Research ArticleVertical Dynamic Response of Pile Embedded in LayeredTransversely Isotropic Soil

Wenbing Wu1 Guosheng Jiang1 Shenggen Huang1 and Chin Jian Leo2

1 Engineering Faculty China University of Geosciences Wuhan Hubei 430074 China2 School of Computing Engineering and Mathematics University of Western Sydney Locked Bag 1797 PenrithSydney NSW 2751 Australia

Correspondence should be addressed to Wenbing Wu zjuwwb1126163com and Shenggen Huang huangshgr163com

Received 15 April 2014 Revised 24 June 2014 Accepted 24 June 2014 Published 17 July 2014

Academic Editor Sarp Adali

Copyright copy 2014 Wenbing Wu et al This 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

The dynamic response of pile embedded in layered transversely isotropic soil and subjected to arbitrary vertical harmonic force isinvestigated Based on the viscoelastic constitutive relations for a transversely isotropicmedium the dynamic governing equation ofthe transversely isotropic soil is obtained in cylindrical coordinates By introducing the fictitious soil pile model and the distributedVoigt model the governing equations of soil-pile system are also derived Firstly the vertical response of the soil layer is solvedby using the Laplace transform technique and the separation of variables technique Secondly the analytical solution of velocityresponse in the frequency domain and its corresponding semianalytical solution of velocity response in the time domain are derivedby means of inverse Fourier transform and convolution theorem Finally based on the obtained solutions a parametric study hasbeen conducted to investigate the influence of the soil anisotropy on the vertical dynamic response of pile It can be seen that theinfluence of the shear modulus of soil in the vertical plane on the dynamic response of pile is more notable than the influence ofthe shear modulus of soil in the horizontal plane on the dynamic response of pile

1 Introduction

The dynamic interaction of soil-pile system is a complicatedcontact problem owing to the complexity of soil layerTherefore many investigators have paid their attention todevelop theoretical models to analyze the soil layer forthe dynamic interaction of soil-pile system Many of thesevarious theoretical models fall into four main categories(1) The dynamic Winkler model [1ndash7] which represents thesoil as a series of distributed springs attached to the pile is thefirst category The stiffness of the soil ldquospringsrdquo is calculatedby assuming that wave energy can only propagate outwardsunder plane-strain conditions [8] (2)The plane-strainmodel[9ndash16] which assumes that the soil is made up of an infinitenumber of infinitesimally thin independent horizontalelastic layers that extend to infinity is the second categoryIt can be noted that the plane-strain model neglects thestrain in the vertical direction and the waves propagate onlyin the horizontal direction (3) Various three-dimensional

axisymmetric continuum models [17ndash22] in which the soillayer is three-dimensional and the wave effect of soil isconsidered is the third category Due to the complexity ofthe three-dimensional axisymmetric continuummodels it isvery difficult to promote the use of these models in layeredsoil (4)The radially inhomogeneousmodel [23ndash26] inwhichthe soil region is assumed as a linear viscoelastic mediumcomposed of two concentric regions an inner annular regionof disturbedmedium and an outer semi-infinite undisturbedregion is the fourth category It can be seen that the modeldoes not simulate the interaction of soil zones rigorously andit is only an approximate simplified model that may lead tothe argument whether the results adequately reflect the realinteraction of the soil zones [27]

Most of the previous investigations on the dynamicinteraction of soil-pile system had treated soil as an isotropicmedium However in practice soil deposits possess a cer-tain degree of anisotropy owing to their deposition historyresulting in properties that are different in the horizontal

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 126916 12 pageshttpdxdoiorg1011552014126916

2 Mathematical Problems in Engineering

and vertical directionThe vertical and horizontal differencescan be simulated by virtue of a transversely isotropic soilmodel [28] which a simple isotropic model cannot allowfor Tsai [29] studied the torsional dynamic response of acircular disk on a transversely isotropic half-space by utilizingintegral transform technique It is worth noting that theanisotropic material constants have obvious influence onthe resonant amplitude and frequency of vibration Liu andNovak [30] investigated the dynamic response of single pileembedded in transversely isotropic layered media by meansof FEM combining with dynamic stiffness matrices of thesoil derived from Greenrsquos functions for ring loads Chenet al [31] proposed a model of transversely isotropic layeredelastic media to study the dynamic horizontal impedancesof double piles embedded in the ground and subjected toharmonic sway-rocking loadings at the head of the two pilesChen et al [32] studied the transient torsional dynamicresponse of a pile embedded in transversely isotropic sat-urated soil by utilizing the Laplace transform Wang et al[33] investigated the dynamic response of an end bearingpile embedded in transversely isotropic saturated soil whenthere was a time-harmonic torsional loading acting on thepile head Notwithstanding the work above it can be seenthat no investigation corresponding to the vertical dynamicresponse of a pile embedded in layered transversely isotropicelastic soil has been reported in existing literatures until now

Based on the above literature review the main objectiveof this paper is to develop an extended soil-pile interactionmodel to simulate the vertical dynamic response of pileembedded in layered transversely isotropic soil Utilizing thefictitious soil pile model to simulate the dynamic interactionbetween the pile and pile end soil [34] the governingequations of soil-pile system are established when the verticalwave effect of surrounding soil is taken into account Thenthe analytical solution of velocity response in the frequencydomain and its corresponding semianalytical solution ofvelocity response in the time domain are derived by meansof integral transform technique and separation of variablestechnique Based on these solutions a parametric study hasbeen conducted to assess the influence of the soil anisotropyon the vibration behavior of the pile

2 Governing Equations

21 Computational Model and Assumptions The problemstudied in this paper is the vertical vibration of viscoelasticpile embedded in layered transversely isotropic soil and thegeometric model is shown in Figure 1 Based on the fictitioussoil pile model the soil-pile system is discretized into atotal of 119898 segments along the vertical direction which arenumbered by 1 2 119895 119898 The thickness of the 119895th (1 le

119895 le 119898) soil-pile segment is denoted by 119897119895 and the depth ofthe 119895th (1 le 119895 le 119898) soil layer top is denoted by ℎ119895 If thenumbers of the soil-pile segments are enough the propertiesof soil and pile within each segment can be assumed to behomogeneous respectively The pile length is denoted by119867119901and the thickness of pile end soil is denoted by119867119904 119902(119905) is anarbitrary vertical harmonic force acting on the pile head

q(t)

Layer m m

hj

Hp lj

Hs Layer 2

Layer 1 1

2

j

o

z

Layer j

Bedrock

PileTransversely

isotropicviscoelastic soil

Fictitious soil pile

r

ksj 120575sj

ksjminus1 120575sjminus1

Figure 1 Schematic of pile-soil interaction model

The following assumptions are adopted during the analy-sis

(1) The surrounding soil of pile is layered transverselyisotropic and viscoelastic The damping force of soilis proportional to the strain rate and the proportionalcoefficient of the 119895th soil layer is denoted by 120578119904119895

(2) The top surface of soil layer is assumed to be freeboundary without normal and shear stresses and thebottom surface of pile end soil is assumed to be rigidboundary without displacements

(3) The dynamic interaction of the adjacent soil layersis simulated by using the distributed Voigt modelThe spring constant and damping coefficient of thedistributedVoigtmodel between the 119895th soil layer andits upper adjacent soil layer are denoted by 119896

119904119895 and

120575119904119895 and the corresponding values between the 119895th soillayer and its lower adjacent soil layer are denoted by119896119904119895minus1 and 120575

119904119895minus1 respectively

(4) During vibration the vertical wave effect of surround-ing soil is taken into account but the radial wave effectof surrounding soil is ignored

(5) Both the pile and the fictitious soil pile are verticalviscoelastic and circular in cross-section and havea perfect contact with the surrounding soil duringvibration

(6) During vibration the soil-pile system is subjected tosmall deformations and strains and the conditionsof displacement continuity and force equilibrium aresatisfied at the interface of the adjacent pile (includingfictitious soil pile) segments

22 Governing Equations of Soil-Pile System

221 Dynamic Equation of Soil Combining with the vis-coelastic constitutive relations for a transversely isotropic

Mathematical Problems in Engineering 3

medium proposed by Ding [35] the dynamic equilibriumequation of the transversely isotropic soil in cylindricalcoordinates can be derived as follows

[11986211 (1205972

1205971199032+

1

119903

120597

120597119903

minus

1

1199032) + 11986266

1

1199032

1205972

1205971205792+ 11986244

1205972

1205971199112] 119906119903

+ [

11986211 minus 11986266

119903

1205972

120597119903 120597120579

minus

11986211 + 11986266

1199032

120597

120597120579

] 119906120579

+ (11986213 + 11986244)1205972119908

120597119903 120597119911

= 120588119904 1205972119906119903

1205971199052

[(11986211 minus 11986266)1

119903

1205972

120597119903 120597120579

+ (11986211 + 11986266)1

1199032

120597

120597120579

] 119906119903

+ (11986213 + 11986244)1

119903

1205972119908

120597120579 120597119911

+ [11986266 (1205972

1205971199032+

1

119903

120597

120597119903

minus

1

1199032) + 11986211

1

1199032

1205972

1205971205792+ 11986244

1205972

1205971199112] 119906120579

= 120588119904 1205972119906120579

1205971199052

(11986213 + 11986244) (1205972

120597119903 120597119911

+

1

119903

120597

120597119911

)119906119903 + (11986213 + 11986244)1

119903

1205972119906120579

120597120579 120597119911

+ [11986244 (1205972

1205971199032+

1

119903

120597

120597119903

+

1

1199032

1205972

1205971205792) + 11986233

1205972

1205971199112]119908

= 120588119904 1205972119908

1205971199052

(1)

where 119906119903 119906120579 and 119908 denote the radial displacement circum-ferential displacement and vertical displacement respec-tively The coefficients of the above equations should satisfythe following equations

