Theoretical basis and numerical simulation ofimpedance log test for evaluating the integrity ofcolumns and piles

Chih-Peng Yu and Shu-Tao Liao

Abstract: In this paper, the theoretical capabilities of the impedance log method in profiling the geometry of a columnor pile along its length are investigated. The main idea of this method is to introduce a transient stress wave into themember and then utilize the reflected signals to obtain the impedance profile. The impedance profile is then used to re-cover the equivalent cross-sectional area along the length of the pile. To gain an insight into this nondestructive evalua-tion technique, this research started with a review of its theoretical background. The dynamic behavior of defectivecolumns or piles with weak zones, bulges, and necks of constant or varying cross sections were then simulated with fi-nite element models. The success in recovering the profiles of this kind of rod-like structural members proved that theimpedance log method is a tool with high potential for evaluating the integrity of piles.

Key words: impedance log method, nondestructive test, stress wave, column, pile.

Résumé : Dans cet article, on étudie la capacité théorique de la méthode « impedance log » pour le profilage de lagéométrie d’une colonne ou d’un pieu le long de sa longueur. L’idée principale de cette méthode est d’introduire uneonde de contrainte transitoire dans le membre et ensuite d’utiliser les signaux réfléchis pour obtenir le profild’impédance. Le profil d’impédance est alors utilisé pour récupérer la surface de la section en travers le long de la lon-gueur du pieu. Pour obtenir une compréhension de cette technique non destructive d’évaluation, cette recherche a dé-buté par une revue du fondement théorique. On a simulé par des modèles en éléments finis le comportementdynamique des colonnes ou pieux défectueux avec des zones faibles, des gonflements et des goulots ayant des sectionsen travers constantes ou variables Le succès dans la récupération des profils de cette sorte de membrure structurale detype tige a prouvé que la méthode « impedance log » est un outil qui offre un grand potentiel pour l’évaluation del’intégrité des pieux.

Mots clés : méthode « impedance log », essai non destructif, onde de contrainte, colonne, pieu.

[Traduit par la Rédaction] Yu and Liao 1248

Background

Evaluating the integrity of drilled shafts or driven pileswith nondestructive testing (NDT) techniques has long beenrecognized as an important means for quality control in theconstruction industry (Davis and Dunn 1974; Baker et al.1991). The essential requirement for all the evaluation workis the acquisition of geometric information for intact or de-fective piles. To this end researchers have been working onmany nondestructive techniques, such as: the impact echo(IE) (Lin et al. 1997; Sadri 2003), the impulse response (IR)(Liao and Roesset 1997a; Finno and Gassman 1998; Davis2003), the sonic echo (SE) (Morgano 1996), the cross-hole

sonic logging (CSL) (Baker et al. 1991), and the parallelseismic (PS) (Stein 1982) methods. These methods are usu-ally classified into two groups, surface reflection and directtransmission methods. Surface reflection methods, such asthe IE, IR, and SE methods, are usually more economicalbecause the dynamic disturbance generator and the receiverare placed at accessible locations. However, these methodsare usually unable to provide a complete assessment of apile with multiple defects. Direct transmission methods,such as the CSL and PS methods, may diagnose a pilethrough its length in a more direct manner but are more ex-pensive because borehole drilling or test coring may be re-quired prior to the evaluation test.

Of the NDT methods mentioned above, the SE and IRmethods are of particular interest here not only because oftheir importance, but also because of their close relationshipwith the impedance log (IL) method. As shown schemati-cally in Fig. 1, both tests involve impacting the top of a pilewith a hammer. The SE method requires only the particle re-sponse history to perform integrity analysis, while in the IRmethod the impact force and the particle velocity responseversus time are both measured on the impacted surface.These two time histories are then transformed to the fre-quency domain using the fast Fourier transform (FFT). The

Can. Geotech. J. 43: 1238–1248 (2006) doi:10.1139/T06-072 © 2006 NRC Canada

1238

Received 7 March 2005. Accepted 7 April 2006. Publishedon the NRC Research Press Web site at http://cgj.nrc.ca on4 January 2007.

C.-P. Yu. Department of Construction Engineering, ChaoyangUniversity of Technology, 168 Jifong E. Rd., Wufong,Taichung County, 41349, Taiwan.S.-T. Liao.1 Department of Civil Engineering, Chung HuaUniversity, 707, Section 2, Wu-Fu Rd., Hsinchu, 30067,Taiwan.

1Corresponding author (e-mail: shutao@chu.edu.tw).

mechanical admittance (or mobility) of the pile is defined asthe ratio of the amplitudes of the particle velocity and theforce in the frequency domain (Davis and Dunn 1974). Fromthe velocity time history at the pile head and the mobilitycurve, one can usually find the existence of embedded de-fects (Davis and Dunn 1974; Liao et al. 1997b), such asnecks or bulges as shown in Fig. 1.

