated in the central coreportion of the bar Hence it will be less sensitive to the surfaceprofile changes
From the analysis it can be drawn that the higher thestrength concrete the more the energy leakage into concretethe faster the attenuation of guidedwaves and the smaller theamplitude ratio of signals at the same position Guided wavesat low frequency mode can propagate very far along the steelbar in the air for the little attenuation so a few meters of steelbar can be monitored by guided waves based technique [40]However while the same steel bar is embedded in ordinary
12 Shock and Vibration
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
Table 3 Attenuation coefficient and the signal amplitude ratio of mode 119871(0 6)
Propagating distance (m) 0 1 15 2 25 3 5Attenuation (dB) 0 minus368 minus552 minus736 minus92 minus1104 minus184Amplitude ratio 1000 00150 00017 00002 000002 30 times 10minus6 62 times 10minus10
050
100150200250300350400450
0 05 1 15 2 25 3
Atte
nuat
ion
(dB
m)
Frequency (MHz)
d = 14mmd = 22mmd = 30mm
Figure 13 Attenuation disperse curves of mode 119871(0 8) in steel barwith different diameter embedded in concrete
concrete the monitoring range is just about twometers Evenmore if it is embedded in high strength concrete the testdistance would be very limited because more and moreenergy leaks into the concrete and with the increasedpropagation the internal energy of steel bar becomes less andits particle vibration becomesmore andmore weak Based onthe above it can be concluded that whether it is embedded inordinary concrete or high strength concrete the attenuationvelocity of guided waves in steel bar is always high and thepropagation distance is always short that is the detectionrange is very limited as the signal is veryweak after twometersof propagation which is the difficulty of nondestructive testapplying guided waves in reinforced concrete
53 Influence on Attenuation of Guided Wave by DifferentBar Diameter For the given reinforced concrete modelthe disperse curves of attenuation of longitudinal mode119871(0 8) for the steel bar embedded in ordinary concrete withthe diameter 119889 = 14mm 119889 = 22mm and 119889 = 30mm arecalculated respectively as shown in Figure 13 in whichthe attenuation curves are denoted by red blue and yellowcurves respectively corresponding to different diameter
As can be seen from Figure 13 the trends of attenuationdisperse curves are undulate and eventually tend to be similarAt the same frequency the value of attenuation decreaseswiththe increase of diameterTherefore for the same steel bar withlarger diameter embedded in the same concrete the energyleaks less into concrete the spread of wave energy attenuationis slower and the spread distance is shorter
6 Conclusions
In reinforced concrete structures the propagation charac-teristics of guided waves in steel bar are influenced greatlyby its surrounded concrete In this paper the impact ofconcrete on the propagation characteristics of guided wavein steel bar is studied by changing the material properties ofconcrete It is found that the higher the excitation frequencythe more obvious the dispersion phenomenon However notthe velocity but attenuation dispersion curves will be affectedby the concrete The shear modulus and density of concretehave no effect on the real parts of wave number that is thepropagating velocity of guided waves in steel bar dependsonly on thematerial properties of steel bar and has nothing todowith the surrounding concreteThe imaginary part ofwavenumber depends on the steel-bar-to-concrete shear modulusratio and density ratio When the concrete has lower shearmodulus or density the spread of thewave attenuation in steelbar is slow and the transmission distance is further
For a given steel-concrete model the attenuation extentin signal propagation can be obtained by the attenuationand frequency dispersion curve It is found that for thesame steel bar with larger diameter embedded in the sameconcrete the energy leaks less into concrete the spread ofwave energy attenuation is slower and the spread distance isshorter Studying the influence on propagation characteristicsof guided wave in steel bars by concrete will increase theaccuracy in judging the structure integrity and promotethe level of defectcorrosion detection for the steel barsembedded in concrete