11986211 = 11986222 =119864119904ℎ (1 minus 120583

119904ℎV120583119904Vℎ)

(1 + 120583119904ℎℎ) (1 minus 120583

119904ℎℎminus 2120583119904ℎV120583119904Vℎ)

11986212 =119864119904ℎ (120583119904ℎℎ + 120583

119904ℎV120583119904Vℎ)

(1 + 120583119904ℎℎ) (1 minus 120583

119904ℎℎminus 2120583119904ℎV120583119904Vℎ)

11986213 = 11986223 =119864119904ℎ120583119904ℎℎ

1 minus 120583119904ℎℎminus 2120583119904ℎV120583119904Vℎ

11986233 =119864V (1 minus 120583

119904ℎℎ)

1 minus 120583119904ℎℎminus 2120583119904ℎV120583119904Vℎ

11986244 = 11986255 = 119866119904V

11986266 =(11986211 minus 11986222)

2

(2)

where 119864119904ℎ and 119864

119904V denote the horizontal and vertical elastic

modulus respectively 119866119904V denotes the shear modulus in thevertical plane 120583119904ℎV is the Poissonrsquos ration in the vertical

direction caused by the horizontal stress and 120583119904Vℎ is the

Poissonrsquos ratio in the horizontal direction caused by thevertical stress and 120583119904ℎV and 120583

119904Vℎ should satisfy 120583

119904Vℎ119864119904V = 120583119904ℎV119864119904ℎ

120583119904ℎℎ is the Poissonrsquos ratio in the orthogonal direction of the

horizontal strain caused by the horizontal stressOwing to the assumption that only the vertical wave

effect of surrounding soil is taken into consideration theequilibrium equation of the transversely isotropic soil for theaxisymmetric problem can be further rewritten as follows

11986233

1205972119908

1205971199112+ 11986244 [

1205972119908

1205971199032+

1

119903

120597119908

120597119903

] = 120588119904 1205972119908

1205971199052 (3)

Based on (3) and taking into account the viscosity of soilthe governing equation of the 119895th transversely isotropic soillayer for the axisymmetric problem can be established as

120575119904119895

1205972119908119895

1205971199112+

120578119904119895

119866119904V119895

120597

120597119905

(

1205972119908119895

1205971199112) + (

1

119903

120597119908119895

120597119903

+

1205972119908119895

1205971199032)

+

120578119904119895

119866119904V119895

120597

120597119905

(

1

119903

120597119908119895

120597119903

+

1205972119908119895

1205971199032) =

120588119904119895

119866119904V119895

1205972119908119895

1205971199052

(4)

where 119908119895 = 119908119895(119903 119911 119905) is the vertical displacement of the119895th soil layer 120578119904119895 120588

119904119895 and 119866

119904V119895 denote the viscous damping

coefficient density and shear modulus in the vertical planeof the 119895th soil layer 120575119904119895 = 2(1 minus 120583

119904ℎℎ119895)(1 + 120583

119904Vℎ119895)(1 minus 120583

119904ℎℎ119895 minus

2(120583119904Vℎ119895)2120581119904119895) 120581119904119895 = (1 + 120583

119904ℎℎ119895)119866119904ℎ119895(1 + 120583

119904Vℎ119895)119866119904V119895 denotes the

ratio of the elastic modulus of the 119895th soil layer in thehorizontal direction to that in the vertical direction 119866119904ℎ119895120583119904Vℎ119895 and 120583

119904ℎℎ119895 denote the shear modulus in the horizontal

plane the Poissonrsquos ratio in the horizontal direction causedby the vertical stress and the Poissonrsquos ratio in the orthogonaldirection of the horizontal strain caused by the horizontalstress of the 119895th soil layer respectively

222 Dynamic Equation of Pile Denoting 119906119895 = 119906119895(119911 119905) to bethe vertical displacement of the 119895th pile (including fictitioussoil pile) segment and according to the Euler-Bernoulli rodtheory the dynamic equilibrium equation of pile can beestablished as

119864119901

119895119860119901

119895

1205972119906119895

1205971199112+ 119860119901

119895120578119901

119895

1205973119906119895

120597119905 1205971199112minus 119898119901

119895

1205972119906119895

1205971199052

minus 2120587119903119901

119895 120591119904119903119911119895 (119903119901

119895 119911 119905) = 0

(5)

where 119864119901119895 = 120588119901

119895 (119881119901

119895 )2 119860119901119895 = 120587(119903

119901

119895 )2 119903119901119895 119898

119901

119895 120588119901

119895 119881119901

119895 and 120578119901

119895

denote the elastic modulus cross-section area radius massper unit length of pile (including fictitious soil pile) densityelastic longitudinal wave velocity and viscous damping coef-ficient of the 119895th pile (including fictitious soil pile) segmentrespectively 120591119904119903119911119895(119903

119901

119895 119911 119905) is the frictional force of the 119895th soillayer acting on the surface of the 119895th pile shaft and can beexpressed as

120591119904119903119911119895 (119903119901

119895 119911 119905) = 119866119904V119895

120597119908119895 (119903119901

119895 119911 119905)

120597119903

+ 120578119904119895

1205972119908119895 (119903119901

119895 119911 119905)

120597119905 120597119903

(6)

4 Mathematical Problems in Engineering

Combining with the assumptions the boundary andinitial conditions of soil-pile system can be established asfollows

(1) Boundary Conditions of Soil At the top surface of the 119895thsoil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= (119896119904119895119908119895 + 120575

119904119895

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(7a)

At the bottom surface of the 119895th soil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus(119896119904119895minus1119908119895 + 120575

119904119895minus1

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(7b)

At an infinite radial distance of the 119895th soil layer

120590119895 (infin 119911) = 0 119908119895 (infin 119911) = 0 (7c)

(2) Boundary Conditions of Pile At the top surface of the 119895thpile (including fictitious soil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= minus

119885119895 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(8a)

At the bottom surface of the 119895th pile (including fictitioussoil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus

119885119895minus1 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(8b)

where 119885119895(119904) and 119885119895minus1(119904) denote the displacement impedancefunction at the top and bottom surfaces of the 119895th pile(including fictitious soil pile) segment 119904 is the Laplacetransform parameter

(3) Boundary C at the Interface of Soil-Pile System

119908(119903119901

119895 119911 119905) = 119906119895 (119911 119905) (9)

(4) Initial Conditions of Soil-Pile System Initial conditions ofthe 119895th soil layer are as follows

119908119895

10038161003816100381610038161003816119905=0

= 0

120597119908119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10a)

Initial conditions of the 119895th pile (including fictitious soilpile) segment are as follows

119906119895

10038161003816100381610038161003816119905=0

= 0

120597119906119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10b)

3 Solutions of the Governing Equations

31 Vibrations of the Soil Layer Denoting 119882119895(119903 119911 119904) =

int

+infin

0119908119895(119903 119911 119905)119890

minus119904119905d119905 to be the Laplace transform with respectto time of119908119895(119903 119911 119905) and associating with the initial condition(10a) (4) is transformed by using the Laplace transformtechnique as follows

(120575119904119895 +

120578119904119895 sdot 119904

119866119904V119895

)

1205972119882119895

1205971199112

+ (1 +

120578119904119895 sdot 119904

119866119904V119895

)(

1

119903

120597119882119895

120597119903

+

1205972119882119895

1205971199032)

= (

119904

119881119904V119895)

2

119882119895

(11)

where119881119904V119895 = radic119866119904V119895120588119904119895 is the shear wave velocity of the 119895th soil

layer in the vertical directionBy virtue of the separation of variables technique and

denoting 119882119895(119903 119911 119904) = 119877119895(119903 119904)119885119895(119911 119904) (11) can be decoupledas follows

d2119877119895 (119903 119904)d1199032

+

1

119903

d119877119895 (119903 119904)d119903

minus 1205852119895119877119895 (119903 119904) = 0

(12)

d2119885119895 (119911 119904)d1199112

+ 1205732119895119885119895 (119911 119904) = 0

(13)

where constants 120585119895 and 120573119895must satisfy the following equation

1205852119895 =

(120575119904119895 + 120578119904119895 sdot 119904119866

119904V119895) 1205732119895 + (119904119881

119904V119895)2

(1 + 120578119904119895 sdot 119904119866

119904V119895)

(14)

It can be seen that (12) is Bessel equation and (13) isordinary differential equation of second order whose generalsolutions can be easily obtained Associating with thesegeneral solutions the vertical displacement of the 119895th soillayer119882119895(119903 119911 119904) can be derived as

119882119895 (119903 119911 119904) = [1198601198951198700 (120585119895119903) + 1198611198951198680 (120585119895119903)]

times [119862119895 sin (120573119895119911) + 119863119895 cos (120573119895119911)] (15)

where 1198680(sdot) and 1198700(sdot) denote the modified Bessel functionsof order zero of the first and second kind respectively 119860119895119861119895 119862119895 and 119863119895 are constants determined by the boundaryconditions

Converting 119911 = ℎ119895 and 119911 = ℎ119895+119897119895 in the global coordinatesinto 119911

1015840= 0 and 119911

1015840= 119897119895 in the local coordinates (7a)

Mathematical Problems in Engineering 5

(7b) and (7c) are transformed by using the Laplace transformtechnique and can be rewritten as follows

[

[

(119896119904119895 + 120575119904119895 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) minus

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=0

= 0 (16a)

[

[

(119896119904119895minus1 + 120575

119904119895minus1 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) +

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=119897119895

= 0

(16b)