In the last decade, a relatively new method for evaluatingthe integrity of piles, the IL method, was proposed andbriefly described by Paquet (1991). The main capability ofthis method is the recovery of changes of the cross-sectionalarea of the pile by obtaining its impedance as a function ofthe distance from the pile head. In other words, the methodprovides the geometric profile of the pile and its variationwith depth. However, research results related to this method,either of a theoretical or an experimental nature (Davis andHertlein 1991; Paquet 1992; Rix et al. 1993), are scarce. Theobjective of this paper is to initiate an in-depth study of theIL method from an academic viewpoint. The theoretical der-ivation and the capability of this method in profiling the ge-ometry of a column or pile along its length is explored inthis paper. With this type of study, it is hoped that the ILmethod can be applied to evaluating the quality of newlybuilt piles and to assessing whether the piles of many exist-ing foundations are damaged following an earthquake event.

Implementation of IL method

Referring to Fig. 2, the reported procedure for applicationof the IL method can be summarized as follows (Rix et al.1993):(1) Perform the IR test on the pile to be evaluated. Use the

particle velocity response and the force functions in thefrequency domain, V1(f) and F1(f), to obtain the mobilitycurve of this target pile, as denoted by V1(f)/F1(f).

(2) Generate the theoretical mobility curve of a fictitious (orideal) pile that has the same nominal diameter and samegeometric configuration as the target pile but is defect-free and infinitely long. The properties of the soil inwhich the tested pile is embedded may be obtainedthrough a site investigation prior to construction. Thiscurve is denoted by V2(f)/F2(f).

(3) A new mobility curve, V3(f)/F3(f), which contains onlythe effect of reflections from defects and from the endof the shaft, can then be obtained by subtracting the ex-perimental and theoretical mobility functions.

V3(f)/F3(f) = V1(f)/F1(f) – V2(f)/F2(f)

(4) Perform the inverse fast Fourier transform (IFFT) onV3(f)/F3(f) to obtain the relative reflectogram X(t) in thetime domain. The relative reflectogram is then scaled toget the reflection coefficient, r(t). The impedance of thepile, as a function of time, can then be determinedthrough

[1] Z t Z rt

( ) exp[ ( ) ]= ∫o 0d2 τ τ

where Zo = EoAo/Cp is the impedance at the pile head,Eo and Ao are Young’s modulus and the cross-sectionalarea of the pile, respectively, and Cp is the wave propa-gation velocity in the pile.

(5) Finally, the time function of the impedance Z(t) can beconverted to a space function Z(x) using the equationx = Cpt, where x is the distance measured from the pilehead. As will be shown, the cross-sectional area of thepile at any distance from the top can thus be recoveredfrom the impedance function Z(x).

Earlier work had concluded that some modificationsshould be made to the reported procedure for specific cases(Yu and Liao 1999). In the following, the theoretical back-ground of the IL method is first derived and then comparedwith the results obtained with a discrete solution in the timedomain using finite element modeling and a continuous so-lution in the frequency domain. A number of case studiesare then solved, starting for simplicity with a pile withoutsurrounding soil (a column) and different types of defects,and proceeding finally to a pile with surrounding soil.

Theoretical derivation

Consider a transient wave propagating in a rod-like com-posite medium, as shown in Fig. 3. At the interface, let I, B,and T denote the amplitudes of the particle motions associ-ated with the incident, reflected, and transmitted waves, re-spectively. Their relationship can then be expressed by

[2] BZ ZZ Z

I= −+

1 2

1 2

[3] TZ

Z ZI=

+2 1

1 2

where Z1 and Z2 are the impedances of the two media, con-taining the incident and the transmitted wave, respectively.The impedance is defined as Z = EA/Cp, E and A being theYoung’s modulus and cross-sectional area of the pile, re-

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Yu and Liao 1239

Fig. 1. Sonic echo and impulse response (SE/IR) test of a pile.

spectively, and Cp the propagating speed of the wave. Inrod-like structures, Cp can be approximated by E /ρ, whereρ is the mass density of the material.

To extract the wave signals reflected at all interfaces withchanging impedances, the original response time history ob-tained in the field test is subtracted by the theoretical re-sponse of an idealized, defect-free, and infinitely long pilewith the same nominal diameter. In the frequency domainthis can be expressed as

[4] [ ( )] [ ( )] [ ( )]V V VRω ω ω= −Actual Ideal

where V is the particle velocity at the pile head as a functionof the circular frequency ω, and the subscript R refers to thereflection. When normalized by the applied force P(ω) inthe frequency domain, the result in eq. [4] can be expressedin the form of transfer function (or mobility) as follows:

[5]VP

VP

VP

R

( )( )

( )( )

( )( )

ωω

ωω

ωω

=

−

Actual

Ideal

This is equivalent to the response due to a Dirac-delta func-tion force δ(t) applied in the time domain, i.e., p(t) = δ(t). Inthis case P(ω) is equal to unity in the frequency domain, andeq. [5] can be expressed as

[6] [ ( )] [ ( )] [ ( )]V V VRω ω ωδ δ δ− − −= −Actual Ideal

where the subscript δ stands for the response associated withthe idealized impulse δ(t). Now define the frequency-dependent reflection coefficient, R(ω), as the ratio of theamplitude of the reflected wave to that of the incident wave,i.e.,