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is supported by Key Science and TechnologyProject from Fujian Province China (no 2013Y0079) andthe Research Fund SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University China
References
[1] T R Hay and J L Rose ldquoInterfacing guided wave ultrasoundwith wireless technologyrdquo in Smart Structures and MaterialsmdashSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 5391 of Proceedings of the SPIE pp314ndash320 San Diego Calif USA March 2004
Shock and Vibration 13
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
[2] A Raghavan andC E S Cesnik ldquoReview of guided-wave struc-tural healthmonitoringrdquo Shock andVibration Digest vol 39 no2 pp 91ndash114 2007
[3] H Sohn D Dutta J Y Yang et al ldquoDelamination detection incomposites through guided wave field image processingrdquo Com-posites Science and Technology vol 71 no 9 pp 1250ndash1256 2011
[4] G B Song Y L Mo K Otero and H Gu ldquoHealth monitoringand rehabilitation of a concrete structure using intelligentmaterialsrdquo Journal of Smart Materials and Structures vol 15 no2 pp 309ndash314 2006
[5] R L Wang H Gu Y L Mo and G Song ldquoProof-of-conceptexperimental study of damage detection of concrete piles usingembedded piezoceramic transducersrdquo Smart Materials andStructures vol 22 no 4 Article ID 042001 2013
[6] V Giurgiutiu ldquoLamb wave generation with piezoelectric waferactive sensors for structural health monitoringrdquo in SmartStructures and Materials 2003 Smart Structures and IntegratedSystems vol 5056 of Proceedings of SPIE San Diego Calif USAMarch 2002
[7] X Wang P W Tse C K Mechefske andM Hua ldquoExperimen-tal investigation of reflection in guided wave-based inspectionfor the characterization of pipeline defectsrdquo NDT amp E Interna-tional vol 43 no 4 pp 365ndash374 2010
[8] A C Cobb H Kwun L Caseres and G Janega ldquoTorsionalguided wave attenuation in piping from coating temperatureand large-area corrosionrdquo NDT and E International vol 47 pp163ndash170 2012
[9] R Ahmad and T Kundu ldquoInfluence of water flow throughpipe networks for damage detection using guided wavesrdquo inNondestructive Testing of Materials and Structures vol 6 ofRILEM Bookseries pp 681ndash687 2013
[10] M D Beard M J S Lowe and P Cawley ldquoInspection of steeltendons in concrete using guidedwavesrdquoReview of QuantitativeNondestructive Evaluation vol 22 pp 1139ndash1147 2003
[11] X P Zhu P Rizzo A Marzani and J Bruck ldquoUltrasonicguided waves for nondestructive evaluationstructural healthmonitoring of trussesrdquo Journal of Measurement Science andTechnology vol 21 no 4 Article ID 045701 2010
[12] Z P Zheng Y Lei and X Xue ldquoNumerical simulation of mon-itoring corrosion in reinforced concrete based on ultrasonicguided wavesrdquoThe ScientificWorld Journal vol 2014 Article ID752494 9 pages 2014
[13] T Yi H Li and M Gu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013
[14] F Moser L J Jacobs and J Qu ldquoModeling elastic wave pro-pagation inwaveguides with the finite elementmethodrdquo Journalof NDT amp E International vol 32 no 4 pp 225ndash234 1999
[15] Z B Cheng Numerical simulation and experimental investiga-tion on crack detection in pipes using ultrasonic guided waves [ME thesis] Taiyuan University of Technology 2004
[16] C H He Y X Sun Z Liu XWang and BWu ldquoFinite elementanalysis of defect detection in curved pipes using ultrasonicguided wavesrdquo Journal of Beijing University of Technology vol32 no 4 pp 289ndash294 2006 (Chinese)
[17] G L Huang F Song and X Wang ldquoQuantitative model-ing of coupled piezo-elastodynamic behavior of piezoelectricactuators bonded to an elastic medium for structural healthmonitoring a reviewrdquo Sensors vol 10 no 4 pp 3681ndash3702 2010