120590119895 (infin 1199111015840) = 0 119882119895 (infin 119911

1015840) = 0 (16c)

According to the properties of the modified Bessel func-tions that is if 119903 rarr infin then 119868119899(sdot) rarr infin and 119870119899(sdot) rarr 0it can be obtained that 119861119895 = 0 from boundary conditions(16c) By means of boundary conditions (16a) and (16b) thefollowing equation can be obtained

tan (120573119895119897119895)

= (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895 +

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895)120573119895119897119895

times ((120573119895119897119895)2minus (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895)(

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895))

minus1

=

(119870119895 + 119870

1015840

119895) 120573119895119897119895

(120573119895119897119895)2minus 119870119895119870

1015840

119895

(17)

where 119870119895 = ((119896119904119895 + 120575

119904119895 sdot 119904)119864

119904V119895)119897119895 and 119870

1015840

119895 = ((119896119904119895minus1 +

120575119904119895minus1 sdot 119904)119864

119904V119895)119897119895 denote the dimensionless complex stiffness

of the upper surface and lower surface of the 119895th soil layerSubstituting 119904 = 119894120596 into (17) and solving it by using bisectionmethod in the frequency domain a series of eigenvalues 120573119895119899can be derived 120585119895119899 can also be derived by substituting120573119895119899 into(14)

Then the vertical displacement of the 119895th soil layer can berewritten as

119882119895 (119903 1199111015840 119904) =

infin

sum

119899=1

1198601198951198991198700 (120585119895119899119903) sin (1205731198951198991199111015840+ 120601119895119899) (18)

where120601119895119899 = arctan(120573119895119899119897119895119870119895) and119860119895119899 are a series of constantsdetermined by the boundary conditions which reflect thedynamic interaction of soil and pile

32 Vibrations of the Pile Denoting119880119895(119911 119904) to be the Laplacetransform with respect to time of 119906119895(119911 119905) (5) is transformed

by using the Laplace transform technique and can be rewrit-ten combining with (6) and (18)

(119881119901

119895 )2(1 +

120578119901

119895

119864119901

119895

sdot 119904)

1205972119880119895

12059711991110158402

minus 1199042119880119895

minus

2120587119903119901

119895

120588119901

119895 119860119901

119895

(119866119904V119895 + 120578119904V119895 sdot 119904)

infin

sum

119899=1

1198601198951198991205851198951198991198701 (120585119895119899119903119901

119895 )

times sin (1205731198951198991199111015840+ 120601119895119899) = 0

(19)

It is not difficult to obtain that the general solution of (19)can be expressed as

119880119895 = 119872119895 [cos(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

+ 119873119895 [sin(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

(20)

where

1205941015840119895119899 = 120594119895119899

[

[

cos (120573119895119899 + 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 + 120582119895

+

cos (120573119895119899 minus 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 minus 120582119895

]

]

12059410158401015840119895119899 = 120594119895119899

[

[

sin (120573119895119899 + 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 + 120582119895

minus

sin (120573119895119899 minus 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 minus 120582119895

]

]

120594119895119899 =

(119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 119897119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895] 120593119895119899119871119895119899

120593119895119899 = 1198700 (120585119895119899119903119901

119895 ) +

2 (119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 1198972119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895]

119871119895119899 = int

119897119895

0

sin2 (1205731198951198991199111015840+ 120601119895119899) d119911

1015840

(21)

where 120582119895 = radicminus11990421199052119895(1 + (120578

119901

119895 119864119901

119895 ) sdot 119904) 120573119895119899 = 120573119895119899119897119895 120585119895119899 = 120585119895119899119897119895and 119903119901

119895 = 119903119901

119895 119897119895 are all dimensionless parameters 119905119895 = 119897119895119881119901

119895

denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment1198701(sdot) is the modified Bessel functions oforder one of the second kind

6 Mathematical Problems in Engineering

Combining with the boundary conditions (8a) and (8b)the displacement impedance function at the head of the 119895thpile segment can be derived in the local coordinates as follows

119885119895 (119904) =

minus119864119901

119895119860119901

119895 (1205971198801198951205971199111015840)

100381610038161003816100381610038161199111015840=0

119880119895

100381610038161003816100381610038161199111015840=0

= minus

119864119901

119895119860119901

119895

119897119895

times (

119872119895

119873119895

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos120601119895119899

+ 120582119895 +

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos120601119895119899)

times (

119872119895

119873119895

(1 +

infin

sum

119899=1

1205941015840119895119899 sin120601119895119899)

+

infin

sum

119899=1

12059410158401015840119895119899 sin120601119895119899)

minus1

(22)

where

119872119895

119873119895

= (

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) + 120582119895 cos 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[sin 120582119895 +infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899 + 120601119895119899)])

times (

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) minus 120582119895 sin 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[cos 120582119895 +infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899 + 120601119895119899)])

minus1

(23)

where 119885119895minus1(119904) denotes the displacement impedance functionat the head of the (119895 minus 1)th pile segmentwhich can be obtainedby using boundary conditionsThen following themethod ofrecursion typically used in the transfer function techniquethe displacement impedance function at the head of pile canbe derived as

119885119898 (119904) =

minus119864119901119898119860119901119898 (120597119880119898120597119911

1015840)

100381610038161003816100381610038161199111015840=0

11988011989810038161003816100381610038161199111015840=0

= minus

119864119901119898119860119901119898

119897119898

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

minus1

(24)

where

119872119898

119873119898

= minus(

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) + 120582119898 cos 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[sin 120582119898 +infin

sum

119899=1

12059410158401015840119898119899 sin (120573119898119899 + 120601119898119899)])

times (

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) minus 120582119898 sin 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[cos 120582119898 +infin

sum

119899=1

1205941015840119898119899 sin (120573119898119899 + 120601119898119899)])

minus1

1205941015840119898119899 = 120594119898119899 [

cos (120573119898119899 + 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 + 120582119898

+

cos (120573119898119899 minus 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 minus 120582119898

]

12059410158401015840119898119899 = 120594119898119899 [

sin (120573119898119899 + 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 + 120582119898

minus

sin (120573119898119899 minus 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 minus 120582119898

]

120594119898119899 =

(119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

120588119901119898119897119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898] 120593119898119899119871119898119899

120593119898119899 = 1198700 (120585119898119899119903119901119898)

+

2 (119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

1205881199011198981198972119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898]

119871119898119899 = int

119897119898

0

sin2 (1205731198981198991199111015840+ 120601119898119899) d119911

1015840

(25)

where 120582119898 = radicminus11990421199052119898(1 + (120578

119901119898119864119901119898) sdot 119904) 120573119898119899 = 120573119898119899119897119898 120585119898119899 =

120585119898119899119897119898 and 119903119901119898 = 119903119901119898119897119898 are all dimensionless parameters 119905119898 =

119897119898119881119901119898 denotes the propagation time of elastic longitudinal

Mathematical Problems in Engineering 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

2 Mathematical Problems in Engineering

and vertical directionThe vertical and horizontal differencescan be simulated by virtue of a transversely isotropic soilmodel [28] which a simple isotropic model cannot allowfor Tsai [29] studied the torsional dynamic response of acircular disk on a transversely isotropic half-space by utilizingintegral transform technique It is worth noting that theanisotropic material constants have obvious influence onthe resonant amplitude and frequency of vibration Liu andNovak [30] investigated the dynamic response of single pileembedded in transversely isotropic layered media by meansof FEM combining with dynamic stiffness matrices of thesoil derived from Greenrsquos functions for ring loads Chenet al [31] proposed a model of transversely isotropic layeredelastic media to study the dynamic horizontal impedancesof double piles embedded in the ground and subjected toharmonic sway-rocking loadings at the head of the two pilesChen et al [32] studied the transient torsional dynamicresponse of a pile embedded in transversely isotropic sat-urated soil by utilizing the Laplace transform Wang et al[33] investigated the dynamic response of an end bearingpile embedded in transversely isotropic saturated soil whenthere was a time-harmonic torsional loading acting on thepile head Notwithstanding the work above it can be seenthat no investigation corresponding to the vertical dynamicresponse of a pile embedded in layered transversely isotropicelastic soil has been reported in existing literatures until now

Based on the above literature review the main objectiveof this paper is to develop an extended soil-pile interactionmodel to simulate the vertical dynamic response of pileembedded in layered transversely isotropic soil Utilizing thefictitious soil pile model to simulate the dynamic interactionbetween the pile and pile end soil [34] the governingequations of soil-pile system are established when the verticalwave effect of surrounding soil is taken into account Thenthe analytical solution of velocity response in the frequencydomain and its corresponding semianalytical solution ofvelocity response in the time domain are derived by meansof integral transform technique and separation of variablestechnique Based on these solutions a parametric study hasbeen conducted to assess the influence of the soil anisotropyon the vibration behavior of the pile

2 Governing Equations

21 Computational Model and Assumptions The problemstudied in this paper is the vertical vibration of viscoelasticpile embedded in layered transversely isotropic soil and thegeometric model is shown in Figure 1 Based on the fictitioussoil pile model the soil-pile system is discretized into atotal of 119898 segments along the vertical direction which arenumbered by 1 2 119895 119898 The thickness of the 119895th (1 le

119895 le 119898) soil-pile segment is denoted by 119897119895 and the depth ofthe 119895th (1 le 119895 le 119898) soil layer top is denoted by ℎ119895 If thenumbers of the soil-pile segments are enough the propertiesof soil and pile within each segment can be assumed to behomogeneous respectively The pile length is denoted by119867119901and the thickness of pile end soil is denoted by119867119904 119902(119905) is anarbitrary vertical harmonic force acting on the pile head

q(t)

