Research ArticleDynamic Response of an Inhomogeneous Viscoelastic Pile ina Multilayered Soil to Transient Axial Loading

Zhiqing Zhang12 Jian Zhou1 Kuihua Wang3 Qiang Li4 and Kaifu Liu5

1Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education Tongji University Shanghai 200092 China2College of Urban Construction Zhejiang Shuren University Hangzhou 310015 China3Key Laboratory of Soft Soils andGeoenvironmental EngineeringMinistry of Education ZhejiangUniversity Hangzhou 310027 China4Department of Civil Engineering Zhejiang Ocean University Zhoushan 316004 China5School of Civil Engineering and Architecture Zhejiang Sci-Tech University Hangzhou 310018 China

Correspondence should be addressed to Jian Zhou tjugezhoujiantongjieducn

Received 15 October 2014 Accepted 15 April 2015

Academic Editor Igor Andrianov

Copyright copy 2015 Zhiqing Zhang et alThis 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

A quasi-analytical solution is developed in this paper to investigate the mechanism of one-dimensional longitudinal wavepropagating in inhomogeneous viscoelastic pile embedded in layered soil and subjected to a transient axial loading At first the pile-soil system is subdivided into several layers along the depth direction in consideration of the variation of cross-sectional acousticimpedance of the pile or differences in soil properties Then the dynamic governing equation of arbitrary soil layer is establishedin cylindrical coordinates and arbitrary viscoelastic pile segment is modeled using a single Voigt model By using the Laplacetransform and boundary conditions of the pile-soil system the vertical impedance at the top of arbitrary pile segment is defined in aclosed form in the frequency domainThen by utilizing the method of recursion typically used in the Transfer Function techniquethe vertical impedance at the pile top can be derived in the frequency domain and the velocity response of an inhomogeneousviscoelastic pile subjected to a semi-sine wave exciting force is obtained in a semi-analytical form in the time domain Selectednumerical results are obtained to study the mechanism of longitudinal wave propagating in a pile with a single defect or doubledefects

1 Introduction

Pile vibration theory can provide valuable guidance for boththe dynamic design of embedded foundations and dynamicnondestructive integrity testing of piles For the dynamicdesign of pile foundations the most concerned problem isthe study of vibration characteristics of embedded piles in thelow frequency range and accordingly the dynamic reactionfrom the surrounding soil is an essential component in therelevant study In light of this many soil models such assimplified continuum model [1ndash3] continuum model [4ndash6] and finite element model [7] have been developed toinvestigate the pile-soil dynamic interaction For the dynamicnondestructive integrity testing of piles low strain integritytesting technique has received wide application in assessingthe construction quality of piles due to its relatively low-cost

and simplicity This testing technique is mainly based on thestress wave propagation theory through a bar As a result thetheoretical study of a pile embedded in the soil and subjectedto a dynamic vertical load under small deformation conditionis increasingly becomingmore important For instance Davisand Dunn [8] firstly proposed the mechanical admittance (ormobility function)method to determine the length and cross-sectional area of a pile by applying a steady-state harmonicexcitation at a specified set of frequencies on the pile top Insubsequent studies Davis and Robertson [9] extended themechanical admittance method to determine the pile headstiffness Accounting for the convenience of transient excita-tion Higgs [10] modified the mechanical admittance methodby applying a vertical impact load on the pile top insteadof steady-state harmonic excitation After that Lin et al [11]introduced the impact-echo method to assess the integrity

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 495253 13 pageshttpdxdoiorg1011552015495253

2 Mathematical Problems in Engineering

of piles by using the amplitude Fourier spectrum of thedisplacement record at the pile head instead of the mobilityfunction Watson et al [12] developed a wavelet transformsignal processingmethod instead of the conventional Fourierbased methods to locate the position of the pile tip It isnoted that in the low strain integrity pile testing where thehammer is relatively small compared to the pile dimensionRayleigh and shearwaveswill radiate from the impact loadingand the effects of three-dimensional (3D) waves on the nearfield responses are obvious Subsequently several researchersconducted relevant studies and proposed several methods todiminish the effects of 3D waves on the dynamic responseof the pile top Liao and Roesset [13 14] investigated theinfluence of 3D waves on the dynamic response at the top ofintact and defective pile by comparing one-dimensional (1D)wave theory and 3D axisymmetric finite element simulationresults It is shown from their studies that 3D effects aremainly influenced by the frequency and are more stronglymanifested at high frequencies Chow et al [15] found thatthe velocity response curves resemble that of a pile with adefect near the pile head when considering 3D effects andfurther proposed that the potential source of error can beremoved by maintaining a distance between hammer andreceiver that is greater than 50 of the pile radius Chai etal [16] found that when the ratio of the characteristic lengthof an impact pulse to the pile radius is large enough thecomponents of Rayleigh waves in the wave field at the piletop are diminished In this study Chai et al still proposedthat the receiver should be placed at positions between 05119877and 075119877 (119877 = pile radius) from the pile axis to diminishthe influence of themultireflections Lu et al [17] investigatedthe 3D characteristics of wave propagation in pipe-pile usingelastodynamic finite integration technique and found that theinterferences of Rayleigh waves are weakest at an angle of90∘ from where hammer hits Furthermore for the drilledpiles with high slenderness ratio it is difficult to detect thepile length and deep flaw from the traditional low strain pileintegrity testing technique due to insufficient impact energytesting signal decay and soil-pile interaction To solve thisproblem Ni et al [18] adjusted the testing devices for acquir-ing a lower frequency signal and developed a new numericalsignal process method to enhance the reflection signals fromthe pile tip It is also shown from the experimental results thatthe testing signal identification abilities can be improved bythe modified method

Most of the previous studies on the low strain pile integ-rity testing did not consider the effect of pile material damp-ing abrupt variation of surrounding soil properties andmul-tidefects in pile on the dynamic response It is worth notingthat the material damping indeed exists in a pile the proper-ties of the surrounding soil may change greatly in certainembedment depth and a pile may contain several defectsWang et al [19] investigated the vertical dynamic response ofan inhomogeneous viscoelastic pile and analyzed the effectof pile material damping and soil properties on the mechani-cal admittance and velocity response of the pile top Howeverin this study the surrounding soil reaction on the pile isapproximately simulated by a general Voigt model which

Layered soilj

r0

1

P(t)

N

H

hN

hj

z

h1

Hj Hjminus1

H1

ksb1cpb kpbksb1middot middot middot middot middot middot middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

Figure 1 Model of pile-soil interaction

cannot veritably and accurately reflect the pile-soil interac-tion Therefore the objective of this paper is to develop apractical solution to evaluate the theoretical capabilities ofthe nondestructive dynamic response method in detectingthe existence and location of single or double defects in aviscoelastic pile embedded in a multilayered soil Using thesolution developed a parametric study has been undertakento investigate the mechanism of one-dimensional elasticlongitudinal wave propagating in a defective pile Finallythe theoretical model developed in the present paper isvalidated by comparison of the theoretical fitted curve andfield measured curve of velocity response

2 Formulation of the Problem

21 Geometry and Basic Assumption The system exam-ined is an inhomogeneous viscoelastic pile embedded ina multilayered soil and the geometric model is shown inFigure 1 To portray the variation of cross-sectional acousticimpedance (the product of density cross-sectional area andthe one-dimensional elastic longitudinal wave velocity) of apile or differences in soil properties the pile-soil system issubdivided into a total of 119873 segments (layers) numbered by1 2 119895 119873 from pile tip to pile head The thickness ofthe 119895th (1 le 119895 le 119873) soil layer is equal to the length ofthe 119895th (1 le 119895 le 119873) pile segment and is denoted by ℎ

119895 In

order to derive an analytical or quasi-analytical solution forthis problem the assumptions are made as follows (1) thesurrounding soil is a linearly viscoelastic layer and the pileis vertical elastic and circular in cross-section The pile andsoil layer properties are assumed to be homogeneous withineach segment or layer respectively but may change fromsegment to segment or layer to layer (2) the pile-soil systemis subjected to small deformations and strains during thevibration the pile has a perfect contact with the surroundingsoil during the vibration (3) the free surface of the soil hasno normal and shear stresses and the soil is infinite in theradial direction (4) the soil at the base of the pile is modeledusing a spring with elastic constant 119896pb and a dashpot with

Mathematical Problems in Engineering 3

r

z

0

jth soil layerjth pilesegmenth

j

ksbj ksbj

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

kstj kstj

Plowastj u

lowastj

Plowastjminus1 u

lowastjminus1

flowastj (z)

Figure 2 Pile segment used in derivation of impedance (in the localcoordinate system)

damping coefficient 119888pb (5) the contact traction acting at the119895th soil layer due to 119895 minus 1th and 119895 + 1th soil layers is treated asthe distributed Winkler subgrade model independent of theradial distance (see Figure 2)

22 Basic Equations and Solutions The axisymmetricdynamic response of a linear viscoelastic soil is consideredhere and the governing equation of the 119895th soil layer can beexpressed in terms of vertical displacement as follows [20]

[1205782

119895

+ 119894 (119863V119895 (1205782

119895

minus 2) + 2119863119904119895)]

1205972

119908119895(119903 119911 119905)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908119895(119903 119911 119905)

=1

1198812

119895

1205972

119908119895(119903 119911 119905)

1205971199052

(1)

where 120578119895= radic(120582

119895+ 2119866119895)119866119895= radic2(1 minus 120583

119895)(1 minus 2120583

119895) 119881119895=

radic119866119895120588119895is the shear wave velocity in the 119895th soil layer 120583

119895

is Poisson ratio 120582119895and 119866

119895are Lamersquos elastic constant and

shear modulus of the 119895th soil layer respectively 1205821015840119895

and 1198661015840119895

are corresponding viscosity coefficients about Lamersquos elasticconstant and shear modulus respectively 119863

119904119895= 1198661015840

119895

119866119895and

119863V119895 = 1205821015840

119895

120582119895are hysteretic type dampingwhich is irrespective

with frequency 119908119895(119903 119911 119905) is the vertical displacement of the

119895th soil layer 119894 = radicminus1If the pile material is modeled as a single Voigt model the

axial displacement of the 119895th pile segment is governed by thefollowing one-dimensional equation of motion

119864119901119895

120588119901119895

1205972

119906119895(119911 119905)

1205971199112+

120578119901119895

120588119901119895

1205973

119906119895(119911 119905)

1205971199112120597119905+

2120587119903119895

120588119901119895119860119901119895

119891119895(119911 119905)

=

1205972

119906119895(119911 119905)

1205971199052

(2)

where 119903119895 120588119901119895 119864119901119895 119860119901119895 120578119901119895 and 119906

119895(119911 119905) are the radius mass

density Youngrsquos modulus cross-sectional area viscoelastic

damping coefficient and axial displacement of the 119895th pilesegment respectively 119891

119895(119911 119905) denotes the contact traction

along the 119895th pile-soil interface due to the surrounding soil

23 Initial Conditions of the Pile-Soil System The pile-soilsystem is stationary in the initial state (119905 = 0) Accordinglythe corresponding initial conditions can be written as

119908119895(119903 119911 119905)

10038161003816100381610038161003816119905=0 = 0

120597119908119895(119903 119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(119895 = 1 2 119873)

(3)

119906119895(119911 119905)

10038161003816100381610038161003816119905=0 = 0

120597119906119895(119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(119895 = 1 2 119873)

(4)

24 Boundary and Continuity Conditions of the Pile-SoilSystem Equations (1) and (2) cannot be solved analyticallyin the time domain thus they are solved by using Laplacetransform In order to solve the pile-soil interaction problemthe boundary and continuity conditions of the pile-soilsystem will be formulated in the Laplace transform spaceand expressed in the local coordinate system (as shown inFigure 2) The Laplace transform of a function 119891(119903 119911 119905) withrespect to 119905 is defined as

119891lowast

(119903 119911 119904) = int

infin

0

119891 (119903 119911 119905) 119890minus119904119905

119889119905 (5)

where 119904 is the Laplace transform parameter

241 The Boundary and Continuity Conditions of the SoilLayer The boundary conditions of the 119895th (1 le 119895 le 119873) soillayer can be written in the local coordinate system as

119908lowast

119895

(119903 997888rarrinfin 119911 119904) = 0 (6)

[

120597119908lowast

119895

(119903 119911 119904)

120597119911minus

119896st119895

119864119904119895

119908lowast

119895

(119903 119911 119904)]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= 0 (7)

[

120597119908lowast

119895

(119903 119911 119904)

120597119911+

119896sb119895

119864119904119895

119908lowast

119895

(119903 119911 119904)]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0 (8)

where 119896st119895 and 119896sb119895 denote the distributed reaction coefficientof the 119895+ 1th and 119895minus 1th soil layer acting on the 119895th soil layerrespectively For the 119873th soil layer 119896st119873 is equal to zero dueto the free surface of the soil

The continuity condition of the interface between the 119895thand 119895 + 1th (1 le 119895 le 119873 minus 1) soil layer can be written as

119896st119895 = 119896sb(119895+1) (9)

4 Mathematical Problems in Engineering

242 The Boundary Conditions of the Pile Segment Theboundary condition at the top of the 119895th (1 le 119895 le 119873) pilesegment can be expressed in the local coordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= minus

119875lowast

119895

(119904)