[7] R V VR I( ) [ ( )] / [ ( )]ω ω ωδ δ≡ − −

Before substituting eq. [6] into eq. [7], we must first define[V(ω)]I–δ, which is in fact the mobility of a member of infi-nite length [V(ω)]Ideal–δ. In the special case where there is nosoil, this value is 1/Zo (= Cp/EoAo) with the subscript “o”representing the properties at the pile head (x = 0). In thiscase, the reflection coefficient R(ω) can be rewritten as

[8] R V V Z VR R( ) [ ( )] / [ ( )] [ ( )]ω ω ω ωδ δ δ≡ =− − −Ideal o

= −−Z Vo Actual[ ( )]ω δ 1

where Zo = Z(0) is the impedance at the pile head. The nor-malized reflection coefficient in the time domain, denotedby r(t) in this paper, can then be obtained by convertingeq. [8] using the IFFT, i.e., r(t) = IFFT [R(ω)]. This coeffi-cient is the normalized reflectogram associated with the re-flected signals received at the pile head due to an idealizedimpact. Referring to eq. [2], the sum of the reflection coeffi-cient r(t) in an infinitesimal time step should also be equal todZ/2Z because

[9] r t tBI

Z ZZ Z

Z Z ZZ Z Z

ZZ

( )( )

( )d

dd

d≡ − = −+

= + −+ +

≅2 1

1 2

1 1

1 1 2

The negative sign in the definition −B I/ for the reflection co-efficient is due to a positive velocity resulting from a nega-tive change in the impedance. Integrating eq. [9] over thetime interval of 0 to t leads to

[10] rZ

Zt

Z

Z t

Z

Z t

0 0 0

12

12∫ ∫= =( ) ln

( )

( )

( )

( )τ τd d

Finally the impedance of the rod can be expressed as a func-tion of time by

[11] Z t Z rt

( ) ( ) exp ( )=

∫0 2

0τ τd or

Z t Z rt

( ) exp ( )=

∫o d2

0τ τ

This time function Z(t) can also be converted to a space func-tion Z(x) using the relationship that x = Cpt, where x is the

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1240 Can. Geotech. J. Vol. 43, 2006

Fig. 2. SE/IR tests on (a) a practical pile and (b) a fictitious defect-free pile of infinite length.

Fig. 3. Reflection and transmission of waves at an interface.

distance measured from the pile head. The equivalent cross-sectional area of the pile at any distance from the top canthus be recovered using the impedance function Z(x).

Some modifications

After working through some numerical studies it wasfound that some modifications to the regular procedure wererequired to obtain reasonable results. The associated consid-erations and modifications are described in the followingsections.

Numerical consideration for δ (t)As stated above, for a rod-like structure such as a column

subjected to an idealized impulse of Dirac-delta functionδ (t), the reflection coefficient can be expressed as shown ineq. [8]. This may not be feasible for numerical simulationbecause of the idealized force δ (t). In fact, a half-sine im-pulse is usually selected as a good approximation in prac-tice. In this case, some modifications must be made to obtainthe appropriate normalized reflection coefficient r(t). Letrm(t) denote the reflectogram associated with an idealized si-nusoidal impulse p t p f tm o( ) ( )= in the time domain with porepresenting the impulse amplitude. This idealized impulsedoes not necessarily represent the realistic impulse in theimpact test. The function f(t) of unit duration can be treatedas a kernel function to be used in converting the normalizedfrequency reflection coefficient R(ω) to the reflectogramrm(t) in time domain. Let Rm(ω) and Pm(ω) (= poF(ω)) standfor the Fourier transform counterparts of the two time do-main quantities rm(t) and pm(t), respectively. Thereflectogram can be obtained through the convolution inte-gral of r(t) and pm(t) or, more efficiently, from the inverseconversion of the product between the two frequency func-tions R(ω) and Pm(ω), as stated by

[12] r t R R p Fm m oIFFT IFFT( ) [ ( )] [ ( ) ( )]= =ω ω ω= IFFT o R[ ( )]Z V ω

in which VR(ω) = [VR(ω)]R–δPm(ω) is the reflection velocityresponse recorded at the pile head associated with the ideal-ized impulse. In cases where the impulse duration is rela-tively small, the impulse pm(t) can be reasonablyapproximated by poAmδ (t), with Am being the area under thetime function f(t) in the force, because the frequency ampli-tude associated with F(ω) is about unity in the frequencyrange of interest. For example, a half-sine impulse with aduration TD and a unit amplitude results in a value of 2TD/πfor Am. Equation [12] can therefore be approximated byeq. [13] as follows:

[13a] r t p A Z V p A r tRm o m o o mIFFT( ) [ [ ( )] ] ( )≅ ω δ− =

and thus

[13b] r tp A

r t Z V R( ) ( ) {[ ( )] }≅ = −1

o mm o IFFT ω δ

=

Zp A

v to

o mR( )

The computation of the normalized reflection coefficientfrom the recorded velocity response can thus be carried outaccordingly. It is apparent that the normalized reflection co-efficient can be estimated from the velocity time history re-corded in the SE method or from the mobility data computedin the IR method. The reflectogram rm(t) can be obtainedsubtracting an estimate of the direct signal from the recordedvelocity time history, while the frequency reflection re-sponse VR(ω) can be derived by deducting a value of Zofrom the mobility of the recorded data, as shown in eq. [4].It should be noted here that [V(ω)]Ideal is equivalent to 1/Zo.In both procedures, the area of an appropriate idealized im-pulse, poAm, has to be determined.