Layer m m

hj

Hp lj

Hs Layer 2

Layer 1 1

2

j

o

z

Layer j

Bedrock

PileTransversely

isotropicviscoelastic soil

Fictitious soil pile

r

ksj 120575sj

ksjminus1 120575sjminus1

Figure 1 Schematic of pile-soil interaction model

The following assumptions are adopted during the analy-sis

(1) The surrounding soil of pile is layered transverselyisotropic and viscoelastic The damping force of soilis proportional to the strain rate and the proportionalcoefficient of the 119895th soil layer is denoted by 120578119904119895

(2) The top surface of soil layer is assumed to be freeboundary without normal and shear stresses and thebottom surface of pile end soil is assumed to be rigidboundary without displacements

(3) The dynamic interaction of the adjacent soil layersis simulated by using the distributed Voigt modelThe spring constant and damping coefficient of thedistributedVoigtmodel between the 119895th soil layer andits upper adjacent soil layer are denoted by 119896

119904119895 and

120575119904119895 and the corresponding values between the 119895th soillayer and its lower adjacent soil layer are denoted by119896119904119895minus1 and 120575

119904119895minus1 respectively

(4) During vibration the vertical wave effect of surround-ing soil is taken into account but the radial wave effectof surrounding soil is ignored

(5) Both the pile and the fictitious soil pile are verticalviscoelastic and circular in cross-section and havea perfect contact with the surrounding soil duringvibration

(6) During vibration the soil-pile system is subjected tosmall deformations and strains and the conditionsof displacement continuity and force equilibrium aresatisfied at the interface of the adjacent pile (includingfictitious soil pile) segments

22 Governing Equations of Soil-Pile System

221 Dynamic Equation of Soil Combining with the vis-coelastic constitutive relations for a transversely isotropic

Mathematical Problems in Engineering 3

medium proposed by Ding [35] the dynamic equilibriumequation of the transversely isotropic soil in cylindricalcoordinates can be derived as follows

[11986211 (1205972

1205971199032+

1

119903

120597

120597119903

minus

1

1199032) + 11986266

1

1199032

1205972

1205971205792+ 11986244

1205972

1205971199112] 119906119903

+ [

11986211 minus 11986266

119903

1205972

120597119903 120597120579

minus

11986211 + 11986266

1199032

120597

120597120579

] 119906120579

+ (11986213 + 11986244)1205972119908

120597119903 120597119911

= 120588119904 1205972119906119903

1205971199052

[(11986211 minus 11986266)1

119903

1205972

120597119903 120597120579

+ (11986211 + 11986266)1

1199032

120597

120597120579

] 119906119903

+ (11986213 + 11986244)1

119903

1205972119908

120597120579 120597119911

+ [11986266 (1205972

1205971199032+

1

119903

120597

120597119903

minus

1

1199032) + 11986211

1

1199032

1205972

1205971205792+ 11986244

1205972

1205971199112] 119906120579

= 120588119904 1205972119906120579

1205971199052

(11986213 + 11986244) (1205972

120597119903 120597119911

+

1

119903

120597

120597119911

)119906119903 + (11986213 + 11986244)1

119903

1205972119906120579

120597120579 120597119911

+ [11986244 (1205972

1205971199032+

1

119903

120597

120597119903

+

1

1199032

1205972

1205971205792) + 11986233

1205972

1205971199112]119908

= 120588119904 1205972119908

1205971199052

(1)

where 119906119903 119906120579 and 119908 denote the radial displacement circum-ferential displacement and vertical displacement respec-tively The coefficients of the above equations should satisfythe following equations

11986211 = 11986222 =119864119904ℎ (1 minus 120583

119904ℎV120583119904Vℎ)

(1 + 120583119904ℎℎ) (1 minus 120583

119904ℎℎminus 2120583119904ℎV120583119904Vℎ)

11986212 =119864119904ℎ (120583119904ℎℎ + 120583

119904ℎV120583119904Vℎ)

(1 + 120583119904ℎℎ) (1 minus 120583

119904ℎℎminus 2120583119904ℎV120583119904Vℎ)

11986213 = 11986223 =119864119904ℎ120583119904ℎℎ

1 minus 120583119904ℎℎminus 2120583119904ℎV120583119904Vℎ

11986233 =119864V (1 minus 120583

119904ℎℎ)

1 minus 120583119904ℎℎminus 2120583119904ℎV120583119904Vℎ

11986244 = 11986255 = 119866119904V

11986266 =(11986211 minus 11986222)

2

(2)

where 119864119904ℎ and 119864

119904V denote the horizontal and vertical elastic

modulus respectively 119866119904V denotes the shear modulus in thevertical plane 120583119904ℎV is the Poissonrsquos ration in the vertical

direction caused by the horizontal stress and 120583119904Vℎ is the

Poissonrsquos ratio in the horizontal direction caused by thevertical stress and 120583119904ℎV and 120583

119904Vℎ should satisfy 120583

119904Vℎ119864119904V = 120583119904ℎV119864119904ℎ

120583119904ℎℎ is the Poissonrsquos ratio in the orthogonal direction of the

horizontal strain caused by the horizontal stressOwing to the assumption that only the vertical wave

effect of surrounding soil is taken into consideration theequilibrium equation of the transversely isotropic soil for theaxisymmetric problem can be further rewritten as follows

11986233

1205972119908

1205971199112+ 11986244 [

1205972119908

1205971199032+

1

119903

120597119908

120597119903

] = 120588119904 1205972119908

1205971199052 (3)

Based on (3) and taking into account the viscosity of soilthe governing equation of the 119895th transversely isotropic soillayer for the axisymmetric problem can be established as

120575119904119895

1205972119908119895

1205971199112+

120578119904119895

119866119904V119895

120597

120597119905

(

1205972119908119895

1205971199112) + (

1

119903

120597119908119895

120597119903

+

1205972119908119895

1205971199032)

+

120578119904119895

119866119904V119895

120597

120597119905

(

1

119903

120597119908119895

120597119903

+

1205972119908119895

1205971199032) =

120588119904119895

119866119904V119895

1205972119908119895

1205971199052

(4)

where 119908119895 = 119908119895(119903 119911 119905) is the vertical displacement of the119895th soil layer 120578119904119895 120588

119904119895 and 119866

119904V119895 denote the viscous damping

coefficient density and shear modulus in the vertical planeof the 119895th soil layer 120575119904119895 = 2(1 minus 120583

119904ℎℎ119895)(1 + 120583

119904Vℎ119895)(1 minus 120583

119904ℎℎ119895 minus

2(120583119904Vℎ119895)2120581119904119895) 120581119904119895 = (1 + 120583

119904ℎℎ119895)119866119904ℎ119895(1 + 120583

119904Vℎ119895)119866119904V119895 denotes the

ratio of the elastic modulus of the 119895th soil layer in thehorizontal direction to that in the vertical direction 119866119904ℎ119895120583119904Vℎ119895 and 120583

119904ℎℎ119895 denote the shear modulus in the horizontal

plane the Poissonrsquos ratio in the horizontal direction causedby the vertical stress and the Poissonrsquos ratio in the orthogonaldirection of the horizontal strain caused by the horizontalstress of the 119895th soil layer respectively

222 Dynamic Equation of Pile Denoting 119906119895 = 119906119895(119911 119905) to bethe vertical displacement of the 119895th pile (including fictitioussoil pile) segment and according to the Euler-Bernoulli rodtheory the dynamic equilibrium equation of pile can beestablished as

119864119901

119895119860119901

119895

1205972119906119895

1205971199112+ 119860119901

119895120578119901

119895

1205973119906119895

120597119905 1205971199112minus 119898119901

119895

1205972119906119895

1205971199052

minus 2120587119903119901

119895 120591119904119903119911119895 (119903119901

119895 119911 119905) = 0

(5)

where 119864119901119895 = 120588119901

119895 (119881119901

119895 )2 119860119901119895 = 120587(119903

119901

119895 )2 119903119901119895 119898

119901

119895 120588119901

119895 119881119901

119895 and 120578119901

119895

denote the elastic modulus cross-section area radius massper unit length of pile (including fictitious soil pile) densityelastic longitudinal wave velocity and viscous damping coef-ficient of the 119895th pile (including fictitious soil pile) segmentrespectively 120591119904119903119911119895(119903

119901

119895 119911 119905) is the frictional force of the 119895th soillayer acting on the surface of the 119895th pile shaft and can beexpressed as

120591119904119903119911119895 (119903119901

119895 119911 119905) = 119866119904V119895

120597119908119895 (119903119901

119895 119911 119905)

120597119903

+ 120578119904119895

1205972119908119895 (119903119901

119895 119911 119905)

120597119905 120597119903

(6)

4 Mathematical Problems in Engineering

Combining with the assumptions the boundary andinitial conditions of soil-pile system can be established asfollows

(1) Boundary Conditions of Soil At the top surface of the 119895thsoil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= (119896119904119895119908119895 + 120575

119904119895

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(7a)

At the bottom surface of the 119895th soil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus(119896119904119895minus1119908119895 + 120575

119904119895minus1

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(7b)

At an infinite radial distance of the 119895th soil layer

120590119895 (infin 119911) = 0 119908119895 (infin 119911) = 0 (7c)

(2) Boundary Conditions of Pile At the top surface of the 119895thpile (including fictitious soil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= minus

119885119895 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(8a)

At the bottom surface of the 119895th pile (including fictitioussoil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus

119885119895minus1 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(8b)

where 119885119895(119904) and 119885119895minus1(119904) denote the displacement impedancefunction at the top and bottom surfaces of the 119895th pile(including fictitious soil pile) segment 119904 is the Laplacetransform parameter