119864119901119895119860119901119895

(10)

where 119875lowast119895

(119904) denotes the axial force of 119895 + 1th pile segmentacting on the top of the 119895th pile segment It is noted that119875lowast

119873

(119904) = 119875lowast

(119904) for the119873th pile segmentAllowing for the continuity conditions of the vertical

displacement and axial force at the interface of adjacent pilesegments the boundary condition at the base of the 119895th (1 le119895 le 119873) pile segment can then be expressed in the localcoordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911+

119906lowast

119895

(119911 119904) 119885119895minus1

119864119901119895119860119901119895

]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0

(11)

where 119885119895minus1 denotes the vertical impedance at the top of the

119895 minus 1th pile segment It is worth noting that 1198850 = 119896pb + 119904119888pbfor the 1st pile segmentMoreover soil response at the pile toeis approximated by the response of a vertically vibrating rigiddisk on the surface of an elastic half-spaceThe soil spring anddamping coefficient at the pile toe can be given as follows [21]

119896pb =411986601199031

1 minus 1205830

119888pb =341199032

1

radic12058801198660

1 minus 1205830

(12)

where1198660 and 1205830 denote the shearmodulus and Poissonrsquos ratioof the soil underlying the pile toe

243 The Continuity Conditions of the Displacement andStress of the 119895th (1 le 119895 le 119873) Pile-Soil Interface Considerthe following

119908lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119906lowast

119895

(119911 119904) (13)

119891lowast

119895

(119911 119904) = 120591lowast

119903119911119895

(119903 119911 119904)

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

(14)

25 Solution of the 119895th Soil Layer Combining (1) with theinitial conditions given in (3) and applying the Laplacetransform yield

1205782

119895

+ 119894 [119863V119895 (1205782

119895

minus 2) + 2119863119904119895]

1205972

119908lowast

119895

(119903 119911 119904)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908lowast

119895

(119903 119911 119904)

= (119904

119881119895

)

2

119908lowast

119895

(119903 119911 119904)

(15)

To solve (15) a single-variable function 119908lowast

119895

(119903 119911 119904) =

119877119895(119903)119885119895(119911) is introduced and substituting119908lowast

119895

(119903 119911 119904) into (15)yields

1198892

119885119895(119911)

1198891199112+ 1198692

119895

119885119895(119911) = 0

1198892

119877119895(119903)

1198891199032+1

119903

119889119877119895(119903)

119889119903minus 1199022

119895

119877119895(119903) = 0

(16)

where 1199022119895

= (1205782

119895

+119894[119863V119895(1205782

119895

minus2)+2119863119904119895]1198692

119895

+(119904119881119895)2

)(1+119894119863119904119895)

The general solutions of (16) can be written as

119885119895(119911) = 119862

119895sin (119869119895119911) +119863

119895cos (119869

119895119911) (17)

119877119895(119903) = 119860

1198951198700 (119902119895119903) + 1198611198951198680 (119902119895119903) (18)

where 1198680(119902119895119903) and 1198700(119902119895119903) are the modified Bessel functionsof the first and second kinds of zero order respectively119860119895 119861119895 119862119895 and 119863

119895are the constants which remain to be

determined later from the boundary conditionsSince the function 1198680(119902119895119903) rarr infin when 119903 rarr infin

the constant 119861119895in (18) should vanish to zero to satisfy the

boundary condition given in (6) Then the substitution ofboundary conditions given in (7) and (8) into (17) results in

tan (ℎ119895119869119895) =

(119896sb119895 + 119896st119895) ℎ119895119869119895

(ℎ119895119869119895)2

minus 119896sb119895119896st119895

(19)

where 119896sb119895 = 119896sb119895ℎ119895119864119904119895 and 119896st119895 = 119896st119895ℎ119895119864119904119895 denote thedimensionless reaction coefficients at the base and top of the119895th soil layer respectively

Then the solution of (15) can be written in a seriesexpansion as

119908lowast

119895

(119903 119911 119904) =

infin

sum

119898=11198601198981198951198700 (119902119898119895119903) sin (119869119898119895119911 +120593119898119895) (20)

where 120593119898119895

= arctan(119869119898119895ℎ119895119896st119895) 119902

2

119898119895

= ([1205782

119895

+ 119894(119863V119895(1205782

119895

minus 2) +

2119863119904119895)]1198692

119898119895

+ (119904119881119895)2

)(1 + 119894119863119904119895)

Based on continuity condition given in (14) the contacttraction along the 1st pile-soil interface can be expressed as

119891lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

= minus119866119895(1 + 119894119863

119904119895)

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(21)

where 1198701(119902119898119895119903) denotes the modified Bessel functions of thesecond kind of the first order

Mathematical Problems in Engineering 5

26 Solution of the 119895th Pile Segment Based on the initialconditions given in (4) applying the Laplace transform into(2) yields

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112+

2120587119903119895

120588119901119895119860119901119895

119891lowast

119895

(119911 119904)

= 1199042

119906lowast

119895

(119911 119904)

(22)

For the 119895th pile segment by utilizing the stress continuitycondition that is given in (14) substituting (21) into (22) gives

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112minus 1199042

119906lowast

119895

(119911 119904)

=

2120587119903119895119866119895(1 + 119894119863

119904119895)

120588119901119895119860119901119895

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(23)

The solution of (23) can be written as

119906lowast

119895

(119911 119904) = 1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911

+

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

(24)

where 120595119898119895

= minus2119866119895(1 + 119894119863

119904119895)1198601198981198951199021198981198951198701(119902119898119895119903119895)(119903119895[119864119901119895(1 +

120578119901119895

119904119905119888)1198692

119898119895

+ 1205881199011198951199042

]) 120574119895= radic120588

1199011198951199042[119864119901119895(1 + 120578

119901119895

119904119905119888)] 1205721119895 and

1205722119895 are undetermined constants 120578119901119895

= 120578119901119895(119864119901119895119905119888) is the

dimensionless damping coefficient of 119895th pile segment 119905119888=

sum119873

119895=1

(ℎ119895119881119901119895) is the time of the elastic longitudinal wave

propagating from the pile top to pile tip119881119901119895= radic119864

119901119895120588119901119895is the

1D elastic longitudinal wave velocity in the 119895th pile segmentBy using the displacement continuity condition that is

given in (13) substituting (20) and (24) into (13) results in

1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911 +

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

=

infin

sum

119898=11198601198981198951198700 (119902119898119895119903119895) sin (119869119898119895119911 +120593119898119895)

(25)

By invoking the orthogonality of eigenfunctionssin(119869119898119895119911 + 120593119898119895) (119898 = 1 2 3 ) multiplying sin(119869

119898119895119911 + 120593119898119895)

on both sides of (25) and then integrating over the interval119911 = [0 ℎ

119895] the undetermined coefficient 119860

119898119895is found to be

119860119898119895=

1

119871119898119895119864119898119895

int

ℎ119895

0

119876119895sin (119869119898119895119911 +120593119898119895) 119889119911 (26)

where 119871119898119895= ℎ1198952 minus (sin(2119869

119898119895ℎ119895+ 2120593119898119895) minus sin(2120593

119898119895))(4119869

119898119895)

119876119895= 1205721119895119890120574119895119911 + 120572

2119895119890minus120574119895119911

119864119898119895=

2119866119895(1 + 119894119863

119904119895) 1199021198981198951198701(119902119898119895119903119895)

119903119895[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 1205881199011198951199042]

+1198700(119902119898119895119903119895) (27)

It is mathematically convenient at this stage to introducethe following dimensionless variables

119871119898119895=

119871119898119895

ℎ119895

119902119898119895

= 119902119898119895ℎ119895

119903119895=

119903119895

ℎ119895

119869119898119895= 119869119898119895ℎ119895

120574119895

= 120574119895ℎ119895

(28)

The amplitude of the vertical displacement of the 119895th pilesegment is then given by

119906lowast

119895

(119911 119904) = 1205721119895[119890120574119895119911 +

infin

sum

119898=1

120585119898119895

sin (119869119898119895119911 +120593119898119895)]

+ 1205722119895[119890minus120574119895119911 +

infin

sum

119898=1

120577119898119895

sin (119869119898119895119911 +120593119898119895)]

(29)

where

120585119898119895= V119898119895

120574119895

[119890120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

120577119898119895= V119898119895

minus120574119895

[119890minus120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890minus120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

V119898119895=

minus2119866119895(1 + 119894119863

119904119895) 119902119898119895

1198701(119902119898119895

119903119895)

[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 120588119901119895ℎ2

119895

1199042] 119903119895119871119898119895119864119898119895

(30)

6 Mathematical Problems in Engineering

Based on the boundary conditions of the 119895th pile segmentthen substituting (10) and (11) into (29) the variables 120572

1119895 1205722119895

are obtained and the vertical impedance function at the topend of the 119895th pile segment can be written as

119885119895(119904) =

119875119895(119904)

119906119895(119911 = 0 119904)

= minus119864119901119895(119860119901119895+ 120578119901119895

119904119905119888119860119901119895)

(12057211198951205722119895) [120574119895+ suminfin

119898=1

120585119898119895119869119898119895

cos (120593119898119895)] + [minus120574

119895+ suminfin

119898=1

120577119898119895119869119898119895

cos (120593119898119895)]

(12057211198951205722119895) [1 + sum

infin

119898=1

120585119898119895

sin (120593119898119895)] + [1 + sum

infin

119898=1

120577119898119895

sin (120593119898119895)]

(31)

where

1205721119895

1205722119895= minus

(1 + 120578119901119895

119904119905119888) [minus120574119895

119890minus120574119895 + sum

infin

119898=1 120577119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890minus120574119895 + sum

infin

119898=1 120577119898119895 sin (119869119898119895 + 120593119898119895)]

(1 + 120578119901119895

119904119905119888) [120574119895

119890120574119895 + sum

infin

119898=1 120585119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890120574119895 + sum

infin

119898=1 120585119898119895 sin (119869119898119895 + 120593119898119895)]

119885119895minus1

=

119885119895minus1ℎ119895

119864119901119895119860119901119895

(32)

For the 1st pile segment1198850= (119896pb+119904119888pb)ℎ1(11986411990111198601199011) denotes

the dimensionless reaction coefficient at the pile baseThrough recursion impedance function from the 1st pile

segment to the 119873th pile segment the vertical impedancefunction at the pile head can be expressed as

119885119873(119904) =

119875119873(119904)

119906119873(119911 = 0 119904)

= minus119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)(12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)](12057211198731205722119873) [1 + sum

infin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)]

(33)

Letting 119904 = 119894120596 the frequency response function ofdisplacement (admittance function of displacement) at thepile head can be written as

119867119906(120596) =

1119885119873

= minus1

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)]

(34)

Then the response function of velocity (or admittancefunction of velocity) at the pile head can be written as

119867V (120596) = 119894120596119867119906 (120596) =1

120588119901119873119860119901119873119881119901119873

1198671015840

V (120596) (35)

where 1198671015840

V(120596) is the dimensionless response function ofvelocity at the pile head and

1198671015840

V (120596) = minus119894120596120588119901119873119860119901119873119881119901119873

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)] (36)

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of piles by using the amplitude Fourier spectrum of thedisplacement record at the pile head instead of the mobilityfunction Watson et al [12] developed a wavelet transformsignal processingmethod instead of the conventional Fourierbased methods to locate the position of the pile tip It isnoted that in the low strain integrity pile testing where thehammer is relatively small compared to the pile dimensionRayleigh and shearwaveswill radiate from the impact loadingand the effects of three-dimensional (3D) waves on the nearfield responses are obvious Subsequently several researchersconducted relevant studies and proposed several methods todiminish the effects of 3D waves on the dynamic responseof the pile top Liao and Roesset [13 14] investigated theinfluence of 3D waves on the dynamic response at the top ofintact and defective pile by comparing one-dimensional (1D)wave theory and 3D axisymmetric finite element simulationresults It is shown from their studies that 3D effects aremainly influenced by the frequency and are more stronglymanifested at high frequencies Chow et al [15] found thatthe velocity response curves resemble that of a pile with adefect near the pile head when considering 3D effects andfurther proposed that the potential source of error can beremoved by maintaining a distance between hammer andreceiver that is greater than 50 of the pile radius Chai etal [16] found that when the ratio of the characteristic lengthof an impact pulse to the pile radius is large enough thecomponents of Rayleigh waves in the wave field at the piletop are diminished In this study Chai et al still proposedthat the receiver should be placed at positions between 05119877and 075119877 (119877 = pile radius) from the pile axis to diminishthe influence of themultireflections Lu et al [17] investigatedthe 3D characteristics of wave propagation in pipe-pile usingelastodynamic finite integration technique and found that theinterferences of Rayleigh waves are weakest at an angle of90∘ from where hammer hits Furthermore for the drilledpiles with high slenderness ratio it is difficult to detect thepile length and deep flaw from the traditional low strain pileintegrity testing technique due to insufficient impact energytesting signal decay and soil-pile interaction To solve thisproblem Ni et al [18] adjusted the testing devices for acquir-ing a lower frequency signal and developed a new numericalsignal process method to enhance the reflection signals fromthe pile tip It is also shown from the experimental results thatthe testing signal identification abilities can be improved bythe modified method