Free end effect at column or pile headIf the response is obtained at a column or pile head, where

a free-end boundary condition is encountered, the amplitudeof the recorded response is actually twice that of the re-flected wave. In this case, eq. [11] should be further modi-fied to

[14] Z t Z rt

( ) ( ) exp ( )=

∫0

0τ τd or

Z t Z rt

( ) exp ( )=

∫o d

0τ τ

Preliminary study on the soil effectFor pile problems where soil effects must be taken into

account, radiation damping associated with the pile–soil in-terface causes the reflections to attenuate rapidly with time.This phenomenon can be clearly seen in the last case studyto be presented in this work. As a result, the direct contribu-tion of the reflectogram computed with eq. [14] will be in-sufficient to recover realistic profiles. To compensate for thiseffect and to reproduce the profiles more accurately, a sim-ple exponential function w(t) was selected as the weightfunction to adjust the reflectogram as implied in the follow-ing eq. [15]:

[15] Z t Z r wt

( ) exp ( ) ( )=

∫o d

0τ τ τ

with

[16] w tqt

t( ) exp=

R

where

[17] qvv

=

ln 1

R

and vI and vR stand for the maximum amplitudes of the ve-locity responses at the pile head associated with the timewhen the impact force is applied and when a reflection ar-rives at the pile head. The time period tR is the traveling timeof the reflection. Therefore, the decay of the stress wavepropagation may be estimated directly from the ratio of theresponses measured at the pile head, and then the logarithmof the inverse of this parameter may serve as the augmenta-tion factor as implied in the constant q. The use of this sig-

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Yu and Liao 1241

nal processing technique will be demonstrated in Case 4 tobe discussed later.

Simulations

To simulate the dynamic behavior of a single pile withcircular cross section, one-, two-, and three-dimensional(1D, 2D, 3D) finite element models were developed in a pre-vious study using 1D rod, 2D plane stress, and 3D axi-symmetric elements (Liao and Roesset 1997a). The resultsof that study indicated that all three models could simulatethe response of a single pile with or without defects as longas there was no pile cap. Based on these results, the 1Dmodel using rod elements was selected here to further studythe IL method. Shown in Fig. 4 is a 1D finite element modelthat simulates the response of piles subjected to the dynamicloads applied at the pile head. Standard linear elements un-der axial deformation were selected for this model. Each ele-ment has two nodes with a linear variation in displacements.

To simulate the effect of the soil on the lateral surface ofthe pile, distributed springs with elastic stiffness k anddashpots with damping coefficient c were applied to eachsegment below the ground surface. The soil at the base ofthe pile was modeled using a spring with elastic constant Kand a dashpot with damping coefficient C. The constants cand C are intended to represent the radiation (or geometric)damping due to soil–structure interaction. The internaldamping in the pile and the soil are assumed to be negligiblysmall under the very small amplitude vibrations of a nonde-structive loading. Consistent foundation matrices (includingboth the springs and dashpots) were developed for these ele-ments. Let a denote the radius of the pile, the values of k, c,K, and C per unit length of the pile were determined throughthe following equations (Lysmer and Richart 1966; Novak1977; Simons and Randolph 1985):

[18] k G= 2.3 N ms2( / ), c = 2πρs sV a (kg/m·s)

KG a=−

4

1s

s

N mν

( / ), C KaV

=

0.85

s

(kg/s)

where

GE

ss

s

N m=+2 1

2

( )( / )

νis the shear modulus

VG

ss

s

m s=ρ

( / ) is the shear wave velocity

and Es, νs, and ρs are Young’s modulus, Poisson’s ratio, andthe mass density of the soil, respectively.

The above equations were derived assuming a rigid diskin an elastic plane. The values of K and C correspond to acircular footing resting on top of the soil, an assumption thatis not strictly correct for the base of a pile. However, the er-ror introduced by this simplifying assumption is small.Lumped masses were used at each nodal point.

To study the relative importance of the soil interaction andchanges in impedance, it is convenient to introduce anondimensional index χ, as defined by

[19] χπ ρ

ρρρ

= = =c aZ

a G

A E

GE

2 44

2

o

s s s s

where A, E, and ρ are the area, the Young’s modulus, and themass density of the concrete pile in this paper. The reasonfor including only the dashpot effect without the spring com-ponent is that the interaction with the soil is dominated bythe dashpot.

As an alternative solution, the dynamic response can beobtained from the continuous solution of the governing dif-ferential equation in the frequency domain. This alternativeprovides a more efficient way in numerical computation.Furthermore, the mobility can be directly evaluated so thatthe FFT processing for the time histories can be avoided.Consequently, the simulation results in the frequency do-main can serve as a benchmark for the finite element modelin the time domain to assess the ability of the proposed for-mulae for the IL method. The program used in this study toobtain the solutions in the frequency domain was previouslydeveloped by the authors. The exact solution associated witha rod theory of second order was used as the basis for the

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1242 Can. Geotech. J. Vol. 43, 2006

Fig. 4. 1D finite element model for a pile partially embedded in soil.

dynamic problem. The formulation is straightforward andcan be found for instance in previous articles by the authors(Yu and Roesset 1998; Yu and Liao 2000).