(3) Boundary C at the Interface of Soil-Pile System

119908(119903119901

119895 119911 119905) = 119906119895 (119911 119905) (9)

(4) Initial Conditions of Soil-Pile System Initial conditions ofthe 119895th soil layer are as follows

119908119895

10038161003816100381610038161003816119905=0

= 0

120597119908119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10a)

Initial conditions of the 119895th pile (including fictitious soilpile) segment are as follows

119906119895

10038161003816100381610038161003816119905=0

= 0

120597119906119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10b)

3 Solutions of the Governing Equations

31 Vibrations of the Soil Layer Denoting 119882119895(119903 119911 119904) =

int

+infin

0119908119895(119903 119911 119905)119890

minus119904119905d119905 to be the Laplace transform with respectto time of119908119895(119903 119911 119905) and associating with the initial condition(10a) (4) is transformed by using the Laplace transformtechnique as follows

(120575119904119895 +

120578119904119895 sdot 119904

119866119904V119895

)

1205972119882119895

1205971199112

+ (1 +

120578119904119895 sdot 119904

119866119904V119895

)(

1

119903

120597119882119895

120597119903

+

1205972119882119895

1205971199032)

= (

119904

119881119904V119895)

2

119882119895

(11)

where119881119904V119895 = radic119866119904V119895120588119904119895 is the shear wave velocity of the 119895th soil

layer in the vertical directionBy virtue of the separation of variables technique and

denoting 119882119895(119903 119911 119904) = 119877119895(119903 119904)119885119895(119911 119904) (11) can be decoupledas follows

d2119877119895 (119903 119904)d1199032

+

1

119903

d119877119895 (119903 119904)d119903

minus 1205852119895119877119895 (119903 119904) = 0

(12)

d2119885119895 (119911 119904)d1199112

+ 1205732119895119885119895 (119911 119904) = 0

(13)

where constants 120585119895 and 120573119895must satisfy the following equation

1205852119895 =

(120575119904119895 + 120578119904119895 sdot 119904119866

119904V119895) 1205732119895 + (119904119881

119904V119895)2

(1 + 120578119904119895 sdot 119904119866

119904V119895)

(14)

It can be seen that (12) is Bessel equation and (13) isordinary differential equation of second order whose generalsolutions can be easily obtained Associating with thesegeneral solutions the vertical displacement of the 119895th soillayer119882119895(119903 119911 119904) can be derived as

119882119895 (119903 119911 119904) = [1198601198951198700 (120585119895119903) + 1198611198951198680 (120585119895119903)]

times [119862119895 sin (120573119895119911) + 119863119895 cos (120573119895119911)] (15)

where 1198680(sdot) and 1198700(sdot) denote the modified Bessel functionsof order zero of the first and second kind respectively 119860119895119861119895 119862119895 and 119863119895 are constants determined by the boundaryconditions

Converting 119911 = ℎ119895 and 119911 = ℎ119895+119897119895 in the global coordinatesinto 119911

1015840= 0 and 119911

1015840= 119897119895 in the local coordinates (7a)

Mathematical Problems in Engineering 5

(7b) and (7c) are transformed by using the Laplace transformtechnique and can be rewritten as follows

[

[

(119896119904119895 + 120575119904119895 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) minus

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=0

= 0 (16a)

[

[

(119896119904119895minus1 + 120575

119904119895minus1 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) +

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=119897119895

= 0

(16b)

120590119895 (infin 1199111015840) = 0 119882119895 (infin 119911

1015840) = 0 (16c)

According to the properties of the modified Bessel func-tions that is if 119903 rarr infin then 119868119899(sdot) rarr infin and 119870119899(sdot) rarr 0it can be obtained that 119861119895 = 0 from boundary conditions(16c) By means of boundary conditions (16a) and (16b) thefollowing equation can be obtained

tan (120573119895119897119895)

= (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895 +

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895)120573119895119897119895

times ((120573119895119897119895)2minus (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895)(

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895))

minus1

=

(119870119895 + 119870

1015840

119895) 120573119895119897119895

(120573119895119897119895)2minus 119870119895119870

1015840

119895

(17)

where 119870119895 = ((119896119904119895 + 120575

119904119895 sdot 119904)119864

119904V119895)119897119895 and 119870

1015840

119895 = ((119896119904119895minus1 +

120575119904119895minus1 sdot 119904)119864

119904V119895)119897119895 denote the dimensionless complex stiffness

of the upper surface and lower surface of the 119895th soil layerSubstituting 119904 = 119894120596 into (17) and solving it by using bisectionmethod in the frequency domain a series of eigenvalues 120573119895119899can be derived 120585119895119899 can also be derived by substituting120573119895119899 into(14)

Then the vertical displacement of the 119895th soil layer can berewritten as

119882119895 (119903 1199111015840 119904) =

infin

sum

119899=1

1198601198951198991198700 (120585119895119899119903) sin (1205731198951198991199111015840+ 120601119895119899) (18)

where120601119895119899 = arctan(120573119895119899119897119895119870119895) and119860119895119899 are a series of constantsdetermined by the boundary conditions which reflect thedynamic interaction of soil and pile

32 Vibrations of the Pile Denoting119880119895(119911 119904) to be the Laplacetransform with respect to time of 119906119895(119911 119905) (5) is transformed

by using the Laplace transform technique and can be rewrit-ten combining with (6) and (18)

(119881119901

119895 )2(1 +

120578119901

119895

119864119901

119895

sdot 119904)

1205972119880119895

12059711991110158402

minus 1199042119880119895

minus

2120587119903119901

119895

120588119901

119895 119860119901

119895

(119866119904V119895 + 120578119904V119895 sdot 119904)

infin

sum

119899=1

1198601198951198991205851198951198991198701 (120585119895119899119903119901

119895 )

times sin (1205731198951198991199111015840+ 120601119895119899) = 0

(19)

It is not difficult to obtain that the general solution of (19)can be expressed as

119880119895 = 119872119895 [cos(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

+ 119873119895 [sin(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

(20)

where

1205941015840119895119899 = 120594119895119899

[

[

cos (120573119895119899 + 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 + 120582119895

+

cos (120573119895119899 minus 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 minus 120582119895

]

]

12059410158401015840119895119899 = 120594119895119899

[

[

sin (120573119895119899 + 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 + 120582119895

minus

sin (120573119895119899 minus 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 minus 120582119895

]

]

120594119895119899 =

(119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 119897119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895] 120593119895119899119871119895119899

120593119895119899 = 1198700 (120585119895119899119903119901

119895 ) +

2 (119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 1198972119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895]

119871119895119899 = int

119897119895

0

sin2 (1205731198951198991199111015840+ 120601119895119899) d119911

1015840

(21)

where 120582119895 = radicminus11990421199052119895(1 + (120578

119901

119895 119864119901

119895 ) sdot 119904) 120573119895119899 = 120573119895119899119897119895 120585119895119899 = 120585119895119899119897119895and 119903119901

119895 = 119903119901

119895 119897119895 are all dimensionless parameters 119905119895 = 119897119895119881119901

119895

denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment1198701(sdot) is the modified Bessel functions oforder one of the second kind

6 Mathematical Problems in Engineering

Combining with the boundary conditions (8a) and (8b)the displacement impedance function at the head of the 119895thpile segment can be derived in the local coordinates as follows

119885119895 (119904) =

minus119864119901

119895119860119901

119895 (1205971198801198951205971199111015840)

100381610038161003816100381610038161199111015840=0

119880119895

100381610038161003816100381610038161199111015840=0

= minus

119864119901

119895119860119901

119895

119897119895

times (

119872119895

119873119895

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos120601119895119899

+ 120582119895 +

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos120601119895119899)

times (

119872119895

119873119895

(1 +

infin

sum

119899=1

1205941015840119895119899 sin120601119895119899)

+

infin

sum

119899=1

12059410158401015840119895119899 sin120601119895119899)

minus1

(22)

where

119872119895

119873119895

= (

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) + 120582119895 cos 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[sin 120582119895 +infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899 + 120601119895119899)])

times (

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) minus 120582119895 sin 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[cos 120582119895 +infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899 + 120601119895119899)])

minus1

(23)

where 119885119895minus1(119904) denotes the displacement impedance functionat the head of the (119895 minus 1)th pile segmentwhich can be obtainedby using boundary conditionsThen following themethod ofrecursion typically used in the transfer function techniquethe displacement impedance function at the head of pile canbe derived as

119885119898 (119904) =

minus119864119901119898119860119901119898 (120597119880119898120597119911

1015840)

100381610038161003816100381610038161199111015840=0

11988011989810038161003816100381610038161199111015840=0

= minus

119864119901119898119860119901119898

119897119898

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

minus1

(24)

where

119872119898

119873119898

= minus(

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) + 120582119898 cos 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[sin 120582119898 +infin

sum

119899=1

12059410158401015840119898119899 sin (120573119898119899 + 120601119898119899)])

times (

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) minus 120582119898 sin 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[cos 120582119898 +infin

sum

119899=1

1205941015840119898119899 sin (120573119898119899 + 120601119898119899)])

minus1

1205941015840119898119899 = 120594119898119899 [

cos (120573119898119899 + 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 + 120582119898

+

cos (120573119898119899 minus 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 minus 120582119898

]

12059410158401015840119898119899 = 120594119898119899 [

sin (120573119898119899 + 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 + 120582119898

minus

sin (120573119898119899 minus 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 minus 120582119898

]

120594119898119899 =

(119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

120588119901119898119897119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898] 120593119898119899119871119898119899