Most of the previous studies on the low strain pile integ-rity testing did not consider the effect of pile material damp-ing abrupt variation of surrounding soil properties andmul-tidefects in pile on the dynamic response It is worth notingthat the material damping indeed exists in a pile the proper-ties of the surrounding soil may change greatly in certainembedment depth and a pile may contain several defectsWang et al [19] investigated the vertical dynamic response ofan inhomogeneous viscoelastic pile and analyzed the effectof pile material damping and soil properties on the mechani-cal admittance and velocity response of the pile top Howeverin this study the surrounding soil reaction on the pile isapproximately simulated by a general Voigt model which

Layered soilj

r0

1

P(t)

N

H

hN

hj

z

h1

Hj Hjminus1

H1

ksb1cpb kpbksb1middot middot middot middot middot middot middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

Figure 1 Model of pile-soil interaction

cannot veritably and accurately reflect the pile-soil interac-tion Therefore the objective of this paper is to develop apractical solution to evaluate the theoretical capabilities ofthe nondestructive dynamic response method in detectingthe existence and location of single or double defects in aviscoelastic pile embedded in a multilayered soil Using thesolution developed a parametric study has been undertakento investigate the mechanism of one-dimensional elasticlongitudinal wave propagating in a defective pile Finallythe theoretical model developed in the present paper isvalidated by comparison of the theoretical fitted curve andfield measured curve of velocity response

2 Formulation of the Problem

21 Geometry and Basic Assumption The system exam-ined is an inhomogeneous viscoelastic pile embedded ina multilayered soil and the geometric model is shown inFigure 1 To portray the variation of cross-sectional acousticimpedance (the product of density cross-sectional area andthe one-dimensional elastic longitudinal wave velocity) of apile or differences in soil properties the pile-soil system issubdivided into a total of 119873 segments (layers) numbered by1 2 119895 119873 from pile tip to pile head The thickness ofthe 119895th (1 le 119895 le 119873) soil layer is equal to the length ofthe 119895th (1 le 119895 le 119873) pile segment and is denoted by ℎ

119895 In

order to derive an analytical or quasi-analytical solution forthis problem the assumptions are made as follows (1) thesurrounding soil is a linearly viscoelastic layer and the pileis vertical elastic and circular in cross-section The pile andsoil layer properties are assumed to be homogeneous withineach segment or layer respectively but may change fromsegment to segment or layer to layer (2) the pile-soil systemis subjected to small deformations and strains during thevibration the pile has a perfect contact with the surroundingsoil during the vibration (3) the free surface of the soil hasno normal and shear stresses and the soil is infinite in theradial direction (4) the soil at the base of the pile is modeledusing a spring with elastic constant 119896pb and a dashpot with

Mathematical Problems in Engineering 3

r

z

0

jth soil layerjth pilesegmenth

j

ksbj ksbj

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

kstj kstj

Plowastj u

lowastj

Plowastjminus1 u

lowastjminus1

flowastj (z)

Figure 2 Pile segment used in derivation of impedance (in the localcoordinate system)

damping coefficient 119888pb (5) the contact traction acting at the119895th soil layer due to 119895 minus 1th and 119895 + 1th soil layers is treated asthe distributed Winkler subgrade model independent of theradial distance (see Figure 2)

22 Basic Equations and Solutions The axisymmetricdynamic response of a linear viscoelastic soil is consideredhere and the governing equation of the 119895th soil layer can beexpressed in terms of vertical displacement as follows [20]

[1205782

119895

+ 119894 (119863V119895 (1205782

119895

minus 2) + 2119863119904119895)]

1205972

119908119895(119903 119911 119905)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908119895(119903 119911 119905)

=1

1198812

119895

1205972

119908119895(119903 119911 119905)

1205971199052

(1)

where 120578119895= radic(120582

119895+ 2119866119895)119866119895= radic2(1 minus 120583

119895)(1 minus 2120583

119895) 119881119895=

radic119866119895120588119895is the shear wave velocity in the 119895th soil layer 120583

119895

is Poisson ratio 120582119895and 119866

119895are Lamersquos elastic constant and

shear modulus of the 119895th soil layer respectively 1205821015840119895

and 1198661015840119895

are corresponding viscosity coefficients about Lamersquos elasticconstant and shear modulus respectively 119863

119904119895= 1198661015840

119895

119866119895and

119863V119895 = 1205821015840

119895

120582119895are hysteretic type dampingwhich is irrespective

with frequency 119908119895(119903 119911 119905) is the vertical displacement of the

119895th soil layer 119894 = radicminus1If the pile material is modeled as a single Voigt model the

axial displacement of the 119895th pile segment is governed by thefollowing one-dimensional equation of motion

119864119901119895

120588119901119895

1205972

119906119895(119911 119905)

1205971199112+

120578119901119895

120588119901119895

1205973

119906119895(119911 119905)

1205971199112120597119905+

2120587119903119895

120588119901119895119860119901119895

119891119895(119911 119905)

=

1205972

119906119895(119911 119905)

1205971199052

(2)

where 119903119895 120588119901119895 119864119901119895 119860119901119895 120578119901119895 and 119906

119895(119911 119905) are the radius mass

density Youngrsquos modulus cross-sectional area viscoelastic

damping coefficient and axial displacement of the 119895th pilesegment respectively 119891

119895(119911 119905) denotes the contact traction

along the 119895th pile-soil interface due to the surrounding soil

23 Initial Conditions of the Pile-Soil System The pile-soilsystem is stationary in the initial state (119905 = 0) Accordinglythe corresponding initial conditions can be written as

119908119895(119903 119911 119905)

10038161003816100381610038161003816119905=0 = 0

120597119908119895(119903 119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(119895 = 1 2 119873)

(3)

119906119895(119911 119905)

10038161003816100381610038161003816119905=0 = 0

120597119906119895(119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(119895 = 1 2 119873)

(4)

24 Boundary and Continuity Conditions of the Pile-SoilSystem Equations (1) and (2) cannot be solved analyticallyin the time domain thus they are solved by using Laplacetransform In order to solve the pile-soil interaction problemthe boundary and continuity conditions of the pile-soilsystem will be formulated in the Laplace transform spaceand expressed in the local coordinate system (as shown inFigure 2) The Laplace transform of a function 119891(119903 119911 119905) withrespect to 119905 is defined as

119891lowast

(119903 119911 119904) = int

infin

0

119891 (119903 119911 119905) 119890minus119904119905

119889119905 (5)

where 119904 is the Laplace transform parameter

241 The Boundary and Continuity Conditions of the SoilLayer The boundary conditions of the 119895th (1 le 119895 le 119873) soillayer can be written in the local coordinate system as

119908lowast

119895

(119903 997888rarrinfin 119911 119904) = 0 (6)

[

120597119908lowast

119895

(119903 119911 119904)

120597119911minus

119896st119895

119864119904119895

119908lowast

119895

(119903 119911 119904)]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= 0 (7)

[

120597119908lowast

119895

(119903 119911 119904)

120597119911+

119896sb119895

119864119904119895

119908lowast

119895

(119903 119911 119904)]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0 (8)

where 119896st119895 and 119896sb119895 denote the distributed reaction coefficientof the 119895+ 1th and 119895minus 1th soil layer acting on the 119895th soil layerrespectively For the 119873th soil layer 119896st119873 is equal to zero dueto the free surface of the soil

The continuity condition of the interface between the 119895thand 119895 + 1th (1 le 119895 le 119873 minus 1) soil layer can be written as

119896st119895 = 119896sb(119895+1) (9)

4 Mathematical Problems in Engineering

242 The Boundary Conditions of the Pile Segment Theboundary condition at the top of the 119895th (1 le 119895 le 119873) pilesegment can be expressed in the local coordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= minus

119875lowast

119895

(119904)

119864119901119895119860119901119895

(10)

where 119875lowast119895

(119904) denotes the axial force of 119895 + 1th pile segmentacting on the top of the 119895th pile segment It is noted that119875lowast

119873

(119904) = 119875lowast

(119904) for the119873th pile segmentAllowing for the continuity conditions of the vertical

displacement and axial force at the interface of adjacent pilesegments the boundary condition at the base of the 119895th (1 le119895 le 119873) pile segment can then be expressed in the localcoordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911+

119906lowast

119895

(119911 119904) 119885119895minus1

119864119901119895119860119901119895

]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0

(11)

where 119885119895minus1 denotes the vertical impedance at the top of the

119895 minus 1th pile segment It is worth noting that 1198850 = 119896pb + 119904119888pbfor the 1st pile segmentMoreover soil response at the pile toeis approximated by the response of a vertically vibrating rigiddisk on the surface of an elastic half-spaceThe soil spring anddamping coefficient at the pile toe can be given as follows [21]

119896pb =411986601199031

1 minus 1205830

119888pb =341199032

1

radic12058801198660

1 minus 1205830

(12)

where1198660 and 1205830 denote the shearmodulus and Poissonrsquos ratioof the soil underlying the pile toe

243 The Continuity Conditions of the Displacement andStress of the 119895th (1 le 119895 le 119873) Pile-Soil Interface Considerthe following

119908lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119906lowast

119895

(119911 119904) (13)

119891lowast

119895

(119911 119904) = 120591lowast

119903119911119895

(119903 119911 119904)

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

(14)

25 Solution of the 119895th Soil Layer Combining (1) with theinitial conditions given in (3) and applying the Laplacetransform yield

1205782

119895

+ 119894 [119863V119895 (1205782

119895

minus 2) + 2119863119904119895]

1205972

119908lowast

119895

(119903 119911 119904)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908lowast

119895

(119903 119911 119904)

= (119904

119881119895

)

2

119908lowast

119895

(119903 119911 119904)

(15)

To solve (15) a single-variable function 119908lowast

119895

(119903 119911 119904) =

119877119895(119903)119885119895(119911) is introduced and substituting119908lowast

119895

(119903 119911 119904) into (15)yields

1198892

119885119895(119911)

1198891199112+ 1198692

119895

119885119895(119911) = 0

1198892

119877119895(119903)

1198891199032+1

119903

119889119877119895(119903)

119889119903minus 1199022

119895

119877119895(119903) = 0

(16)

where 1199022119895

= (1205782

119895

+119894[119863V119895(1205782

119895

minus2)+2119863119904119895]1198692

119895

+(119904119881119895)2

)(1+119894119863119904119895)

The general solutions of (16) can be written as

119885119895(119911) = 119862

119895sin (119869119895119911) +119863

119895cos (119869

119895119911) (17)

119877119895(119903) = 119860

1198951198700 (119902119895119903) + 1198611198951198680 (119902119895119903) (18)

where 1198680(119902119895119903) and 1198700(119902119895119903) are the modified Bessel functionsof the first and second kinds of zero order respectively119860119895 119861119895 119862119895 and 119863

119895are the constants which remain to be

determined later from the boundary conditionsSince the function 1198680(119902119895119903) rarr infin when 119903 rarr infin

the constant 119861119895in (18) should vanish to zero to satisfy the

boundary condition given in (6) Then the substitution ofboundary conditions given in (7) and (8) into (17) results in

tan (ℎ119895119869119895) =

(119896sb119895 + 119896st119895) ℎ119895119869119895

(ℎ119895119869119895)2

minus 119896sb119895119896st119895

(19)

where 119896sb119895 = 119896sb119895ℎ119895119864119904119895 and 119896st119895 = 119896st119895ℎ119895119864119904119895 denote thedimensionless reaction coefficients at the base and top of the119895th soil layer respectively

Then the solution of (15) can be written in a seriesexpansion as

119908lowast

119895

(119903 119911 119904) =

infin

sum

119898=11198601198981198951198700 (119902119898119895119903) sin (119869119898119895119911 +120593119898119895) (20)

where 120593119898119895

= arctan(119869119898119895ℎ119895119896st119895) 119902

2

119898119895

= ([1205782

119895

+ 119894(119863V119895(1205782

119895

minus 2) +

2119863119904119895)]1198692

119898119895

+ (119904119881119895)2

)(1 + 119894119863119904119895)

Based on continuity condition given in (14) the contacttraction along the 1st pile-soil interface can be expressed as

119891lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

= minus119866119895(1 + 119894119863

119904119895)

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(21)

where 1198701(119902119898119895119903) denotes the modified Bessel functions of thesecond kind of the first order

Mathematical Problems in Engineering 5

26 Solution of the 119895th Pile Segment Based on the initialconditions given in (4) applying the Laplace transform into(2) yields

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112+

2120587119903119895

120588119901119895119860119901119895

119891lowast

119895

(119911 119904)

= 1199042

119906lowast

119895

(119911 119904)

(22)

For the 119895th pile segment by utilizing the stress continuitycondition that is given in (14) substituting (21) into (22) gives

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112minus 1199042

119906lowast

119895

(119911 119904)

=

2120587119903119895119866119895(1 + 119894119863

119904119895)

120588119901119895119860119901119895

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(23)

The solution of (23) can be written as

119906lowast

119895

(119911 119904) = 1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911

+

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

(24)

where 120595119898119895

= minus2119866119895(1 + 119894119863

119904119895)1198601198981198951199021198981198951198701(119902119898119895119903119895)(119903119895[119864119901119895(1 +