Simulation results

In all of the following case studies the numerical solutionsobtained, using both the discrete time domain and the con-tinuous frequency domain formulation approaches, wereidentical to each other except that the mobility plots derivedfrom the time domain model resulted in spurious peaks atthe frequencies associated with the zeroes in the frequencyamplitude of the half-sine impulse function. Thus, the nu-merical results shown in this work were first obtained withthe finite element approach in time domain and then verifiedusing the frequency domain approach. It should be noted,however, that the mobility curves can be directly computedfrom the dynamic stiffness formulae using the frequency ap-proach.

Case 1: Columns with straight bulges and necksConsider an axisymmetric column as shown in Fig. 5. The

radius and total length of the column are 0.5 m and 12 m, re-spectively. The Young’s modulus and mass density of thisconcrete column are 3.31 × 1010 N/m2 and 2300 kg/m3, re-spectively. A bulge starts at a distance of 4 m from the headand ends at 5 m. Between 8 m and 9 m from the head thereis also a neck. The radii of the column in the bulge and neckportions are 0.6 m and 0.4 m, respectively. A half-sine com-pressive impulse with a duration of 0.25 ms and peak forceamplitude of 1 N is applied at the column head to introducea transient stress wave into the column. The column is as-sumed to rest on a soil surface at its far end. Throughout thispaper, the Young’s modulus, mass density, and Poisson’s ra-tio of the soil are 1.8 × 108 N/m2, 1924 kg/m3, and 0.4, re-spectively, unless otherwise specified.

Using the 1D finite element model, the particle velocity atthe head of this defective column and the corresponding mo-bility curve were obtained as shown in Figs. 6a and 6b, re-spectively. Both the reflections from the defects and fromthe pile bottom can be clearly identified in Fig. 6a. FromFig. 6b, the pile length (12 m) can also be recognized fromthe uniform frequency intervals between peaks (about150 Hz on average), but the frequency intervals associatedwith the locations of the two defects seem much harder toidentify in this case. Using eqs. [13] and [14] with the halfsine of 0.25 ms duration as the kernel function, the corre-sponding reflectogram and the recovered impedance profilewere obtained, as shown in Figs. 6c and 6d. Figure 6c showsthat the reflectogram is a scaled velocity history excludingthe first response impulse occuring in the period of time ofimpacting. The variation of the impedance and the equiva-lent diameter ratios with respect to those at the pile head areplotted in Fig. 6d. In this figure the light line represents theimpedance profile and the dark line stands for the equivalentdiameter ratio. Equation [14] indicates that the impedanceprofile is computed by integrating the reflectogram, which isa velocity function. Therefore, the recovered impedance pro-file should exhibit a similar trend to the integration of theparticle velocity response, i.e., the particle displacement re-sponse. As expected and illustrated in Fig. 6e, the particle

displacement time history shows identical shape to the im-pedance profile in Fig. 6d. To demonstrate the effect of theduration of the ideal impulse (or the kernel function) on thecomputed profiles, three different durations, 0.10 ms,0.25 ms, and 0.50 ms were used in the kernel functions, andthe results are shown in Fig. 6f. It is obvious that the shorterthe duration is, the sharper the profile is. In other words,abrupt impedance changes can only be well recovered whena relatively short duration is used in the kernel function. An-other observation from Fig. 6f is that kernels of longer dura-tion usually result in not only smoother profile changes butalso larger extensions of the defects.

Case 2: Column with bulge and neck of linearlyvarying diameters

In Fig. 5, the cross-sectional areas of the column are con-stant within each of the defective zones. A nonuniform bulgeand neck are considered here to test the capabilities of the ILmethod. As shown in Fig. 7, a bulge starts at a distance of4 m from the head and extends for 1 m. The radius of thecolumn increases linearly from 0.5 m to 0.6 m in this region.There also exists a neck of varying diameters between 8 mand 9 m from the head. The radius of the column increasesgradually from 0.4 m to 0.5 m in this region. Using the re-sults from finite element analysis, the mobility curve and thereflectogram of this defective column can be obtained andthen plotted in Figs. 8a and 8b, respectively. As in the previ-ous case, the mobility curve clearly indicates the frequencyintervals associated with the pile length instead of the loca-tions of the defects. Although the time domain response, i.e.,reflectogram in this case, is capable of predicting abruptchanges of impedance at x = 5 m and 8 m, it seems insensi-tive to the gradual changes in sectional impedance. Figure 8cshows the impedance and diameter ratio profiles recoveredby eq. [14]. In this figure the impedance was drawn using alight line and the diameter ratio with a darker line. This fig-ure illustrates that the IL method profiles a defective columnvery well, even with defects of gradual changes. However,small disturbances may arise, as can be seen in the regionfollowing the neck in Fig. 8c.

Case 3: Column with a weak zone, a straight bulge,and a neck

To study the capability of the IL method to detect regionsof poor quality in a column, a weak zone near the columnhead was included in the defective column used in Case 1. Asshown in Fig. 9, the weak zone extends between 1 m and 2 mfrom the column head and the Young’s modulus in this regionis reduced from 3.31 × 1010 N/m2 to 2.68 × 1010 N/m2.