120593119898119899 = 1198700 (120585119898119899119903119901119898)

+

2 (119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

1205881199011198981198972119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898]

119871119898119899 = int

119897119898

0

sin2 (1205731198981198991199111015840+ 120601119898119899) d119911

1015840

(25)

where 120582119898 = radicminus11990421199052119898(1 + (120578

119901119898119864119901119898) sdot 119904) 120573119898119899 = 120573119898119899119897119898 120585119898119899 =

120585119898119899119897119898 and 119903119901119898 = 119903119901119898119897119898 are all dimensionless parameters 119905119898 =

119897119898119881119901119898 denotes the propagation time of elastic longitudinal

Mathematical Problems in Engineering 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

medium proposed by Ding [35] the dynamic equilibriumequation of the transversely isotropic soil in cylindricalcoordinates can be derived as follows

[11986211 (1205972

1205971199032+

1

119903

120597

120597119903

minus

1

1199032) + 11986266

1

1199032

1205972

1205971205792+ 11986244

1205972

1205971199112] 119906119903

+ [

11986211 minus 11986266

119903

1205972

120597119903 120597120579

minus

11986211 + 11986266

1199032

120597

120597120579

] 119906120579

+ (11986213 + 11986244)1205972119908

120597119903 120597119911

= 120588119904 1205972119906119903

1205971199052

[(11986211 minus 11986266)1

119903

1205972

120597119903 120597120579

+ (11986211 + 11986266)1

1199032

120597

120597120579

] 119906119903

+ (11986213 + 11986244)1

119903

1205972119908

120597120579 120597119911

+ [11986266 (1205972

1205971199032+

1

119903

120597

120597119903

minus

1

1199032) + 11986211

1

1199032

1205972

1205971205792+ 11986244

1205972

1205971199112] 119906120579

= 120588119904 1205972119906120579

1205971199052

(11986213 + 11986244) (1205972

120597119903 120597119911

+

1

119903

120597

120597119911

)119906119903 + (11986213 + 11986244)1

119903

1205972119906120579

120597120579 120597119911

+ [11986244 (1205972

1205971199032+

1

119903

120597

120597119903

+

1

1199032

1205972

1205971205792) + 11986233

1205972

1205971199112]119908

= 120588119904 1205972119908

1205971199052

(1)

where 119906119903 119906120579 and 119908 denote the radial displacement circum-ferential displacement and vertical displacement respec-tively The coefficients of the above equations should satisfythe following equations

11986211 = 11986222 =119864119904ℎ (1 minus 120583

119904ℎV120583119904Vℎ)

(1 + 120583119904ℎℎ) (1 minus 120583

119904ℎℎminus 2120583119904ℎV120583119904Vℎ)

11986212 =119864119904ℎ (120583119904ℎℎ + 120583

119904ℎV120583119904Vℎ)

(1 + 120583119904ℎℎ) (1 minus 120583

119904ℎℎminus 2120583119904ℎV120583119904Vℎ)

11986213 = 11986223 =119864119904ℎ120583119904ℎℎ

1 minus 120583119904ℎℎminus 2120583119904ℎV120583119904Vℎ

11986233 =119864V (1 minus 120583

119904ℎℎ)

1 minus 120583119904ℎℎminus 2120583119904ℎV120583119904Vℎ

11986244 = 11986255 = 119866119904V

11986266 =(11986211 minus 11986222)

2

(2)

where 119864119904ℎ and 119864

119904V denote the horizontal and vertical elastic

modulus respectively 119866119904V denotes the shear modulus in thevertical plane 120583119904ℎV is the Poissonrsquos ration in the vertical

direction caused by the horizontal stress and 120583119904Vℎ is the

Poissonrsquos ratio in the horizontal direction caused by thevertical stress and 120583119904ℎV and 120583

119904Vℎ should satisfy 120583

119904Vℎ119864119904V = 120583119904ℎV119864119904ℎ

120583119904ℎℎ is the Poissonrsquos ratio in the orthogonal direction of the

horizontal strain caused by the horizontal stressOwing to the assumption that only the vertical wave

effect of surrounding soil is taken into consideration theequilibrium equation of the transversely isotropic soil for theaxisymmetric problem can be further rewritten as follows

11986233

1205972119908

1205971199112+ 11986244 [

1205972119908

1205971199032+

1

119903

120597119908

120597119903

] = 120588119904 1205972119908

1205971199052 (3)

Based on (3) and taking into account the viscosity of soilthe governing equation of the 119895th transversely isotropic soillayer for the axisymmetric problem can be established as

120575119904119895

1205972119908119895

1205971199112+

120578119904119895

119866119904V119895

120597

120597119905

(

1205972119908119895

1205971199112) + (

1

119903

120597119908119895

120597119903

+

1205972119908119895

1205971199032)

+

120578119904119895

119866119904V119895

120597

120597119905

(

1

119903

120597119908119895

120597119903

+

1205972119908119895

1205971199032) =

120588119904119895

119866119904V119895

1205972119908119895

1205971199052

(4)

where 119908119895 = 119908119895(119903 119911 119905) is the vertical displacement of the119895th soil layer 120578119904119895 120588

119904119895 and 119866

119904V119895 denote the viscous damping

coefficient density and shear modulus in the vertical planeof the 119895th soil layer 120575119904119895 = 2(1 minus 120583

119904ℎℎ119895)(1 + 120583

119904Vℎ119895)(1 minus 120583

119904ℎℎ119895 minus

2(120583119904Vℎ119895)2120581119904119895) 120581119904119895 = (1 + 120583

119904ℎℎ119895)119866119904ℎ119895(1 + 120583

119904Vℎ119895)119866119904V119895 denotes the

ratio of the elastic modulus of the 119895th soil layer in thehorizontal direction to that in the vertical direction 119866119904ℎ119895120583119904Vℎ119895 and 120583

119904ℎℎ119895 denote the shear modulus in the horizontal

plane the Poissonrsquos ratio in the horizontal direction causedby the vertical stress and the Poissonrsquos ratio in the orthogonaldirection of the horizontal strain caused by the horizontalstress of the 119895th soil layer respectively

222 Dynamic Equation of Pile Denoting 119906119895 = 119906119895(119911 119905) to bethe vertical displacement of the 119895th pile (including fictitioussoil pile) segment and according to the Euler-Bernoulli rodtheory the dynamic equilibrium equation of pile can beestablished as

119864119901

119895119860119901

119895

1205972119906119895

1205971199112+ 119860119901

119895120578119901

119895

1205973119906119895

120597119905 1205971199112minus 119898119901

119895

1205972119906119895

1205971199052

minus 2120587119903119901

119895 120591119904119903119911119895 (119903119901

119895 119911 119905) = 0

(5)

where 119864119901119895 = 120588119901

119895 (119881119901

119895 )2 119860119901119895 = 120587(119903

119901

119895 )2 119903119901119895 119898

119901

119895 120588119901

119895 119881119901

119895 and 120578119901

119895

denote the elastic modulus cross-section area radius massper unit length of pile (including fictitious soil pile) densityelastic longitudinal wave velocity and viscous damping coef-ficient of the 119895th pile (including fictitious soil pile) segmentrespectively 120591119904119903119911119895(119903

119901

119895 119911 119905) is the frictional force of the 119895th soillayer acting on the surface of the 119895th pile shaft and can beexpressed as

120591119904119903119911119895 (119903119901

119895 119911 119905) = 119866119904V119895

120597119908119895 (119903119901

119895 119911 119905)

120597119903

+ 120578119904119895

1205972119908119895 (119903119901

119895 119911 119905)

120597119905 120597119903

(6)

4 Mathematical Problems in Engineering

Combining with the assumptions the boundary andinitial conditions of soil-pile system can be established asfollows

(1) Boundary Conditions of Soil At the top surface of the 119895thsoil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= (119896119904119895119908119895 + 120575

119904119895

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(7a)

At the bottom surface of the 119895th soil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus(119896119904119895minus1119908119895 + 120575

119904119895minus1

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(7b)

At an infinite radial distance of the 119895th soil layer

120590119895 (infin 119911) = 0 119908119895 (infin 119911) = 0 (7c)

(2) Boundary Conditions of Pile At the top surface of the 119895thpile (including fictitious soil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= minus

119885119895 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(8a)

At the bottom surface of the 119895th pile (including fictitioussoil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus

119885119895minus1 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(8b)

where 119885119895(119904) and 119885119895minus1(119904) denote the displacement impedancefunction at the top and bottom surfaces of the 119895th pile(including fictitious soil pile) segment 119904 is the Laplacetransform parameter

(3) Boundary C at the Interface of Soil-Pile System

119908(119903119901

119895 119911 119905) = 119906119895 (119911 119905) (9)

(4) Initial Conditions of Soil-Pile System Initial conditions ofthe 119895th soil layer are as follows

119908119895

10038161003816100381610038161003816119905=0

= 0

120597119908119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10a)

Initial conditions of the 119895th pile (including fictitious soilpile) segment are as follows

119906119895

10038161003816100381610038161003816119905=0

= 0

120597119906119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10b)

3 Solutions of the Governing Equations

31 Vibrations of the Soil Layer Denoting 119882119895(119903 119911 119904) =

int

+infin

0119908119895(119903 119911 119905)119890

minus119904119905d119905 to be the Laplace transform with respectto time of119908119895(119903 119911 119905) and associating with the initial condition(10a) (4) is transformed by using the Laplace transformtechnique as follows

(120575119904119895 +

120578119904119895 sdot 119904

119866119904V119895

)

1205972119882119895

1205971199112

+ (1 +

120578119904119895 sdot 119904

119866119904V119895

)(

1

119903

120597119882119895

120597119903

+

1205972119882119895

1205971199032)