[18] Y Lei and Z P Zheng ldquoReview of physical based monitoringtechniques for condition assessment of corrosion in reinforcedconcreterdquo Mathematical Problems in Engineering vol 2013Article ID 953930 14 pages 2013
[19] W Na T Kundu and M R Ehsani ldquoUltrasonic guided wavesfor steel bar concrete interface testingrdquo Journal of MaterialEvaluation vol 60 no 3 pp 437ndash444 2002
[20] W Na T Kundu and M R Ehsani ldquoLamb waves for detectingdelamination between steel bars and concreterdquoComputer-AidedCivil and Infrastructure Engineering vol 18 no 1 pp 58ndash632003
[21] W Na T Kundu and M R Ehsani ldquoA comparison ofsteelconcrete and glass fiber reinforced polymersconcreteinterface testing by guided wavesrdquo Journal of Material Evalua-tion vol 61 no 2 pp 155ndash161 2003
[22] H Reis B L Ervin D A Kuchma and J T BernhardldquoEstimation of corrosion damage in steel reinforced mortarusing guided wavesrdquo Journal of Pressure Vessel Technology vol127 no 3 pp 255ndash261 2005
[23] F Wu and F Chang ldquoDebond detection using embeddedpiezoelectric elements in reinforced concrete structures part Iexperimentrdquo Structural HealthMonitoring vol 5 no 1 pp 5ndash152006
[24] F Wu and F-K Chang ldquoDebond detection using embeddedpiezoelectric elements for reinforced concrete structuresmdashpartII analysis and algorithmrdquo Structural Health Monitoring vol 5no 1 pp 17ndash28 2006
[25] C He J K van Velsor C M Lee and J L Rose ldquoHealthmonitoring of rock bolts using ultrasonic guided waves quan-titative nondestructive evaluationrdquo in Proceedings of the AIPConference pp 195ndash201 AIP Reston Va USA 2006
[26] L Dongsheng R Tao and Y Junhui ldquoInspection of reinforcedconcrete interface delamination using ultrasonic guided wavenon-destructive test techniquerdquo Science China TechnologicalSciences vol 55 no 10 pp 2893ndash2901 2012
[27] B L Ervin and H Reis ldquoLongitudinal guided waves for moni-toring corrosion in reinforced mortarrdquo Journal of MeasurementScience and Technology vol 19 no 5 Article ID 055702 2008
[28] B L Ervin D A Kuchma J T Bernhard and H ReisldquoMonitoring corrosion of rebar embedded in mortar usinghigh-frequency guided ultrasonicwavesrdquo Journal of EngineeringMechanics vol 135 no 1 pp 9ndash19 2009
[29] L Pochhammer ldquoUber die fortpflanzungsgeschwindigkeitenkleiner schwingungen in einem unbegrenzten isotropen kreis-zylinderrdquo Journal fur die Reine und Angewandte Mathematikvol 81 pp 324ndash336 1876
[30] C Chree ldquoThe equations of an isotropic elastic solid in polarand cylindrical coordinates their solutions and applicationsrdquoTransactions of the Cambridge Philosophical Society vol 14 pp250ndash369 1889
[31] D Bancroft ldquoThe velocity of longitudinal waves in cylindricalbarsrdquo Physical Review vol 59 no 7 pp 588ndash593 1941
[32] J L Rose Ultrasonic Waves in Solid Media Cambridge Univer-sity Press Cambridge UK 1999
[33] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 no 3-4 pp 345ndash361 2013
[34] T H Yi and H N Li ldquoMethodology developments in sen-sor placement for health monitoring of civil infrastructuresrdquo
14 Shock and Vibration
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
International Journal of Distributed Sensor Networks vol 2012Article ID 612726 11 pages 2012
[35] A A Hanifah A theoretical evaluation of guided waves in deepfoundations [Dissertation] Northwestern University EvanstonIll USA 1999
[36] B N Pavlakovic and M Lowe Disperse Users Manual Version2016B University of London 2003
[37] R NThurston ldquoElastic waves in rods and clad rodsrdquo Journal ofthe Acoustical Society of America vol 64 no 1 pp 1ndash37 1978
[38] Z P Zheng Y Lei X P Cui and Y Song ldquoNon-destructive testof the steel bar by using piezoceramics sheetsrdquo in Proceeds of 3rdInternational Conference on Advanced Measurement and TestXiamen China 2013
[39] D Liu A Zhou and Y Liu ldquoTorsional wave in finite lengthintact pilerdquo Chinese Journal of Applied Mechanics vol 2 no 2pp 258ndash263 2005
[40] J L Rose ldquoA baseline and vision of ultrasonic guided waveinspection potentialrdquo Journal of Pressure Vessel TechnologyTransactions of the ASME vol 124 no 3 pp 273ndash282 2002