The mobility and the reflectogram curves of this defectivecolumn are plotted in Figs. 10a and 10b, respectively. Fromthe mobility curve the pile length can be easily recovered.However, no reliable information about the defects can befound from the curve. On the other hand, the reflectogramcan usually reveal the relative time lags of the reflection sig-nals associated with defects, and therefore more informationabout defects may be deduced from the curve. As comparedto those shown in Fig. 6 for Case 1, it can be noted that theexistence of the third defect, i.e., the weak zone, results in amore complicated multiple-reflection pattern associated withnoticeable secondary reflection signals. This may be caused

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Yu and Liao 1243

by the trapped waves between defects. As a result, the re-covered diameter profile may contain several minor changesof step-like shapes, as illustrated by the dark line in Fig. 10c.For comparison purpose, the diameter ratio profile from

Case 1, in which no weak zone exists, is also plotted with alight line in Fig. 10c. It is noted that the recovered profilesuccessfully exposed all defects, including the weak zone.With the IL method, however, a weak zone in practice may

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1244 Can. Geotech. J. Vol. 43, 2006

Fig. 6. The (a) velocity, (b) mobility, (c) reflectogram, (d) impedance profile, (e) displacement, and (f) diameter ratio profile of a col-umn with straight bulge and neck.

Fig. 7. Geometric configuration of a defective column with a bulge and a neck of varying diameters.

Fig. 5. Geometric configuration of a defective column with a straight bulge and a straight neck.

be revealed by an equivalent reduction in the cross-sectionalarea at the same location. It is also noted that a weak zonewill result in a time delay of the reflection signals, and thusthe pile length may be slightly overestimated, as can be seenfrom the figure. From this particular case it is concluded thatcertain modifications should be made to the reported proce-dures in cases where significant secondary reflections exist.In addition, further studies may be required for distinguish-ing defects of different types that have the same impedancechanges. For example, a 19% reduction in cross-sectionalimpedance may result from several causes, for example aneck with a loss of 10% in the radius, or a normal cross sec-tion with a reduction in Young’s modulus, or even a combi-nation of these two causes.

Case 4: Pile with trapezoidal bulge and neckTo study the effect of soil on the results of the IL analysis,

consider a member with a bulge and a neck as shown inFig. 11. As in the previous cases, these two abnormal re-gions start at 4 m and 8 m from the head, respectively. Tofocus more on soil effect however, trapezoidal shapes wereconsidered for both defects so that no abrupt changes in im-pedance were made in this case. From the figure it can be

seen that the radius in the bulge increases linearly from0.5 m to 0.6 m within a longitudinal distance of 0.25 m,stays constant for a similar distance, and then decreases backto 0.5 m in the same distance again. A similar configurationwas applied to the neck except that the constant radius waschanged to 0.4 m.

The results for the reflectogram and diameter ratio profileof the pile without soil are shown in Figs. 12a and 12b, re-spectively, with kernel functions of three different durations,i.e., 0.10, 0.25, and 0.50 ms. Both figures reveal similar re-sults as seen in previous cases where abrupt changes in im-pedance were studied. It seems that the smoothing effect ofthe kernel function with a moderate duration of 0.25 mshelps to recover a better profile that matches well with therealistic trapezoidal bulge and trench neck. However, it isalso seen that using kernel functions of different durationsmay result in different predictions in the length (longitudinalextension) of the defects. This error can be corrected by con-sidering the distance that the stress wave travels within ahalf time of the duration of the kernel. For example, thelength of the bulge predicted by using the kernel with a du-ration of 0.25 ms is 1.2 m. The distance that the stress wavetravels within a half of the duration is 0.125 ms ×3793 m/s = 0.47 m. Therefore, the length of the bulgeshould be corrected to be 1.2 m – 0.47 m = 0.73 m, which isvery close to the real value, 0.75 m.

Assume that the member shown in Fig. 11 is embedded insoil so that the ground surface is leveled at the pile head.The Young’s modulus, mass density, and Poisson’s ratio ofthe soil are identical to those for the bottom soil. The shearwave velocity in the soil is then about 305 m/s, and thereforethe relative importance index defined in eq. [19] is χ = 0.27.The reflectogram corresponding to this defective pile isshown by a solid line in Fig. 13a. It can be seen that the re-flections from the defects and the bottom are extremelysmall relative to those with the dashed line associated withno soil (i.e., a column). The diameter ratio profile cannot berecovered properly on the basis of this reflectogram usingthe reported procedures of the IL method. The computedprofile is shown with a solid line in Fig. 13b. The predicteddiameter ratio for the bulge region is far from the correctvalue, i.e., 1.2. To better illustrate the effect of the soil, aless stiff soil with a shear wave velocity of 150 m/s was con-sidered. In this case, the relative importance index is χ =0.13. The recovered diameter ratio profile is also plottedwith a dashed line in Fig. 13b. It can be seen that the resultsare better in this case. A number of parametric studies sug-gested that for piles the errors in recovering the profiles de-pend on the relative stiffness of the pile to the soil. Thehigher the relative stiffness, the more accurate the predictionwill be. This is due to the fact that the radiation damping isproportional to the shear wave velocity of the soil, which ismodeled by a spring-and-dashpot system in this work. Thisphenomenon also demonstrates the usage of the relative im-portance index χ, which is a very good measure for the ef-fect of soil interaction on this method.