= (

119904

119881119904V119895)

2

119882119895

(11)

where119881119904V119895 = radic119866119904V119895120588119904119895 is the shear wave velocity of the 119895th soil

layer in the vertical directionBy virtue of the separation of variables technique and

denoting 119882119895(119903 119911 119904) = 119877119895(119903 119904)119885119895(119911 119904) (11) can be decoupledas follows

d2119877119895 (119903 119904)d1199032

+

1

119903

d119877119895 (119903 119904)d119903

minus 1205852119895119877119895 (119903 119904) = 0

(12)

d2119885119895 (119911 119904)d1199112

+ 1205732119895119885119895 (119911 119904) = 0

(13)

where constants 120585119895 and 120573119895must satisfy the following equation

1205852119895 =

(120575119904119895 + 120578119904119895 sdot 119904119866

119904V119895) 1205732119895 + (119904119881

119904V119895)2

(1 + 120578119904119895 sdot 119904119866

119904V119895)

(14)

It can be seen that (12) is Bessel equation and (13) isordinary differential equation of second order whose generalsolutions can be easily obtained Associating with thesegeneral solutions the vertical displacement of the 119895th soillayer119882119895(119903 119911 119904) can be derived as

119882119895 (119903 119911 119904) = [1198601198951198700 (120585119895119903) + 1198611198951198680 (120585119895119903)]

times [119862119895 sin (120573119895119911) + 119863119895 cos (120573119895119911)] (15)

where 1198680(sdot) and 1198700(sdot) denote the modified Bessel functionsof order zero of the first and second kind respectively 119860119895119861119895 119862119895 and 119863119895 are constants determined by the boundaryconditions

Converting 119911 = ℎ119895 and 119911 = ℎ119895+119897119895 in the global coordinatesinto 119911

1015840= 0 and 119911

1015840= 119897119895 in the local coordinates (7a)

Mathematical Problems in Engineering 5

(7b) and (7c) are transformed by using the Laplace transformtechnique and can be rewritten as follows

[

[

(119896119904119895 + 120575119904119895 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) minus

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=0

= 0 (16a)

[

[

(119896119904119895minus1 + 120575

119904119895minus1 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) +

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=119897119895

= 0

(16b)

120590119895 (infin 1199111015840) = 0 119882119895 (infin 119911

1015840) = 0 (16c)

According to the properties of the modified Bessel func-tions that is if 119903 rarr infin then 119868119899(sdot) rarr infin and 119870119899(sdot) rarr 0it can be obtained that 119861119895 = 0 from boundary conditions(16c) By means of boundary conditions (16a) and (16b) thefollowing equation can be obtained

tan (120573119895119897119895)

= (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895 +

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895)120573119895119897119895

times ((120573119895119897119895)2minus (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895)(

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895))

minus1

=

(119870119895 + 119870

1015840

119895) 120573119895119897119895

(120573119895119897119895)2minus 119870119895119870

1015840

119895

(17)

where 119870119895 = ((119896119904119895 + 120575

119904119895 sdot 119904)119864

119904V119895)119897119895 and 119870

1015840

119895 = ((119896119904119895minus1 +

120575119904119895minus1 sdot 119904)119864

119904V119895)119897119895 denote the dimensionless complex stiffness

of the upper surface and lower surface of the 119895th soil layerSubstituting 119904 = 119894120596 into (17) and solving it by using bisectionmethod in the frequency domain a series of eigenvalues 120573119895119899can be derived 120585119895119899 can also be derived by substituting120573119895119899 into(14)

Then the vertical displacement of the 119895th soil layer can berewritten as

119882119895 (119903 1199111015840 119904) =

infin

sum

119899=1

1198601198951198991198700 (120585119895119899119903) sin (1205731198951198991199111015840+ 120601119895119899) (18)

where120601119895119899 = arctan(120573119895119899119897119895119870119895) and119860119895119899 are a series of constantsdetermined by the boundary conditions which reflect thedynamic interaction of soil and pile

32 Vibrations of the Pile Denoting119880119895(119911 119904) to be the Laplacetransform with respect to time of 119906119895(119911 119905) (5) is transformed

by using the Laplace transform technique and can be rewrit-ten combining with (6) and (18)

(119881119901

119895 )2(1 +

120578119901

119895

119864119901

119895

sdot 119904)

1205972119880119895

12059711991110158402

minus 1199042119880119895

minus

2120587119903119901

119895

120588119901

119895 119860119901

119895

(119866119904V119895 + 120578119904V119895 sdot 119904)

infin

sum

119899=1

1198601198951198991205851198951198991198701 (120585119895119899119903119901

119895 )

times sin (1205731198951198991199111015840+ 120601119895119899) = 0

(19)

It is not difficult to obtain that the general solution of (19)can be expressed as

119880119895 = 119872119895 [cos(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

+ 119873119895 [sin(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

(20)

where

1205941015840119895119899 = 120594119895119899

[

[

cos (120573119895119899 + 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 + 120582119895

+

cos (120573119895119899 minus 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 minus 120582119895

]

]

12059410158401015840119895119899 = 120594119895119899

[

[

sin (120573119895119899 + 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 + 120582119895

minus

sin (120573119895119899 minus 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 minus 120582119895

]

]

120594119895119899 =

(119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 119897119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895] 120593119895119899119871119895119899

120593119895119899 = 1198700 (120585119895119899119903119901

119895 ) +

2 (119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 1198972119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895]

119871119895119899 = int

119897119895

0

sin2 (1205731198951198991199111015840+ 120601119895119899) d119911

1015840

(21)

where 120582119895 = radicminus11990421199052119895(1 + (120578

119901

119895 119864119901

119895 ) sdot 119904) 120573119895119899 = 120573119895119899119897119895 120585119895119899 = 120585119895119899119897119895and 119903119901

119895 = 119903119901

119895 119897119895 are all dimensionless parameters 119905119895 = 119897119895119881119901

119895

denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment1198701(sdot) is the modified Bessel functions oforder one of the second kind

6 Mathematical Problems in Engineering

Combining with the boundary conditions (8a) and (8b)the displacement impedance function at the head of the 119895thpile segment can be derived in the local coordinates as follows

119885119895 (119904) =

minus119864119901

119895119860119901

119895 (1205971198801198951205971199111015840)

100381610038161003816100381610038161199111015840=0

119880119895

100381610038161003816100381610038161199111015840=0

= minus

119864119901

119895119860119901

119895

119897119895

times (

119872119895

119873119895

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos120601119895119899

+ 120582119895 +

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos120601119895119899)

times (

119872119895

119873119895

(1 +

infin

sum

119899=1

1205941015840119895119899 sin120601119895119899)

+

infin

sum

119899=1

12059410158401015840119895119899 sin120601119895119899)

minus1

(22)

where

119872119895

119873119895

= (

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) + 120582119895 cos 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[sin 120582119895 +infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899 + 120601119895119899)])

times (

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) minus 120582119895 sin 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[cos 120582119895 +infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899 + 120601119895119899)])

minus1

(23)

where 119885119895minus1(119904) denotes the displacement impedance functionat the head of the (119895 minus 1)th pile segmentwhich can be obtainedby using boundary conditionsThen following themethod ofrecursion typically used in the transfer function techniquethe displacement impedance function at the head of pile canbe derived as

119885119898 (119904) =

minus119864119901119898119860119901119898 (120597119880119898120597119911

1015840)

100381610038161003816100381610038161199111015840=0

11988011989810038161003816100381610038161199111015840=0

= minus

119864119901119898119860119901119898

119897119898

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

minus1

(24)

where

119872119898

119873119898

= minus(

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) + 120582119898 cos 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[sin 120582119898 +infin

sum

119899=1

12059410158401015840119898119899 sin (120573119898119899 + 120601119898119899)])

times (

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) minus 120582119898 sin 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[cos 120582119898 +infin

sum

119899=1

1205941015840119898119899 sin (120573119898119899 + 120601119898119899)])

minus1

1205941015840119898119899 = 120594119898119899 [

cos (120573119898119899 + 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 + 120582119898

+

cos (120573119898119899 minus 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 minus 120582119898

]

12059410158401015840119898119899 = 120594119898119899 [

sin (120573119898119899 + 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 + 120582119898

minus

sin (120573119898119899 minus 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 minus 120582119898

]

120594119898119899 =

(119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

120588119901119898119897119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898] 120593119898119899119871119898119899

120593119898119899 = 1198700 (120585119898119899119903119901119898)

+

2 (119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

1205881199011198981198972119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898]

119871119898119899 = int

119897119898

0

sin2 (1205731198981198991199111015840+ 120601119898119899) d119911

1015840

(25)

where 120582119898 = radicminus11990421199052119898(1 + (120578

119901119898119864119901119898) sdot 119904) 120573119898119899 = 120573119898119899119897119898 120585119898119899 =

120585119898119899119897119898 and 119903119901119898 = 119903119901119898119897119898 are all dimensionless parameters 119905119898 =

119897119898119881119901119898 denotes the propagation time of elastic longitudinal

Mathematical Problems in Engineering 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

4 Mathematical Problems in Engineering

Combining with the assumptions the boundary andinitial conditions of soil-pile system can be established asfollows

(1) Boundary Conditions of Soil At the top surface of the 119895thsoil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= (119896119904119895119908119895 + 120575

119904119895

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(7a)

At the bottom surface of the 119895th soil layer

119864119904V119895

120597119908119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus(119896119904119895minus1119908119895 + 120575

119904119895minus1

120597119908119895

120597119905

)

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(7b)