120578119901119895

119904119905119888)1198692

119898119895

+ 1205881199011198951199042

]) 120574119895= radic120588

1199011198951199042[119864119901119895(1 + 120578

119901119895

119904119905119888)] 1205721119895 and

1205722119895 are undetermined constants 120578119901119895

= 120578119901119895(119864119901119895119905119888) is the

dimensionless damping coefficient of 119895th pile segment 119905119888=

sum119873

119895=1

(ℎ119895119881119901119895) is the time of the elastic longitudinal wave

propagating from the pile top to pile tip119881119901119895= radic119864

119901119895120588119901119895is the

1D elastic longitudinal wave velocity in the 119895th pile segmentBy using the displacement continuity condition that is

given in (13) substituting (20) and (24) into (13) results in

1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911 +

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

=

infin

sum

119898=11198601198981198951198700 (119902119898119895119903119895) sin (119869119898119895119911 +120593119898119895)

(25)

By invoking the orthogonality of eigenfunctionssin(119869119898119895119911 + 120593119898119895) (119898 = 1 2 3 ) multiplying sin(119869

119898119895119911 + 120593119898119895)

on both sides of (25) and then integrating over the interval119911 = [0 ℎ

119895] the undetermined coefficient 119860

119898119895is found to be

119860119898119895=

1

119871119898119895119864119898119895

int

ℎ119895

0

119876119895sin (119869119898119895119911 +120593119898119895) 119889119911 (26)

where 119871119898119895= ℎ1198952 minus (sin(2119869

119898119895ℎ119895+ 2120593119898119895) minus sin(2120593

119898119895))(4119869

119898119895)

119876119895= 1205721119895119890120574119895119911 + 120572

2119895119890minus120574119895119911

119864119898119895=

2119866119895(1 + 119894119863

119904119895) 1199021198981198951198701(119902119898119895119903119895)

119903119895[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 1205881199011198951199042]

+1198700(119902119898119895119903119895) (27)

It is mathematically convenient at this stage to introducethe following dimensionless variables

119871119898119895=

119871119898119895

ℎ119895

119902119898119895

= 119902119898119895ℎ119895

119903119895=

119903119895

ℎ119895

119869119898119895= 119869119898119895ℎ119895

120574119895

= 120574119895ℎ119895

(28)

The amplitude of the vertical displacement of the 119895th pilesegment is then given by

119906lowast

119895

(119911 119904) = 1205721119895[119890120574119895119911 +

infin

sum

119898=1

120585119898119895

sin (119869119898119895119911 +120593119898119895)]

+ 1205722119895[119890minus120574119895119911 +

infin

sum

119898=1

120577119898119895

sin (119869119898119895119911 +120593119898119895)]

(29)

where

120585119898119895= V119898119895

120574119895

[119890120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

120577119898119895= V119898119895

minus120574119895

[119890minus120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890minus120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

V119898119895=

minus2119866119895(1 + 119894119863

119904119895) 119902119898119895

1198701(119902119898119895

119903119895)

[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 120588119901119895ℎ2

119895

1199042] 119903119895119871119898119895119864119898119895

(30)

6 Mathematical Problems in Engineering

Based on the boundary conditions of the 119895th pile segmentthen substituting (10) and (11) into (29) the variables 120572

1119895 1205722119895

are obtained and the vertical impedance function at the topend of the 119895th pile segment can be written as

119885119895(119904) =

119875119895(119904)

119906119895(119911 = 0 119904)

= minus119864119901119895(119860119901119895+ 120578119901119895

119904119905119888119860119901119895)

(12057211198951205722119895) [120574119895+ suminfin

119898=1

120585119898119895119869119898119895

cos (120593119898119895)] + [minus120574

119895+ suminfin

119898=1

120577119898119895119869119898119895

cos (120593119898119895)]

(12057211198951205722119895) [1 + sum

infin

119898=1

120585119898119895

sin (120593119898119895)] + [1 + sum

infin

119898=1

120577119898119895

sin (120593119898119895)]

(31)

where

1205721119895

1205722119895= minus

(1 + 120578119901119895

119904119905119888) [minus120574119895

119890minus120574119895 + sum

infin

119898=1 120577119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890minus120574119895 + sum

infin

119898=1 120577119898119895 sin (119869119898119895 + 120593119898119895)]

(1 + 120578119901119895

119904119905119888) [120574119895

119890120574119895 + sum

infin

119898=1 120585119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890120574119895 + sum

infin

119898=1 120585119898119895 sin (119869119898119895 + 120593119898119895)]

119885119895minus1

=

119885119895minus1ℎ119895

119864119901119895119860119901119895

(32)

For the 1st pile segment1198850= (119896pb+119904119888pb)ℎ1(11986411990111198601199011) denotes

the dimensionless reaction coefficient at the pile baseThrough recursion impedance function from the 1st pile

segment to the 119873th pile segment the vertical impedancefunction at the pile head can be expressed as

119885119873(119904) =

119875119873(119904)

119906119873(119911 = 0 119904)

= minus119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)(12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)](12057211198731205722119873) [1 + sum

infin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)]

(33)

Letting 119904 = 119894120596 the frequency response function ofdisplacement (admittance function of displacement) at thepile head can be written as

119867119906(120596) =

1119885119873

= minus1

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)]

(34)

Then the response function of velocity (or admittancefunction of velocity) at the pile head can be written as

119867V (120596) = 119894120596119867119906 (120596) =1

120588119901119873119860119901119873119881119901119873

1198671015840

V (120596) (35)

where 1198671015840

V(120596) is the dimensionless response function ofvelocity at the pile head and

1198671015840

V (120596) = minus119894120596120588119901119873119860119901119873119881119901119873

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)] (36)

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

r

z

0

jth soil layerjth pilesegmenth

j

ksbj ksbj

middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

kstj kstj

Plowastj u

lowastj

Plowastjminus1 u

lowastjminus1

flowastj (z)

Figure 2 Pile segment used in derivation of impedance (in the localcoordinate system)

damping coefficient 119888pb (5) the contact traction acting at the119895th soil layer due to 119895 minus 1th and 119895 + 1th soil layers is treated asthe distributed Winkler subgrade model independent of theradial distance (see Figure 2)

22 Basic Equations and Solutions The axisymmetricdynamic response of a linear viscoelastic soil is consideredhere and the governing equation of the 119895th soil layer can beexpressed in terms of vertical displacement as follows [20]

[1205782

119895

+ 119894 (119863V119895 (1205782

119895

minus 2) + 2119863119904119895)]

1205972

119908119895(119903 119911 119905)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908119895(119903 119911 119905)

=1

1198812

119895

1205972

119908119895(119903 119911 119905)

1205971199052

(1)

where 120578119895= radic(120582

119895+ 2119866119895)119866119895= radic2(1 minus 120583

119895)(1 minus 2120583

119895) 119881119895=

radic119866119895120588119895is the shear wave velocity in the 119895th soil layer 120583

119895

is Poisson ratio 120582119895and 119866

119895are Lamersquos elastic constant and

shear modulus of the 119895th soil layer respectively 1205821015840119895

and 1198661015840119895

are corresponding viscosity coefficients about Lamersquos elasticconstant and shear modulus respectively 119863

119904119895= 1198661015840

119895

119866119895and

119863V119895 = 1205821015840

119895

120582119895are hysteretic type dampingwhich is irrespective

with frequency 119908119895(119903 119911 119905) is the vertical displacement of the

119895th soil layer 119894 = radicminus1If the pile material is modeled as a single Voigt model the

axial displacement of the 119895th pile segment is governed by thefollowing one-dimensional equation of motion

119864119901119895

120588119901119895

1205972

119906119895(119911 119905)

1205971199112+

120578119901119895

120588119901119895

1205973

119906119895(119911 119905)

1205971199112120597119905+

2120587119903119895

120588119901119895119860119901119895

119891119895(119911 119905)

=

1205972

119906119895(119911 119905)

1205971199052

(2)

where 119903119895 120588119901119895 119864119901119895 119860119901119895 120578119901119895 and 119906

119895(119911 119905) are the radius mass

density Youngrsquos modulus cross-sectional area viscoelastic

damping coefficient and axial displacement of the 119895th pilesegment respectively 119891

119895(119911 119905) denotes the contact traction

along the 119895th pile-soil interface due to the surrounding soil

23 Initial Conditions of the Pile-Soil System The pile-soilsystem is stationary in the initial state (119905 = 0) Accordinglythe corresponding initial conditions can be written as

119908119895(119903 119911 119905)

10038161003816100381610038161003816119905=0 = 0

120597119908119895(119903 119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(119895 = 1 2 119873)

(3)

119906119895(119911 119905)

10038161003816100381610038161003816119905=0 = 0

120597119906119895(119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(119895 = 1 2 119873)

(4)

24 Boundary and Continuity Conditions of the Pile-SoilSystem Equations (1) and (2) cannot be solved analyticallyin the time domain thus they are solved by using Laplacetransform In order to solve the pile-soil interaction problemthe boundary and continuity conditions of the pile-soilsystem will be formulated in the Laplace transform spaceand expressed in the local coordinate system (as shown inFigure 2) The Laplace transform of a function 119891(119903 119911 119905) withrespect to 119905 is defined as

119891lowast

(119903 119911 119904) = int

infin

0

119891 (119903 119911 119905) 119890minus119904119905

119889119905 (5)

where 119904 is the Laplace transform parameter

241 The Boundary and Continuity Conditions of the SoilLayer The boundary conditions of the 119895th (1 le 119895 le 119873) soillayer can be written in the local coordinate system as

119908lowast

119895

(119903 997888rarrinfin 119911 119904) = 0 (6)

[

120597119908lowast

119895

(119903 119911 119904)

120597119911minus

119896st119895

119864119904119895

119908lowast

119895

(119903 119911 119904)]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= 0 (7)

[

120597119908lowast

119895

(119903 119911 119904)

120597119911+

119896sb119895

119864119904119895

119908lowast

119895

(119903 119911 119904)]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0 (8)

where 119896st119895 and 119896sb119895 denote the distributed reaction coefficientof the 119895+ 1th and 119895minus 1th soil layer acting on the 119895th soil layerrespectively For the 119873th soil layer 119896st119873 is equal to zero dueto the free surface of the soil

The continuity condition of the interface between the 119895thand 119895 + 1th (1 le 119895 le 119873 minus 1) soil layer can be written as

119896st119895 = 119896sb(119895+1) (9)

4 Mathematical Problems in Engineering

242 The Boundary Conditions of the Pile Segment Theboundary condition at the top of the 119895th (1 le 119895 le 119873) pilesegment can be expressed in the local coordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= minus

119875lowast

119895

(119904)

119864119901119895119860119901119895

(10)

where 119875lowast119895

(119904) denotes the axial force of 119895 + 1th pile segmentacting on the top of the 119895th pile segment It is noted that119875lowast

119873

(119904) = 119875lowast

(119904) for the119873th pile segmentAllowing for the continuity conditions of the vertical

displacement and axial force at the interface of adjacent pilesegments the boundary condition at the base of the 119895th (1 le119895 le 119873) pile segment can then be expressed in the localcoordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911+

119906lowast

119895

(119911 119904) 119885119895minus1

119864119901119895119860119901119895

]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0

(11)

where 119885119895minus1 denotes the vertical impedance at the top of the

119895 minus 1th pile segment It is worth noting that 1198850 = 119896pb + 119904119888pbfor the 1st pile segmentMoreover soil response at the pile toeis approximated by the response of a vertically vibrating rigiddisk on the surface of an elastic half-spaceThe soil spring anddamping coefficient at the pile toe can be given as follows [21]

119896pb =411986601199031

1 minus 1205830

119888pb =341199032

1

radic12058801198660

1 minus 1205830

(12)

where1198660 and 1205830 denote the shearmodulus and Poissonrsquos ratioof the soil underlying the pile toe

243 The Continuity Conditions of the Displacement andStress of the 119895th (1 le 119895 le 119873) Pile-Soil Interface Considerthe following

119908lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119906lowast

119895

(119911 119904) (13)

119891lowast

119895

(119911 119904) = 120591lowast

119903119911119895

(119903 119911 119904)

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

(14)

25 Solution of the 119895th Soil Layer Combining (1) with theinitial conditions given in (3) and applying the Laplacetransform yield

1205782

119895

+ 119894 [119863V119895 (1205782

119895

minus 2) + 2119863119904119895]

1205972

119908lowast

119895

(119903 119911 119904)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908lowast

119895

(119903 119911 119904)

= (119904

119881119895

)

2

119908lowast

119895

(119903 119911 119904)

(15)

To solve (15) a single-variable function 119908lowast

119895

(119903 119911 119904) =

119877119895(119903)119885119895(119911) is introduced and substituting119908lowast

119895

(119903 119911 119904) into (15)yields

1198892

119885119895(119911)

1198891199112+ 1198692

119895

119885119895(119911) = 0

1198892

119877119895(119903)

1198891199032+1

119903

119889119877119895(119903)

119889119903minus 1199022

119895

119877119895(119903) = 0

(16)

where 1199022119895

= (1205782

119895

+119894[119863V119895(1205782

119895

minus2)+2119863119904119895]1198692

119895

+(119904119881119895)2

)(1+119894119863119904119895)

The general solutions of (16) can be written as

119885119895(119911) = 119862

119895sin (119869119895119911) +119863

119895cos (119869

119895119911) (17)

119877119895(119903) = 119860

1198951198700 (119902119895119903) + 1198611198951198680 (119902119895119903) (18)

where 1198680(119902119895119903) and 1198700(119902119895119903) are the modified Bessel functionsof the first and second kinds of zero order respectively119860119895 119861119895 119862119895 and 119863