Another interesting observation can be made from the mo-bility plot of this particular pile. As shown in Fig. 13c, thelocation of the bulge can be recovered using the frequencyintervals between the significant peaks appearing within the

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Yu and Liao 1245

Fig. 8. Numerical results for a column with a bulge and neck ofvarying diameters. (a) Mobility curve. (b) Reflectogram withTD = 0.1 ms. (c) Impedance profile and equivalent diameter ra-tios with TD = 0.1 ms.

frequency of 3000 Hz. That is, using the idea of the IRmethod (Liao and Roesset 1997a, 1997b), the distance LBfrom the pile head to the bulge can be estimated by

[20] LC

fB

p 3800 m/ss

4.3 m= =×

=−2 2 441 1∆

On the other hand, the pile length can be recovered using thefrequency intervals between the significant peaks appearingwithin the frequency range higher than 3000 Hz. That is,

[21] LC

f= =

×=−

p 3800 m/ss

11.9 m2 2 159 1∆

As far as the impedance at the pile head is concerned, theaverage value in the mobility shows very good agreementwith the theoretical value, i.e., 1/Zo = 1.46 × 10–7 m/s/N. Todemonstrate how the predicted profile may be greatly im-proved by simply considering the effect of the radiationdamping due to the soil, the reflection signal associated withfrom the pile bottom was selected to serve as vR in eq. [17].In this case, tR = 6.37 ms. The maximum amplitude of thevelocity response at the pile head associated with the timewhen the impact is applied may be determined directly fromthe velocity response time history. This value will serve asthe parameter vI in eq. [17]. Using eq. [15], a modified pro-file for the pile can then be obtained as shown in Fig. 14. Itcan clearly be seen that the modified profile shows goodagreement at the locations of both defects. However, somesmall difference can also be observed in the regions follow-ing the bulge and the neck. The disturbances caused by theimpedance changes of significant magnitudes are quite visi-ble and thus may not be neglected. To obtain a better agree-ment between the predicted and the real values, furtherstudies must be carried out to enhance the scope of applica-tion of the IL method. Nevertheless, the studies conducted todate indicate that the IL method is a promising technique toidentify possible defects in piles and to obtain their profiles.

Conclusions

In this paper the theoretical background for the IL methodwas first presented and then a series of case studies for col-umns and piles were shown to demonstrate the capability ofthis method in recovering the profiles of defective members.From the available results the ability of this method to evalu-ate the integrity of columns and piles seems reasonablypromising.

In deriving the formulae used in the IL method, some spe-cial modifications should be taken into account for differentcases to recover the correct profiles of the tested member.The factors to be considered include the sign convention ofthe impact force and the effect of free end at column or pileheads. The traditional way of calculating the impedancefunction Z(t) in the IL method is good only for specificcases.

Although the theory was derived assuming the use of anidealized impulse, the reflection coefficient can be obtainedfrom the velocity history at the pile head or from the convo-lution of the mobility function and an appropriate idealizedimpulse. The former approach requires the use of an esti-

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1246 Can. Geotech. J. Vol. 43, 2006

Fig. 9. Geometric configuration of a defective column with a weak layer, a straight bulge, and a neck.

Fig. 10. Numerical results of a defective column with a weaklayer, a straight bulge, and a neck. (a) Mobility curve.(b) Reflectogram with TD = 0.1 ms. (c) Equivalent diameter ra-tios with or without a weak layer with TD = 0.1 ms.

mate of the direct impact response from the velocity re-sponse. The latter provides a more flexible alternative toform the normalized reflection coefficient in which thereflectogram rm(t) can be obtained by use of a kernel func-tion with arbitrary duration TD.

From these studies, it was found that the IL method canaccurately recover the profiles of columns with a variety ofanomalies, such as bulges and necks of constant or varyingcross sections, weak zones, etc. It should be noted that aweak zone also results in an equivalent reduction in the im-pedance of the cross section. Therefore, further investigationshould be conducted to differentiate the impedance changesthat may result from different causes.

For pile problems that involve soil effects, the traditionalapproach of the IL method did not provide good results forthe profiles. Some preliminary studies in this research indi-cated that using some appropriate yet reasonable functionsas simple adjustments to the reflectogram may result in sig-nificant improvement in recovering the profiles.

The use of an idealized impulse to reproduce a desiredreflectogram leads to an effect of smoothing on the com-puted profile. However, using kernel functions of differentdurations may result in different predictions in the length (orlongitudinal extension) of the defects. Fortunately, thesekinds of errors can be corrected by considering the distance

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Yu and Liao 1247

Fig. 11. Geometric configuration of a defective member with trapezoidal bulge and trench neck.

Fig. 12. The (a) reflectogram and (b) diameter ratio profile of a pilewith a trapezoidal bulge and neck but without surrounding soil.

Fig. 13. The (a) reflectogram, (b) diameter ratio profile, and(c) mobility of a pile with a trapezoidal bulge and neck in a soilwith shear wave velocity of 305 m/s.