119895are the constants which remain to be

determined later from the boundary conditionsSince the function 1198680(119902119895119903) rarr infin when 119903 rarr infin

the constant 119861119895in (18) should vanish to zero to satisfy the

boundary condition given in (6) Then the substitution ofboundary conditions given in (7) and (8) into (17) results in

tan (ℎ119895119869119895) =

(119896sb119895 + 119896st119895) ℎ119895119869119895

(ℎ119895119869119895)2

minus 119896sb119895119896st119895

(19)

where 119896sb119895 = 119896sb119895ℎ119895119864119904119895 and 119896st119895 = 119896st119895ℎ119895119864119904119895 denote thedimensionless reaction coefficients at the base and top of the119895th soil layer respectively

Then the solution of (15) can be written in a seriesexpansion as

119908lowast

119895

(119903 119911 119904) =

infin

sum

119898=11198601198981198951198700 (119902119898119895119903) sin (119869119898119895119911 +120593119898119895) (20)

where 120593119898119895

= arctan(119869119898119895ℎ119895119896st119895) 119902

2

119898119895

= ([1205782

119895

+ 119894(119863V119895(1205782

119895

minus 2) +

2119863119904119895)]1198692

119898119895

+ (119904119881119895)2

)(1 + 119894119863119904119895)

Based on continuity condition given in (14) the contacttraction along the 1st pile-soil interface can be expressed as

119891lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

= minus119866119895(1 + 119894119863

119904119895)

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(21)

where 1198701(119902119898119895119903) denotes the modified Bessel functions of thesecond kind of the first order

Mathematical Problems in Engineering 5

26 Solution of the 119895th Pile Segment Based on the initialconditions given in (4) applying the Laplace transform into(2) yields

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112+

2120587119903119895

120588119901119895119860119901119895

119891lowast

119895

(119911 119904)

= 1199042

119906lowast

119895

(119911 119904)

(22)

For the 119895th pile segment by utilizing the stress continuitycondition that is given in (14) substituting (21) into (22) gives

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112minus 1199042

119906lowast

119895

(119911 119904)

=

2120587119903119895119866119895(1 + 119894119863

119904119895)

120588119901119895119860119901119895

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(23)

The solution of (23) can be written as

119906lowast

119895

(119911 119904) = 1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911

+

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

(24)

where 120595119898119895

= minus2119866119895(1 + 119894119863

119904119895)1198601198981198951199021198981198951198701(119902119898119895119903119895)(119903119895[119864119901119895(1 +

120578119901119895

119904119905119888)1198692

119898119895

+ 1205881199011198951199042

]) 120574119895= radic120588

1199011198951199042[119864119901119895(1 + 120578

119901119895

119904119905119888)] 1205721119895 and

1205722119895 are undetermined constants 120578119901119895

= 120578119901119895(119864119901119895119905119888) is the

dimensionless damping coefficient of 119895th pile segment 119905119888=

sum119873

119895=1

(ℎ119895119881119901119895) is the time of the elastic longitudinal wave

propagating from the pile top to pile tip119881119901119895= radic119864

119901119895120588119901119895is the

1D elastic longitudinal wave velocity in the 119895th pile segmentBy using the displacement continuity condition that is

given in (13) substituting (20) and (24) into (13) results in

1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911 +

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

=

infin

sum

119898=11198601198981198951198700 (119902119898119895119903119895) sin (119869119898119895119911 +120593119898119895)

(25)

By invoking the orthogonality of eigenfunctionssin(119869119898119895119911 + 120593119898119895) (119898 = 1 2 3 ) multiplying sin(119869

119898119895119911 + 120593119898119895)

on both sides of (25) and then integrating over the interval119911 = [0 ℎ

119895] the undetermined coefficient 119860

119898119895is found to be

119860119898119895=

1

119871119898119895119864119898119895

int

ℎ119895

0

119876119895sin (119869119898119895119911 +120593119898119895) 119889119911 (26)

where 119871119898119895= ℎ1198952 minus (sin(2119869

119898119895ℎ119895+ 2120593119898119895) minus sin(2120593

119898119895))(4119869

119898119895)

119876119895= 1205721119895119890120574119895119911 + 120572

2119895119890minus120574119895119911

119864119898119895=

2119866119895(1 + 119894119863

119904119895) 1199021198981198951198701(119902119898119895119903119895)

119903119895[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 1205881199011198951199042]

+1198700(119902119898119895119903119895) (27)

It is mathematically convenient at this stage to introducethe following dimensionless variables

119871119898119895=

119871119898119895

ℎ119895

119902119898119895

= 119902119898119895ℎ119895

119903119895=

119903119895

ℎ119895

119869119898119895= 119869119898119895ℎ119895

120574119895

= 120574119895ℎ119895

(28)

The amplitude of the vertical displacement of the 119895th pilesegment is then given by

119906lowast

119895

(119911 119904) = 1205721119895[119890120574119895119911 +

infin

sum

119898=1

120585119898119895

sin (119869119898119895119911 +120593119898119895)]

+ 1205722119895[119890minus120574119895119911 +

infin

sum

119898=1

120577119898119895

sin (119869119898119895119911 +120593119898119895)]

(29)

where

120585119898119895= V119898119895

120574119895

[119890120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

120577119898119895= V119898119895

minus120574119895

[119890minus120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890minus120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

V119898119895=

minus2119866119895(1 + 119894119863

119904119895) 119902119898119895

1198701(119902119898119895

119903119895)

[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 120588119901119895ℎ2

119895

1199042] 119903119895119871119898119895119864119898119895

(30)

6 Mathematical Problems in Engineering

Based on the boundary conditions of the 119895th pile segmentthen substituting (10) and (11) into (29) the variables 120572

1119895 1205722119895

are obtained and the vertical impedance function at the topend of the 119895th pile segment can be written as

119885119895(119904) =

119875119895(119904)

119906119895(119911 = 0 119904)

= minus119864119901119895(119860119901119895+ 120578119901119895

119904119905119888119860119901119895)

(12057211198951205722119895) [120574119895+ suminfin

119898=1

120585119898119895119869119898119895

cos (120593119898119895)] + [minus120574

119895+ suminfin

119898=1

120577119898119895119869119898119895

cos (120593119898119895)]

(12057211198951205722119895) [1 + sum

infin

119898=1

120585119898119895

sin (120593119898119895)] + [1 + sum

infin

119898=1

120577119898119895

sin (120593119898119895)]

(31)

where

1205721119895

1205722119895= minus

(1 + 120578119901119895

119904119905119888) [minus120574119895

119890minus120574119895 + sum

infin

119898=1 120577119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890minus120574119895 + sum

infin

119898=1 120577119898119895 sin (119869119898119895 + 120593119898119895)]

(1 + 120578119901119895

119904119905119888) [120574119895

119890120574119895 + sum

infin

119898=1 120585119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890120574119895 + sum

infin

119898=1 120585119898119895 sin (119869119898119895 + 120593119898119895)]

119885119895minus1

=

119885119895minus1ℎ119895

119864119901119895119860119901119895

(32)

For the 1st pile segment1198850= (119896pb+119904119888pb)ℎ1(11986411990111198601199011) denotes

the dimensionless reaction coefficient at the pile baseThrough recursion impedance function from the 1st pile

segment to the 119873th pile segment the vertical impedancefunction at the pile head can be expressed as

119885119873(119904) =

119875119873(119904)

119906119873(119911 = 0 119904)

= minus119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)(12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)](12057211198731205722119873) [1 + sum

infin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)]

(33)

Letting 119904 = 119894120596 the frequency response function ofdisplacement (admittance function of displacement) at thepile head can be written as

119867119906(120596) =

1119885119873

= minus1

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)]

(34)

Then the response function of velocity (or admittancefunction of velocity) at the pile head can be written as

119867V (120596) = 119894120596119867119906 (120596) =1

120588119901119873119860119901119873119881119901119873

1198671015840

V (120596) (35)

where 1198671015840

V(120596) is the dimensionless response function ofvelocity at the pile head and

1198671015840

V (120596) = minus119894120596120588119901119873119860119901119873119881119901119873

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)] (36)

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

242 The Boundary Conditions of the Pile Segment Theboundary condition at the top of the 119895th (1 le 119895 le 119873) pilesegment can be expressed in the local coordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911]

1003816100381610038161003816100381610038161003816100381610038161003816119911=0

= minus

119875lowast

119895

(119904)

119864119901119895119860119901119895

(10)

where 119875lowast119895

(119904) denotes the axial force of 119895 + 1th pile segmentacting on the top of the 119895th pile segment It is noted that119875lowast

119873

(119904) = 119875lowast

(119904) for the119873th pile segmentAllowing for the continuity conditions of the vertical

displacement and axial force at the interface of adjacent pilesegments the boundary condition at the base of the 119895th (1 le119895 le 119873) pile segment can then be expressed in the localcoordinate system as

[

119889119906lowast

119895

(119911 119904)

119889119911+

120578119901119895119904

119864119901119895

119889119906lowast

119895

(119911 119904)

119889119911+

119906lowast

119895

(119911 119904) 119885119895minus1

119864119901119895119860119901119895

]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= 0

(11)

where 119885119895minus1 denotes the vertical impedance at the top of the

119895 minus 1th pile segment It is worth noting that 1198850 = 119896pb + 119904119888pbfor the 1st pile segmentMoreover soil response at the pile toeis approximated by the response of a vertically vibrating rigiddisk on the surface of an elastic half-spaceThe soil spring anddamping coefficient at the pile toe can be given as follows [21]

119896pb =411986601199031

1 minus 1205830

119888pb =341199032

1

radic12058801198660

1 minus 1205830

(12)

where1198660 and 1205830 denote the shearmodulus and Poissonrsquos ratioof the soil underlying the pile toe

243 The Continuity Conditions of the Displacement andStress of the 119895th (1 le 119895 le 119873) Pile-Soil Interface Considerthe following

119908lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119906lowast

119895

(119911 119904) (13)

119891lowast

119895

(119911 119904) = 120591lowast

119903119911119895

(119903 119911 119904)

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

(14)

25 Solution of the 119895th Soil Layer Combining (1) with theinitial conditions given in (3) and applying the Laplacetransform yield

1205782

119895

+ 119894 [119863V119895 (1205782

119895

minus 2) + 2119863119904119895]

1205972

119908lowast

119895

(119903 119911 119904)

1205971199112

+ (1 + 119894119863119904119895)(

1

119903

120597

120597119903+1205972

1205971199032)119908lowast

119895

(119903 119911 119904)

= (119904

119881119895

)

2

119908lowast

119895

(119903 119911 119904)

(15)

To solve (15) a single-variable function 119908lowast

119895

(119903 119911 119904) =

119877119895(119903)119885119895(119911) is introduced and substituting119908lowast

119895

(119903 119911 119904) into (15)yields

1198892

119885119895(119911)

1198891199112+ 1198692

119895

119885119895(119911) = 0

1198892

119877119895(119903)

1198891199032+1

119903

119889119877119895(119903)

119889119903minus 1199022

119895

119877119895(119903) = 0

(16)

where 1199022119895

= (1205782

119895

+119894[119863V119895(1205782

119895

minus2)+2119863119904119895]1198692

119895

+(119904119881119895)2

)(1+119894119863119904119895)

The general solutions of (16) can be written as

119885119895(119911) = 119862

119895sin (119869119895119911) +119863

119895cos (119869

119895119911) (17)

119877119895(119903) = 119860

1198951198700 (119902119895119903) + 1198611198951198680 (119902119895119903) (18)

where 1198680(119902119895119903) and 1198700(119902119895119903) are the modified Bessel functionsof the first and second kinds of zero order respectively119860119895 119861119895 119862119895 and 119863

119895are the constants which remain to be

determined later from the boundary conditionsSince the function 1198680(119902119895119903) rarr infin when 119903 rarr infin

the constant 119861119895in (18) should vanish to zero to satisfy the

boundary condition given in (6) Then the substitution ofboundary conditions given in (7) and (8) into (17) results in

tan (ℎ119895119869119895) =

(119896sb119895 + 119896st119895) ℎ119895119869119895

(ℎ119895119869119895)2

minus 119896sb119895119896st119895

(19)

where 119896sb119895 = 119896sb119895ℎ119895119864119904119895 and 119896st119895 = 119896st119895ℎ119895119864119904119895 denote thedimensionless reaction coefficients at the base and top of the119895th soil layer respectively

Then the solution of (15) can be written in a seriesexpansion as

119908lowast

119895

(119903 119911 119904) =

infin

sum

119898=11198601198981198951198700 (119902119898119895119903) sin (119869119898119895119911 +120593119898119895) (20)

where 120593119898119895

= arctan(119869119898119895ℎ119895119896st119895) 119902

2

119898119895

= ([1205782

119895

+ 119894(119863V119895(1205782

119895

minus 2) +

2119863119904119895)]1198692

119898119895

+ (119904119881119895)2

)(1 + 119894119863119904119895)

Based on continuity condition given in (14) the contacttraction along the 1st pile-soil interface can be expressed as

119891lowast

119895

(119903 119911 119904)10038161003816100381610038161003816119903=119903119895

= 119866119895(1 + 119894119863

119904119895)

120597119908lowast

119895

(119903 119911 119904)

120597119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119903119895

= minus119866119895(1 + 119894119863

119904119895)

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(21)

where 1198701(119902119898119895119903) denotes the modified Bessel functions of thesecond kind of the first order

Mathematical Problems in Engineering 5

26 Solution of the 119895th Pile Segment Based on the initialconditions given in (4) applying the Laplace transform into(2) yields

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112+

2120587119903119895

120588119901119895119860119901119895

119891lowast

119895

(119911 119904)

= 1199042

119906lowast

119895

(119911 119904)

(22)

For the 119895th pile segment by utilizing the stress continuitycondition that is given in (14) substituting (21) into (22) gives

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112minus 1199042

119906lowast

119895

(119911 119904)

=

2120587119903119895119866119895(1 + 119894119863

119904119895)

120588119901119895119860119901119895

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(23)

The solution of (23) can be written as

119906lowast

119895

(119911 119904) = 1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911

+

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

(24)

where 120595119898119895

= minus2119866119895(1 + 119894119863

119904119895)1198601198981198951199021198981198951198701(119902119898119895119903119895)(119903119895[119864119901119895(1 +

120578119901119895

119904119905119888)1198692

119898119895

+ 1205881199011198951199042

]) 120574119895= radic120588

1199011198951199042[119864119901119895(1 + 120578

119901119895

119904119905119888)] 1205721119895 and

1205722119895 are undetermined constants 120578119901119895

= 120578119901119895(119864119901119895119905119888) is the

dimensionless damping coefficient of 119895th pile segment 119905119888=

sum119873

119895=1

(ℎ119895119881119901119895) is the time of the elastic longitudinal wave

propagating from the pile top to pile tip119881119901119895= radic119864

119901119895120588119901119895is the

1D elastic longitudinal wave velocity in the 119895th pile segmentBy using the displacement continuity condition that is

given in (13) substituting (20) and (24) into (13) results in

1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911 +

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

=

infin

sum

119898=11198601198981198951198700 (119902119898119895119903119895) sin (119869119898119895119911 +120593119898119895)

(25)

By invoking the orthogonality of eigenfunctionssin(119869119898119895119911 + 120593119898119895) (119898 = 1 2 3 ) multiplying sin(119869

119898119895119911 + 120593119898119895)

on both sides of (25) and then integrating over the interval119911 = [0 ℎ

119895] the undetermined coefficient 119860

119898119895is found to be

119860119898119895=

1

119871119898119895119864119898119895

int

ℎ119895

0

119876119895sin (119869119898119895119911 +120593119898119895) 119889119911 (26)

where 119871119898119895= ℎ1198952 minus (sin(2119869

119898119895ℎ119895+ 2120593119898119895) minus sin(2120593

119898119895))(4119869

119898119895)

119876119895= 1205721119895119890120574119895119911 + 120572

2119895119890minus120574119895119911

119864119898119895=

2119866119895(1 + 119894119863

119904119895) 1199021198981198951198701(119902119898119895119903119895)

119903119895[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 1205881199011198951199042]

+1198700(119902119898119895119903119895) (27)

It is mathematically convenient at this stage to introducethe following dimensionless variables

119871119898119895=

119871119898119895

ℎ119895

119902119898119895

= 119902119898119895ℎ119895

119903119895=

119903119895

ℎ119895

119869119898119895= 119869119898119895ℎ119895

120574119895

= 120574119895ℎ119895

(28)

The amplitude of the vertical displacement of the 119895th pilesegment is then given by

119906lowast

119895

(119911 119904) = 1205721119895[119890120574119895119911 +

infin

sum

119898=1

120585119898119895

sin (119869119898119895119911 +120593119898119895)]

+ 1205722119895[119890minus120574119895119911 +

infin

sum

119898=1

120577119898119895

sin (119869119898119895119911 +120593119898119895)]

(29)

where

120585119898119895= V119898119895

120574119895

[119890120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

120577119898119895= V119898119895

minus120574119895

[119890minus120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890minus120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

V119898119895=

minus2119866119895(1 + 119894119863

119904119895) 119902119898119895

1198701(119902119898119895

119903119895)

[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 120588119901119895ℎ2

119895

1199042] 119903119895119871119898119895119864119898119895

(30)

6 Mathematical Problems in Engineering

Based on the boundary conditions of the 119895th pile segmentthen substituting (10) and (11) into (29) the variables 120572

1119895 1205722119895

are obtained and the vertical impedance function at the topend of the 119895th pile segment can be written as

119885119895(119904) =

119875119895(119904)

119906119895(119911 = 0 119904)

= minus119864119901119895(119860119901119895+ 120578119901119895

119904119905119888119860119901119895)

(12057211198951205722119895) [120574119895+ suminfin

119898=1

120585119898119895119869119898119895

cos (120593119898119895)] + [minus120574

119895+ suminfin

119898=1

120577119898119895119869119898119895

cos (120593119898119895)]

(12057211198951205722119895) [1 + sum

infin

119898=1

120585119898119895

sin (120593119898119895)] + [1 + sum

infin

119898=1

120577119898119895

sin (120593119898119895)]

(31)

where

1205721119895

1205722119895= minus

(1 + 120578119901119895

119904119905119888) [minus120574119895

119890minus120574119895 + sum

infin

119898=1 120577119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890minus120574119895 + sum

infin

119898=1 120577119898119895 sin (119869119898119895 + 120593119898119895)]

(1 + 120578119901119895

119904119905119888) [120574119895

119890120574119895 + sum

infin

119898=1 120585119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890120574119895 + sum

infin

119898=1 120585119898119895 sin (119869119898119895 + 120593119898119895)]

119885119895minus1

=

119885119895minus1ℎ119895

119864119901119895119860119901119895

(32)

For the 1st pile segment1198850= (119896pb+119904119888pb)ℎ1(11986411990111198601199011) denotes

the dimensionless reaction coefficient at the pile baseThrough recursion impedance function from the 1st pile

segment to the 119873th pile segment the vertical impedancefunction at the pile head can be expressed as

119885119873(119904) =

119875119873(119904)

119906119873(119911 = 0 119904)

= minus119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)(12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)](12057211198731205722119873) [1 + sum

infin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)]

(33)

Letting 119904 = 119894120596 the frequency response function ofdisplacement (admittance function of displacement) at thepile head can be written as

119867119906(120596) =

1119885119873

= minus1

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)]

(34)

Then the response function of velocity (or admittancefunction of velocity) at the pile head can be written as

119867V (120596) = 119894120596119867119906 (120596) =1

120588119901119873119860119901119873119881119901119873

1198671015840

V (120596) (35)

where 1198671015840

V(120596) is the dimensionless response function ofvelocity at the pile head and

1198671015840

V (120596) = minus119894120596120588119901119873119860119901119873119881119901119873

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)] (36)

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

26 Solution of the 119895th Pile Segment Based on the initialconditions given in (4) applying the Laplace transform into(2) yields

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112+

2120587119903119895

120588119901119895119860119901119895

119891lowast

119895

(119911 119904)

= 1199042

119906lowast

119895

(119911 119904)

(22)

For the 119895th pile segment by utilizing the stress continuitycondition that is given in (14) substituting (21) into (22) gives

(

119864119901119895

120588119901119895

+

120578119901119895119904

120588119901119895

)

1198892

119906lowast

119895

(119911 119904)

1198891199112minus 1199042

119906lowast

119895

(119911 119904)

=

2120587119903119895119866119895(1 + 119894119863

119904119895)

120588119901119895119860119901119895

sdot

infin

sum

119898=1

1198601198981198951199021198981198951198701(119902119898119895119903119895) sin (119869

119898119895119911 +120593119898119895)

(23)

The solution of (23) can be written as

119906lowast

119895

(119911 119904) = 1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911

+

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

(24)

where 120595119898119895

= minus2119866119895(1 + 119894119863

119904119895)1198601198981198951199021198981198951198701(119902119898119895119903119895)(119903119895[119864119901119895(1 +

120578119901119895

119904119905119888)1198692

119898119895

+ 1205881199011198951199042

]) 120574119895= radic120588

1199011198951199042[119864119901119895(1 + 120578

119901119895

119904119905119888)] 1205721119895 and

1205722119895 are undetermined constants 120578119901119895

= 120578119901119895(119864119901119895119905119888) is the

dimensionless damping coefficient of 119895th pile segment 119905119888=

sum119873

119895=1

(ℎ119895119881119901119895) is the time of the elastic longitudinal wave

propagating from the pile top to pile tip119881119901119895= radic119864

119901119895120588119901119895is the

1D elastic longitudinal wave velocity in the 119895th pile segmentBy using the displacement continuity condition that is

given in (13) substituting (20) and (24) into (13) results in

1205721119895119890120574119895119911 +1205722119895119890

minus120574119895119911 +

infin

sum

119898=1120595119898119895

sin (119869119898119895119911 +120593119898119895)

=

infin

sum

119898=11198601198981198951198700 (119902119898119895119903119895) sin (119869119898119895119911 +120593119898119895)

(25)

By invoking the orthogonality of eigenfunctionssin(119869119898119895119911 + 120593119898119895) (119898 = 1 2 3 ) multiplying sin(119869

119898119895119911 + 120593119898119895)

on both sides of (25) and then integrating over the interval119911 = [0 ℎ

119895] the undetermined coefficient 119860

119898119895is found to be

119860119898119895=

1

119871119898119895119864119898119895

int

ℎ119895

0

119876119895sin (119869119898119895119911 +120593119898119895) 119889119911 (26)

where 119871119898119895= ℎ1198952 minus (sin(2119869

119898119895ℎ119895+ 2120593119898119895) minus sin(2120593

119898119895))(4119869

119898119895)

119876119895= 1205721119895119890120574119895119911 + 120572

2119895119890minus120574119895119911

119864119898119895=

2119866119895(1 + 119894119863

119904119895) 1199021198981198951198701(119902119898119895119903119895)

119903119895[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 1205881199011198951199042]

+1198700(119902119898119895119903119895) (27)

It is mathematically convenient at this stage to introducethe following dimensionless variables

119871119898119895=

119871119898119895

ℎ119895

119902119898119895

= 119902119898119895ℎ119895

119903119895=

119903119895

ℎ119895

119869119898119895= 119869119898119895ℎ119895

120574119895

= 120574119895ℎ119895

(28)

The amplitude of the vertical displacement of the 119895th pilesegment is then given by

119906lowast

119895

(119911 119904) = 1205721119895[119890120574119895119911 +

infin

sum

119898=1

120585119898119895

sin (119869119898119895119911 +120593119898119895)]

+ 1205722119895[119890minus120574119895119911 +

infin

sum

119898=1

120577119898119895

sin (119869119898119895119911 +120593119898119895)]

(29)

where

120585119898119895= V119898119895

120574119895

[119890120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

120577119898119895= V119898119895

minus120574119895

[119890minus120574119895 sin (119869

119898119895+ 120593119898119895) minus sin (120593

119898119895)] minus 119869

119898119895[119890minus120574119895 cos (119869

119898119895+ 120593119898119895) minus cos (120593

119898119895)]

1198692

119898119895

+ 1205742

119895

V119898119895=

minus2119866119895(1 + 119894119863

119904119895) 119902119898119895

1198701(119902119898119895

119903119895)

[119864119901119895(1 + 120578

119901119895

119904119905119888) 1198692

119898119895

+ 120588119901119895ℎ2

119895

1199042] 119903119895119871119898119895119864119898119895

(30)

6 Mathematical Problems in Engineering

Based on the boundary conditions of the 119895th pile segmentthen substituting (10) and (11) into (29) the variables 120572

1119895 1205722119895

are obtained and the vertical impedance function at the topend of the 119895th pile segment can be written as

119885119895(119904) =

119875119895(119904)

119906119895(119911 = 0 119904)

= minus119864119901119895(119860119901119895+ 120578119901119895

119904119905119888119860119901119895)

(12057211198951205722119895) [120574119895+ suminfin

119898=1

120585119898119895119869119898119895

cos (120593119898119895)] + [minus120574

119895+ suminfin

119898=1

120577119898119895119869119898119895

cos (120593119898119895)]

(12057211198951205722119895) [1 + sum

infin

119898=1

120585119898119895

sin (120593119898119895)] + [1 + sum

infin

119898=1

120577119898119895

sin (120593119898119895)]

(31)

where

1205721119895

1205722119895= minus

(1 + 120578119901119895

119904119905119888) [minus120574119895

119890minus120574119895 + sum

infin

119898=1 120577119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890minus120574119895 + sum

infin

119898=1 120577119898119895 sin (119869119898119895 + 120593119898119895)]

(1 + 120578119901119895

119904119905119888) [120574119895

119890120574119895 + sum

infin

119898=1 120585119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890120574119895 + sum

infin

119898=1 120585119898119895 sin (119869119898119895 + 120593119898119895)]

119885119895minus1

=

119885119895minus1ℎ119895

119864119901119895119860119901119895

(32)

For the 1st pile segment1198850= (119896pb+119904119888pb)ℎ1(11986411990111198601199011) denotes

the dimensionless reaction coefficient at the pile baseThrough recursion impedance function from the 1st pile

segment to the 119873th pile segment the vertical impedancefunction at the pile head can be expressed as

119885119873(119904) =

119875119873(119904)

119906119873(119911 = 0 119904)

= minus119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)(12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)](12057211198731205722119873) [1 + sum

infin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)]

(33)

Letting 119904 = 119894120596 the frequency response function ofdisplacement (admittance function of displacement) at thepile head can be written as