Fig. 14. Modified diameter ratio profile of a pile with a trape-zoidal bulge and neck in a soil with shear wave velocity of305 m/s.

that the stress wave travels within a half time of the durationin the kernels.

All the responses of the tested members in this paper wereobtained using two different approaches, i.e., finite elementmodeling in the time and in the frequency domains. The for-mulation in the frequency domain was based on the continu-ous solution of the corresponding differential equations andthus is a more efficient computation scheme. Furthermore,the reflectograms can be adjusted and the mobility curvescan be obtained in a more direct and simpler way. As a re-sult, the frequency domain formulation is recommended forthe numerical simulations regarding pile dynamics as well asother related dynamic problems.

Acknowledgements

The work presented in this paper was financially sup-ported by the National Science Council of the Republic ofChina under consecutive Grants NSC88–2625-Z-216–001and NSC89–2625-Z-216–002. Their help is gratefully ac-knowledged.

References

Baker, C.N. Jr., Drumright, E.E., Mensah, F.D., Parikh, G., andEaly, C.D. 1991. Use of nondestructive testing to evaluate de-fects in drilled shafts: results of FHWA research. IntegrityTesting of Foundations, Transportation Research Record, 1331:28–35.

Davis, A.G. 2003. The nondestructive impulse response test inNorth America: 1985–2001. NDT&E International, 36: 185–193.

Davis, A.G., and Dunn, C.S. 1974. From theory to field experiencewith the nondestructive vibration testing of piles. In Proceedingsof the Institution of Civil Engineers, Part 2, pp. 571–593.

Davis, A.G., and Hertlein, B.H. 1991. Development of nondestruc-tive small-strain methods for testing deep foundations: a review.Integrity Testing of Foundations, Transportation Research Re-cord, 1331: 15–20.

Finno, R.J., and Gassman, S.L. 1998. Impulse response evaluationof drilled shafts. Journal of Geotechnical and Environmental En-gineering, ASCE, 124(10): 965–975.

Liao, S.T., and Roesset, J.M. 1997a. Dynamic response of intactpiles to impulse loads. International Journal for Numerical andAnalytical Methods in Geomechanics, 21(4): 255–275.

Liao, S.T., and Roesset, J.M. 1997b. Identification of defects inpiles through dynamic testing. International Journal for Numeri-cal and Analytical Methods in Geomechanics, 21(4): 277–291.

Lin, Y., Sansalone, M., and Carino, N.J. 1997. Impact-echo re-sponse of concrete shafts. Geotechnical Testing Journal, 14(2):121–137.

Lysmer, J., and Richart, F.E. 1966. Dynamic response of footingsto vertical loading. Journal of the Soil Mechanics and Founda-tions Division, ASCE, 92(SM1): 65–91.

Morgano, C.M. 1996. Determining embedment depths of deepfoundations using non-destructive methods. In STRESSWAVE 6Conference, Orlando, Fla.

Novak, M. 1977. Vertical vibration of floating piles. Journal of theEngineering Mechanics Division, ASCE, 103(EM1):153–168.

Paquet, J. 1991. A new method for testing integrity of piles by dy-namic impulse: the impedance log. In International Colloquiumon Deep Foundations, Paris, March. pp. 1–10.

Paquet, J. 1992. Pile integrity testing – the CEMTP reflectogram.In Piling: European practice and worldwide trends. Thomas Tel-ford, London, pp. 206–216.

Rix, J.R., Jacobs, L.J., and Reichert, C.D. 1993. Evaluation of non-destructive test methods for length, diameter, and stiffness mea-surements on drilled shafts. Field Performance of Structure andNondestructive Evaluation of Subsurface Infrastructure, Trans-portation Research Record, 1415: 69–77.

Sadri, A. 2003. Application of impact-echo technique in diagnosesand repair of stone masonry structures. NDT&E International,36: 195–202.

Simons, H.A., and Randolph, M.F. 1985. A new approach to one-dimensional pile driving analysis. In Proceedings of the 5th Inter-national Conference on Numerical Methods in Geomechanics,Nagoya, Vol. 3, pp. 1457–1464.

Stein, R.T. 1982. Integrity testing. Civil Engineering, April: 54–59.Yu, C.P., and Liao, S.T. 1999. A close look at impedance log

method for integrity test for piles. In The 5th Far-East Confer-ence on Nondestructive Testing, 8–11 November 1999, Kenting,Taiwan. Edited by T.-T. Wu, C.-C. Ma, and P.-L. Liu. pp. 227–233.

Yu, C.P., and Liao, S.T. 2000. Dynamic analysis of piles using ex-act dynamic stiffness of an elementary rod with varying crosssections. In Proceedings of The First International Conferenceon Structural Stability And Dynamics, December. Edited by L.J.Leu. National Taiwan University, Taipei, Taiwan, pp. 393–398.

Yu, C.P., and Roesset, J.M. 1998. Dynamic analysis of structuresusing continuous solutions in frequency domain. In Proceedingsof EASEC-6 Conference on Structural Engineering & Construc-tion, 14–16 January 1998, Taipei, Taiwan. Edited by Y.-B. Yangand L.-J. Leu, Vol. 2, pp. 1369–1375.

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