119867119906(120596) =

1119885119873

= minus1

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)]

(34)

Then the response function of velocity (or admittancefunction of velocity) at the pile head can be written as

119867V (120596) = 119894120596119867119906 (120596) =1

120588119901119873119860119901119873119881119901119873

1198671015840

V (120596) (35)

where 1198671015840

V(120596) is the dimensionless response function ofvelocity at the pile head and

1198671015840

V (120596) = minus119894120596120588119901119873119860119901119873119881119901119873

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)] (36)

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Based on the boundary conditions of the 119895th pile segmentthen substituting (10) and (11) into (29) the variables 120572

1119895 1205722119895

are obtained and the vertical impedance function at the topend of the 119895th pile segment can be written as

119885119895(119904) =

119875119895(119904)

119906119895(119911 = 0 119904)

= minus119864119901119895(119860119901119895+ 120578119901119895

119904119905119888119860119901119895)

(12057211198951205722119895) [120574119895+ suminfin

119898=1

120585119898119895119869119898119895

cos (120593119898119895)] + [minus120574

119895+ suminfin

119898=1

120577119898119895119869119898119895

cos (120593119898119895)]

(12057211198951205722119895) [1 + sum

infin

119898=1

120585119898119895

sin (120593119898119895)] + [1 + sum

infin

119898=1

120577119898119895

sin (120593119898119895)]

(31)

where

1205721119895

1205722119895= minus

(1 + 120578119901119895

119904119905119888) [minus120574119895

119890minus120574119895 + sum

infin

119898=1 120577119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890minus120574119895 + sum

infin

119898=1 120577119898119895 sin (119869119898119895 + 120593119898119895)]

(1 + 120578119901119895

119904119905119888) [120574119895

119890120574119895 + sum

infin

119898=1 120585119898119895119869119898119895 cos (119869119898119895 + 120593119898119895)] + 119885119895minus1 [119890120574119895 + sum

infin

119898=1 120585119898119895 sin (119869119898119895 + 120593119898119895)]

119885119895minus1

=

119885119895minus1ℎ119895

119864119901119895119860119901119895

(32)

For the 1st pile segment1198850= (119896pb+119904119888pb)ℎ1(11986411990111198601199011) denotes

the dimensionless reaction coefficient at the pile baseThrough recursion impedance function from the 1st pile

segment to the 119873th pile segment the vertical impedancefunction at the pile head can be expressed as

119885119873(119904) =

119875119873(119904)

119906119873(119911 = 0 119904)

= minus119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)(12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)](12057211198731205722119873) [1 + sum

infin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)]

(33)

Letting 119904 = 119894120596 the frequency response function ofdisplacement (admittance function of displacement) at thepile head can be written as

119867119906(120596) =

1119885119873

= minus1

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)]

(34)

Then the response function of velocity (or admittancefunction of velocity) at the pile head can be written as

119867V (120596) = 119894120596119867119906 (120596) =1

120588119901119873119860119901119873119881119901119873

1198671015840

V (120596) (35)

where 1198671015840

V(120596) is the dimensionless response function ofvelocity at the pile head and

1198671015840

V (120596) = minus119894120596120588119901119873119860119901119873119881119901119873

119864119901119873

(119860119901119873

+ 120578119901119873

119904119905119888119860119901119873)

(12057211198731205722119873) [1 + suminfin

119898=1 120585119898119873 sin (120593119898119873)] + [1 + suminfin

119898=1 120577119898119873 sin (120593119898119873)](12057211198731205722119873) [120574119873 + sum

infin

119898=1 120585119898119873119869119898119873 cos (120593119898119873)] + [minus120574119873 + suminfin

119898=1 120577119898119873119869119898119873 cos (120593119898119873)] (36)

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

H

Weak concrete

(a) (b) (c)

FH

FL

WRBWRN

Figure 3 Geometry of typical defective piles (a) pile with a neck(b) pile with a bulb and (c) pile with a weak concrete

If the Fourier transform of the longitudinal exciting force119875(119905) acting on the pile top is denoted by 119875(120596) the velocityresponse of the pile top in the time domain can be calculatedby convolution theorem as

119881 (119905) = IFT [119875 (120596)119867 (120596)] (37)

When the force at the top of the pile is a half-sine pulse

119875 (119905) =

119876max sin(120587119905

1198790) 119905 lt 1198790

0 119905 ge 1198790(38)

where 1198790 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the excitingforce respectively Then the velocity response of the pile topin the time domain can be further expressed as

119881 (119905) =1

2120587int

+infin

minusinfin

119876max119881119901119873119860119901119873120588119901119873

119867V (120596)

sdot1205871198790

1205872 minus 1198792

0

1205962(1 + 119890

minus1198941205961198790) 1198901198941205961198790119889120596

(39)

To facilitate analysis it is useful to introduce the normal-ized velocity response

1198811015840

(119905) =119881 (119905)

max [119881 (119905)] (40)

where max[119881(119905)] denotes the maximum value of 119881(119905)

3 Problem Definition

The possible defects due to voids or inclusions in a pileafter construction can generally be categorized as neck bulband weak concrete The geometry of typical defective pilesis plotted in Figure 3 119882RN (neck width) and 119882RB (bulbwidth) denote the radius of the neck and bulb respectively119865119867and 119865

119871denote the embedment depth and length of the

defect respectively If a pile contains more than one defect(ie two defects) the geometry of defective pile can be thecombination of Figures 3(a) 3(b) and 3(c)

0000 0004 0008 0012

00

05

10

t (s)

RSTRS2RS1

Neck width (WRN) = 040mNeck width (WRN) = 030m

V998400 (t)

minus05

Figure 4 Variation of the normalized velocity response of the piletop with the neck width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

4 Results and Discussion

In the past the mechanical admittance method has receivedwide application in the field pile integrity test However themechanical admittancemethod is difficult to accurately iden-tify defects when a pile contains more than one flaw Further-more quick attenuation of the amplitude of admittance curvein the high frequency range caused by the pilematerial damp-ing can also make the location of the defect hard to be accu-rately detected As a result in previous few years the methodby virtue of time history of velocity response of the pile tophas then been widely used instead of the mechanical admit-tance method in China due to its simplicity of subsequentparameter analysis Accordingly the influence of geometricand physical characteristics of the defects and pile materialdamping on the velocity response of the pile top will be inves-tigated in the following section Unless otherwise specifiedthe pile properties employed in the following analysis are 119895 =1 2 119873 120588

119901119895= 2500 kgm3 119881

119901119895= 3790ms 120578

119901119895

= 00015119867 = 19m 119903

119895= 05m 1199050 = 15 times 10minus3 s and Poissonrsquos ratio

= 02 the soil properties are 120588119895= 1800 kgm3 119881

119904119895= 120ms

119863119904119895= 119863V119895 = 002 119896sb119895 = 119896st119895 = 001 and Poissonrsquos ratio = 03

41 Pile Containing a Single Defect

411 Effect of Width of Defect Figure 4 shows the influenceof the neck width (119882RN) on the velocity response of the piletop It can be seen from Figure 4 that the first reflective signalfrom the defect (RS1) arrives at 119905 = 119905RS1 = 000300 s (119905RS1is the time of arrival of the first reflective signal from thedefect) Accordingly the embedment depth of the defect canbe calculated as 119865

119867= 119881rod times 119905RS12 = 568m (119881rod is the

1D elastic longitudinal wave velocity in an intact pile error =53) The reflective signal from the pile tip (RST) arrives at119905 = 119905RST = 000967 s (119905RST is the time of arrival of the reflective

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RS1RSTRS2V

998400 (t)

Bulb width (WRB) = 060mBulb width (WRB) = 070m

t (s)

minus05

Figure 5 Variation of the normalized velocity response of the piletop with the bulb width (119873 = 3 ℎ

1

= 125m 119865119871

= ℎ2

= 05m119865119867

= ℎ1

= 60m)

signal from the pile tip) and the length of the pile119867 = 119881rod times119905RST2 = 1832m (error = 36) When the degree of defectis relatively great (119882RN = 03m) (namely the peak of RS1 isabove zero axis) the second reflective signal from the defect(RS2) appears and arrives at 119905 asymp 2119905RS1 (119905 = 000598 s) andthe amplitude of RS2 is much lower than that of RS1 becauseof quick dissipation of the energy It is worth noting that thissignal (RS2) can be used to further check the location of thedefect For instance the embedment of the defect can be cal-culated as119865

119867= 119881rodtimes119905RS24 = 567m(error = 55) It is also

observed from Figure 4 that the amplitude of RS1 increasesgreatly with the decrease of 119882RN This result indicates thatthe higher the amplitude of the reflective signal is the greaterthe degree of defect will be Furthermore for the pile with aneck the phase of RS1 is the same as that of the incident pulsedue to the abrupt decrease of the cross-sectional area

Figure 5 shows the influence of the bulb width (119882RB) onthe velocity response of the pile top It can be seen fromFigure 5 that RS1 and RST arrive at 119905 = 119905RS1 = 000296 sand 119905 = 119905RST = 000973 s respectively Therefore 119865

119867=

119881rodtimes119905RS12 = 561m (error = 65) and119867 = 119881rodtimes119905RST2 =1832m (error = 36) It is also observed from Figure 6 thatthe amplitude of the RS1 increases greatly with the increaseof 119882RB Moreover the phase of RS1 is the opposite to thatof the incident pulse due to the abrupt increase of the cross-sectional area However the phase of RS2 is the same as thatof the incident pulse which is similar to the reflective signalof the pile with a neck As a result RS2 is easily identified asa defect by mistake

Figure 6 shows the influence of the longitudinal wavevelocity in a weak concrete (119881

1199012) on the velocity response

of the pile top It can be seen from Figure 6 that the velocityresponse curve of the pile with a weak concrete is similar tothat of the pilewith a neckThephase of RS1 is the same as thatof the incident pulse due to the abrupt decrease of 119881

1199012and

0000 0004 0008 0012

00

05

10

RS1

RSTRS2

V998400 (t)

Vp2 = 2000msVp2 = 3000ms

t (s)

minus05

Figure 6 Variation of the normalized velocity response of the piletop with the longitudinal wave velocity in weak concrete (119873 = 3ℎ1

= 125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

0000 0004 0008 0012

00

05

10

RST

RS1

V998400 (t)

120578pj = 0

120578pj = 0003

120578pj = 0006

t (s)

minus05

Figure 7 Variation of the normalized velocity response of the piletop with material damping of the pile (119873 = 3119882RN = 030m ℎ

1

=

125m 119865119871

= ℎ2

= 05m 119865119867

= ℎ1

= 60m)

the amplitude of RS1 increases greatly with the decreaseof 1198811199012 Furthermore the time of arrival of RST increases

obviously with the decrease of 1198811199012 The reason for this result

is that RST will need much more time to arrive at the pile topas 1198811199012

decreases

412 Effect of Material Damping of the Pile Figure 7 showsthe influence of material damping of the pile on the velocityresponse of the pile top It can be seen from Figure 7 thatthe material damping of the pile has marked influence onthe velocity response of the pile top The amplitude of RS1and RST decreases markedly with the increasing material

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0000 0004 0008 0012

00

05

10

RS2RS1RSTV

998400 (t)

t (s)

Neck length (FL) = 05mNeck length (FL) = 20m

minus05

Figure 8 Variation of the normalized velocity response of the piletop with neck length (119873 = 3119882RN = 030m 119865

119867

= ℎ1

= 60m)

damping of the pile which results in the fact that the degree ofdefect seems to be smaller than the actual degree Moreoverthe width of RS1 and RST shows obvious increase with theincrease of pile material damping resulting in the marginalof the reflective signal becoming more and more ambiguousAs a result it is difficult to accurately obtain the time ofarrival of the signal reflected from the defect and determinethe location of the defect For instance RS1 arrives at 119905 =

119905RS1 = 000311 s and 119865119867= 119881rod times 119905RS12 = 589m (error =

18) when pile material damping is not considered (120578119895

=

0) However when pile material damping is considered (ie120578119895

= 0006) RS1 arrives at 119905 = 119905RS1 = 000265 s and 119865119867=

119881rod times 119905RS12 = 502m the error can be reached to 163Therefore it is needed to increase the calculated embedmentdepth of the defect when determining the location of thedefect in the field test

413 Effect of Length of Defect Figure 8 shows the influenceof the neck length on the velocity response of the piletop It can be seen from Figure 8 that the neck length hasmarked influence on the velocity response As the neck lengthincreases from 05m to 20m the width and amplitude ofRS1 and RS2 show marked increase Moreover the time ofarrival of RS2 and RST is difficult to accurately identify withthe increase of neck length

Figure 9 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 9that the bulb length has marked influence on the velocityresponse of the pile topWhen the bulb length increases from05m to 20m the width and amplitude of RS1 and RS2 showmarked increase and the time of arrival of RS2 is difficult toobtain However the increase of bulb length has negligibleinfluence on the amplitude and the time of arrival of RST

Figure 10 shows the influence of the bulb length on thevelocity response of the pile top It can be seen from Figure 10that the weak concrete length has marked influence on

0000 0004 0008 0012

00

05

10

RST

RS2

RS1

V998400 (t)

t (s)

Bulb length (FL) = 05mBulb length (FL) = 20m

minus05

Figure 9 Variation of the normalized velocity response of the piletop with blub length (119873 = 3119882RB = 070m 119865

119867

= ℎ1

= 60m)

0000 0005 0010 0015

00

05

10

RSTRS2RS1

V998400 (t)

t (s)

Weak concrete length (FL) = 05mWeak concrete length (FL) = 20m

minus05

Figure 10 Variation of the normalized velocity response of the piletop with weak concrete length (119873 = 3 119881

1198752

= 2000ms 119865119867

= ℎ1

=

60m)

the velocity response of the pile top As the weak concretelength increases from05m to 20m thewidth and amplitudeof RS1 and RS2 showmarked increase Furthermore the timeof arrival of RST increases with the increase of weak concretelength

414 Effect of Surrounding Soil Properties In some casesthe soil properties may change greatly in certain embedmentdepth In order to explicitly identify the influence of thevariation of the surrounding soil properties the pile isassumed to be homogeneous and four cases of soil propertiesare investigated Case 1 the shear velocity of soil layersdecreases along the depth direction (namely 119881

1= 60ms

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0000 0004 0008 0012

00

05

10

RST

Case 1Case 2

Reflective signal fromthe soil interfaceV

998400 (t)

t (s)

minus05

(a)

0000 0004 0008 0012

00

05

10

RSTReflective signal from

the soil interfaceV998400 (t)

t (s)

Case 3Case 4

minus05

(b)

Figure 11 Effect of interface of adjacent soil layers on the normalized velocity response of the pile (119873 = 3 ℎ1

= 120m ℎ2

= 10mℎ3

= 60m)

1198812= 100ms 119881

3= 140ms) Case 2 the shear velocity of

soil layers increases along the depth direction (namely 1198811=

140ms1198812= 100ms119881

3= 60ms) Case 3 the soil has a soft

interlayer (namely1198811= 120ms119881

2= 60ms119881

3= 140ms)

Case 4 the soil has a hard interlayer (namely 1198811= 120ms

1198812= 240ms 119881

3= 140ms) It can be seen from Figure 11(a)

that the velocity curve shows minor upward inclinationas the shear velocity of the soil gradually decreases alongthe depth direction However as the shear velocity of soilgradually increases along the depth direction the velocitycurve shows minor downward inclination Based on theseresults it can be concluded that the gradual variation of thesurrounding soil properties along the depth has negligibleinfluence on the identification of the defect in a pile Itcan be seen from Figure 11(b) that the abrupt variation ofthe surrounding soil properties along the depth has certaininfluence on the velocity response of the pile The phase ofthe reflective signal of the interface of adjacent soil layers isrespectively the same as and opposite to that of the incidentpulse for Case 3 and Case 4 The shape of the reflective signalof the soil with a soft or hard interlayer is in a half-sine formwhich is obviously different from that of the defective pileTherefore this result should be noted in the field test in caseof identifying this kind of signal as a defect by mistake

42 Pile Containing Multidefects In Section 41 the mech-anism of longitudinal wave propagating in a pile with asingle defect has received detailed investigation However inpractical engineering a pile may contain more than one flawalong the pile body It is noted that the reflected signals fromdifferent defects in the velocity curve may interact due to theexistence of the potential second reflected signal from thedefect Therefore the following section will take a pile withdouble defects as an example to illustrate the influence of the

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Overlapped signalRSTReflected signal

from bulb

V998400 (t)

t (s)

r2 = 040m r4 = 070mr2 = 040m r4 = 060m

minus05

Figure 12 Time history of the normalized velocity response of thepile with bulb near the pile top and neck near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal reflected from one defect near the pile top on the signalreflected from the other defect near the pile tip

Figure 12 shows the time history of the velocity responseof the pile with a bulb near the pile top and a neck near thepile tip It can be seen fromFigure 12 that the characteristics ofthe reflected signals from bulb and neck are obvious for 119903

4=

060mwhen considering the relative amplitude of the signalsHowever the time of arrival of the reflected signal from theneck (119905 asymp 000300 s) is very close to that of the secondreflected signal from the bulb (119905 asymp 000591 s) for 119903

4= 070m

accordingly the neck near the pile tip is easy to be identifiedas the second reflected signal from the bulb (referred to inFigure 9)The reason for this result is that the second reflected

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0000 0004 0008 0012

00

05

10

Reflected signal from neck

Reflected signal from neck

Overlapped signal

RSTV998400 (t)

t (s)

r2 = 060m r4 = 040mr2 = 060m r4 = 030m

minus05

Figure 13 Time history of the normalized velocity response of thepile with neck near the pile top and bulb near the pile tip (119873 = 5ℎ1

= 70m ℎ2

= 05m ℎ3

= 45m ℎ4

= 10m ℎ5

= 60m)

signal from the bulb and reflected signal from the neckoverlap in the time domain Therefore if the amplitude andwidth of the first reflected signal from the bulb are great andthe second reflected signal is obvious other testing methodsare suggested to further check the integrity of the pile

Figure 13 shows the time history of the velocity responseof the pile with a neck near the pile top and a bulb near the piletip It can be seen fromFigure 13 that the characteristics of thereflected signals from the neck and bulb are obvious for 119903

4=

040mHowever the reflected signal from the neck cannot beidentified for 119903

4= 030mThe reason for this result is that the

reflected signal from the bulb is overlapped and covered up bythe second reflected signal from the neck It is worth notingthat this case will not influence the result of pile integrity testbecause of the existence of the neck near the pile top

5 Application in Engineering

Figure 14 shows the comparison of the fitted theoreticalcurves and the measured curve from the model pile The pileis an embedded prefabricated nonuniform square pile witha bulb near the pile top and a neck near the pile tip whichwas installed at a site in ChinaThe length and cross-sectionaldimension of the pile are 10m and 400mm times 400mmrespectively Field measurements recorded the propagationtime of the impulse traveling from the pile top to pile tipto pile top as 46ms 1D elastic longitudinal wave velocityof the pile is estimated to be 43478ms The surroundingsoil is soft plastic viscoelastic backfill soil with mass densityof 1700 kgm3 and shear wave velocity of 90ms For thepurpose of analysis the square pile is considered as anequivalent circular pile with radius 119903 = 2255mm Thetheoretical velocity curve has been derived through adjustingthe radius depths andmaterial damping of the pile segmentsas well as the duration of impulse to give a good fit tothe measured curve The fitted curves require the following

0000 0002 0004 0006

00

05

10

RST

Reflected signal from neck

Reflected signalfrom bulb

Measured curve

V998400 (t)

t (s)

minus05

Fitted curve 1 cpb = 442 lowast 104 N middot smFitted curve 2 cpb = 353 lowast 105 N middot sm

Figure 14 Comparison of the fitted theoretical curves andmeasuredcurve from the model pile

input (1) the embedment of the bulb is 49m and the cross-sectional area of the bulb to the normal shaft area is 210(2) the embedment of the neck is 720m and the cross-sectional area of the neck decreased to a ratio of 54 ofthe normal shaft area (3) the impulse width of the excitingforce is 059ms and the dimensionless damping coefficientof the pile 120578

119901119895

= 00015 By comparison the actual conditionof the pile is listed as follows at the depth of 48m theenlarged area ratio is 200 at the depth of 716m the cross-sectional area decreased to a ratio of 55 Moreover it isnoted fromFigure 14 that fitted curves 1 and 2matchwell withthe measured curve when 119905 lt 00045 s When 119905 gt 00045 s inthe case of the damping coefficient at the pile toe 119888pb taken interms of (14) the amplitude of RST in fitted curve 1 is muchhigher than that in measured curve However RST in fittedcurve 2 matches well with that in the measured curve when119888pb is taken as a relatively great valueThe reason for this resultis that the amplitude of RST decreases with the increase of soildamping at the pile toe and the corresponding soil dampingmay be underestimated by virtue of (14) for the square pile

6 Conclusions

(1) For a pile with a single neck the phase of RS1 andRS2 is the same as that of the incident pulse Theamplitude of the signals reflected from the neckincreases markedly with the decrease of the neckwidth

(2) For a pile with a single bulb the phase of RS1 isthe opposite to that of the incident pulse Howeverthe phase of RS2 is the same as that of the incidentpulse which is easily identified as a defect by mistakeThe amplitude of the signals reflected from the bulbincreases greatly with the increase of bulb width

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

(3) For a pile with a single weak concrete the phase of RS1and RS2 is the same as that of the incident pulse dueto the abrupt decrease of119881

1199012and the amplitude of RS1

increases greatly with the decrease of1198811199012 The time of

arrival of RST increases obviously with the decreaseof 1198811199012

(4) The amplitude andwidth of RS1 andRST respectivelydecrease and increase with the increase of pile mate-rial damping which makes the degree of the defectseem to be smaller than the actual degree and thetime of arrival of the signal reflected from the defectdifficult to accurately identify

(5) As the length of single defect (such as neck bulb orweak concrete) increases the width and amplitude ofRS1 and RS2 from the defect show marked increaseFor a pile containing a neck it is noted that the time ofarrival of RST is difficult to identify with the increaseof neck length

(6) The gradual variation of the surrounding soil prop-erties along the depth has negligible influence onidentification of the defect in a pileThe abrupt changein adjacent soil layer properties has certain influenceon the velocity response and this result should benoted in the field test in case of identifying this kindof signal as a defect by mistake

(7) For a pile with a bulb near the pile top and a necknear the pile tip when the amplitude and widthof the first reflected signal from the bulb are greatand the second reflected signal is obvious othertesting methods are suggested to further check thepile integrity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 51378464) the NationalScience Foundation for Post-Doctoral Scientists of China(Grant no 2013M541544) and the Shanghai PostdoctoralSustentation Fund China (Grant no 13R21416200)

References

[1] MNovak TNogami andFAboul-Ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[2] G Militano and R K N D Rajapakse ldquoDynamic response ofa pile in a multi-layered soil to transient torsional and axialloadingrdquo Geotechnique vol 49 no 1 pp 91ndash109 1999

[3] K H Wang D Y Yang Z Q Zhang and C J Leo ldquoA newapproach for vertical impedance in radially inhomogeneous

soil layerrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 36 no 6 pp 697ndash707 2012

[4] S H Lu K H Wang W B Wu and C J Leo ldquoLongitudinalvibration of a pile embedded in layered soil considering thetransverse inertia effect of pilerdquoComputers andGeotechnics vol62 pp 90ndash99 2014

[5] W Wu G Jiang S Huang and C J Leo ldquoVertical dynamicresponse of pile embedded in layered transversely isotropicsoilrdquo Mathematical Problems in Engineering vol 2014 ArticleID 126916 12 pages 2014

[6] D Y Yang K H Wang Z Q Zhang and C J Leo ldquoVerticaldynamic response of pile in a radially heterogeneous soil layerrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 33 no 8 pp 1039ndash1054 2009

[7] W M Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineering and Structural Dynamics vol 23 no 11 pp 1239ndash1257 1994

[8] A G Davis and C S Dunn ldquoFrom theory to field experiencewith the non-destructive vibration testing of pilesrdquo Proceedingsof the Institution of Civil Engineers Part 2 vol 57 no 4 pp 571ndash593 1974

[9] A G Davis and S A Robertson ldquoVibration testing of pilesrdquoStructural Engineer vol 54 no 6 pp A7ndashA10 1976

[10] J S Higgs ldquoIntegrity testing of concrete piles by shockmethodrdquoConcrete vol 13 no 10 pp 31ndash33 1979

[11] Y Lin M Sansalone and N J Carino ldquoImpact-echo responseof concrete shaftsrdquo Geotechnical Testing Journal vol 14 no 2pp 121ndash137 1991

[12] J N Watson P S Addison and A Sibbald ldquoThe de-noising ofsonic echo test data through wavelet transform reconstructionrdquoShock and Vibration vol 6 no 5-6 pp 267ndash272 1999

[13] S T Liao and J M Roesset ldquoDynamic response of intactpiles to impulse loadsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 21 no 4 pp 255ndash2751997

[14] S T Liao and J M Roesset ldquoIdentification of defects in pilesthrough dynamic testingrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 21 no 4 pp 277ndash291 1997

[15] Y K Chow K K Phoon W F Chow and K Y WongldquoLow strain integrity testing of piles three-dimensional effectsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1057ndash1062 2003

[16] H Y Chai K K Phoon and D J Zhang ldquoEffects of thesource on wave propagation in pile integrity testingrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 136 no 9pp 1200ndash1208 2010

[17] Z T Lu Z LWang andD J Liu ldquoStudy on low-strain integritytesting of pipe-pile using the elastodynamic finite integrationtechniquerdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 5 pp 536ndash550 2013

[18] S-H Ni L Lehmann J-J Charng and K-F Lo ldquoLow-strainintegrity testing of drilled piles with high slenderness ratiordquoComputers and Geotechnics vol 33 no 6-7 pp 283ndash293 2006

[19] K H Wang W B Wu Z Q Zhang and C J Leo ldquoVerticaldynamic response of an inhomogeneous viscoelastic pilerdquoComputers and Geotechnics vol 37 no 4 pp 536ndash544 2010

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[20] T Nogami and M Novak ldquoSoil-pile interaction in verticalvibrationrdquo Earthquake Engineering and Structural Dynamicsvol 4 no 3 pp 277ndash293 1976

[21] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadingrdquo Journal of the Soil Mechanics and FoundationsDivision vol 92 no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of