At an infinite radial distance of the 119895th soil layer

120590119895 (infin 119911) = 0 119908119895 (infin 119911) = 0 (7c)

(2) Boundary Conditions of Pile At the top surface of the 119895thpile (including fictitious soil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= minus

119885119895 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(8a)

At the bottom surface of the 119895th pile (including fictitioussoil pile) segment

120597119906119895

120597119911

100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

= minus

119885119895minus1 (119904) 119906119895

119864119901

119895119860119901

119895

10038161003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895+119897119895

(8b)

where 119885119895(119904) and 119885119895minus1(119904) denote the displacement impedancefunction at the top and bottom surfaces of the 119895th pile(including fictitious soil pile) segment 119904 is the Laplacetransform parameter

(3) Boundary C at the Interface of Soil-Pile System

119908(119903119901

119895 119911 119905) = 119906119895 (119911 119905) (9)

(4) Initial Conditions of Soil-Pile System Initial conditions ofthe 119895th soil layer are as follows

119908119895

10038161003816100381610038161003816119905=0

= 0

120597119908119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10a)

Initial conditions of the 119895th pile (including fictitious soilpile) segment are as follows

119906119895

10038161003816100381610038161003816119905=0

= 0

120597119906119895

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0

= 0 (10b)

3 Solutions of the Governing Equations

31 Vibrations of the Soil Layer Denoting 119882119895(119903 119911 119904) =

int

+infin

0119908119895(119903 119911 119905)119890

minus119904119905d119905 to be the Laplace transform with respectto time of119908119895(119903 119911 119905) and associating with the initial condition(10a) (4) is transformed by using the Laplace transformtechnique as follows

(120575119904119895 +

120578119904119895 sdot 119904

119866119904V119895

)

1205972119882119895

1205971199112

+ (1 +

120578119904119895 sdot 119904

119866119904V119895

)(

1

119903

120597119882119895

120597119903

+

1205972119882119895

1205971199032)

= (

119904

119881119904V119895)

2

119882119895

(11)

where119881119904V119895 = radic119866119904V119895120588119904119895 is the shear wave velocity of the 119895th soil

layer in the vertical directionBy virtue of the separation of variables technique and

denoting 119882119895(119903 119911 119904) = 119877119895(119903 119904)119885119895(119911 119904) (11) can be decoupledas follows

d2119877119895 (119903 119904)d1199032

+

1

119903

d119877119895 (119903 119904)d119903

minus 1205852119895119877119895 (119903 119904) = 0

(12)

d2119885119895 (119911 119904)d1199112

+ 1205732119895119885119895 (119911 119904) = 0

(13)

where constants 120585119895 and 120573119895must satisfy the following equation

1205852119895 =

(120575119904119895 + 120578119904119895 sdot 119904119866

119904V119895) 1205732119895 + (119904119881

119904V119895)2

(1 + 120578119904119895 sdot 119904119866

119904V119895)

(14)

It can be seen that (12) is Bessel equation and (13) isordinary differential equation of second order whose generalsolutions can be easily obtained Associating with thesegeneral solutions the vertical displacement of the 119895th soillayer119882119895(119903 119911 119904) can be derived as

119882119895 (119903 119911 119904) = [1198601198951198700 (120585119895119903) + 1198611198951198680 (120585119895119903)]

times [119862119895 sin (120573119895119911) + 119863119895 cos (120573119895119911)] (15)

where 1198680(sdot) and 1198700(sdot) denote the modified Bessel functionsof order zero of the first and second kind respectively 119860119895119861119895 119862119895 and 119863119895 are constants determined by the boundaryconditions

Converting 119911 = ℎ119895 and 119911 = ℎ119895+119897119895 in the global coordinatesinto 119911

1015840= 0 and 119911

1015840= 119897119895 in the local coordinates (7a)

Mathematical Problems in Engineering 5

(7b) and (7c) are transformed by using the Laplace transformtechnique and can be rewritten as follows

[

[

(119896119904119895 + 120575119904119895 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) minus

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=0

= 0 (16a)

[

[

(119896119904119895minus1 + 120575

119904119895minus1 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) +

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=119897119895

= 0

(16b)

120590119895 (infin 1199111015840) = 0 119882119895 (infin 119911

1015840) = 0 (16c)

According to the properties of the modified Bessel func-tions that is if 119903 rarr infin then 119868119899(sdot) rarr infin and 119870119899(sdot) rarr 0it can be obtained that 119861119895 = 0 from boundary conditions(16c) By means of boundary conditions (16a) and (16b) thefollowing equation can be obtained

tan (120573119895119897119895)

= (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895 +

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895)120573119895119897119895

times ((120573119895119897119895)2minus (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895)(

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895))

minus1

=

(119870119895 + 119870

1015840

119895) 120573119895119897119895

(120573119895119897119895)2minus 119870119895119870

1015840

119895

(17)

where 119870119895 = ((119896119904119895 + 120575

119904119895 sdot 119904)119864

119904V119895)119897119895 and 119870

1015840

119895 = ((119896119904119895minus1 +

120575119904119895minus1 sdot 119904)119864

119904V119895)119897119895 denote the dimensionless complex stiffness

of the upper surface and lower surface of the 119895th soil layerSubstituting 119904 = 119894120596 into (17) and solving it by using bisectionmethod in the frequency domain a series of eigenvalues 120573119895119899can be derived 120585119895119899 can also be derived by substituting120573119895119899 into(14)

Then the vertical displacement of the 119895th soil layer can berewritten as

119882119895 (119903 1199111015840 119904) =

infin

sum

119899=1

1198601198951198991198700 (120585119895119899119903) sin (1205731198951198991199111015840+ 120601119895119899) (18)

where120601119895119899 = arctan(120573119895119899119897119895119870119895) and119860119895119899 are a series of constantsdetermined by the boundary conditions which reflect thedynamic interaction of soil and pile

32 Vibrations of the Pile Denoting119880119895(119911 119904) to be the Laplacetransform with respect to time of 119906119895(119911 119905) (5) is transformed

by using the Laplace transform technique and can be rewrit-ten combining with (6) and (18)

(119881119901

119895 )2(1 +

120578119901

119895

119864119901

119895

sdot 119904)

1205972119880119895

12059711991110158402

minus 1199042119880119895

minus

2120587119903119901

119895

120588119901

119895 119860119901

119895

(119866119904V119895 + 120578119904V119895 sdot 119904)

infin

sum

119899=1

1198601198951198991205851198951198991198701 (120585119895119899119903119901

119895 )

times sin (1205731198951198991199111015840+ 120601119895119899) = 0

(19)

It is not difficult to obtain that the general solution of (19)can be expressed as

119880119895 = 119872119895 [cos(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

+ 119873119895 [sin(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

(20)

where

1205941015840119895119899 = 120594119895119899

[

[

cos (120573119895119899 + 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 + 120582119895

+

cos (120573119895119899 minus 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 minus 120582119895

]

]

12059410158401015840119895119899 = 120594119895119899

[

[

sin (120573119895119899 + 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 + 120582119895

minus

sin (120573119895119899 minus 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 minus 120582119895

]

]

120594119895119899 =

(119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 119897119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895] 120593119895119899119871119895119899

120593119895119899 = 1198700 (120585119895119899119903119901

119895 ) +

2 (119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 1198972119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895]

119871119895119899 = int

119897119895

0

sin2 (1205731198951198991199111015840+ 120601119895119899) d119911

1015840

(21)

where 120582119895 = radicminus11990421199052119895(1 + (120578

119901

119895 119864119901

119895 ) sdot 119904) 120573119895119899 = 120573119895119899119897119895 120585119895119899 = 120585119895119899119897119895and 119903119901

119895 = 119903119901

119895 119897119895 are all dimensionless parameters 119905119895 = 119897119895119881119901

119895

denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment1198701(sdot) is the modified Bessel functions oforder one of the second kind

6 Mathematical Problems in Engineering

Combining with the boundary conditions (8a) and (8b)the displacement impedance function at the head of the 119895thpile segment can be derived in the local coordinates as follows

119885119895 (119904) =

minus119864119901

119895119860119901

119895 (1205971198801198951205971199111015840)

100381610038161003816100381610038161199111015840=0

119880119895

100381610038161003816100381610038161199111015840=0

= minus

119864119901

119895119860119901

119895

119897119895

times (

119872119895

119873119895

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos120601119895119899

+ 120582119895 +

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos120601119895119899)

times (

119872119895

119873119895

(1 +

infin

sum

119899=1

1205941015840119895119899 sin120601119895119899)

+

infin

sum

119899=1

12059410158401015840119895119899 sin120601119895119899)

minus1

(22)

where

119872119895

119873119895

= (

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) + 120582119895 cos 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[sin 120582119895 +infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899 + 120601119895119899)])

times (

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) minus 120582119895 sin 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[cos 120582119895 +infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899 + 120601119895119899)])

minus1

(23)

where 119885119895minus1(119904) denotes the displacement impedance functionat the head of the (119895 minus 1)th pile segmentwhich can be obtainedby using boundary conditionsThen following themethod ofrecursion typically used in the transfer function techniquethe displacement impedance function at the head of pile canbe derived as

119885119898 (119904) =

minus119864119901119898119860119901119898 (120597119880119898120597119911

1015840)

100381610038161003816100381610038161199111015840=0

11988011989810038161003816100381610038161199111015840=0

= minus

119864119901119898119860119901119898

119897119898

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

minus1

(24)

where

119872119898

119873119898

= minus(

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) + 120582119898 cos 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[sin 120582119898 +infin

sum

119899=1

12059410158401015840119898119899 sin (120573119898119899 + 120601119898119899)])

times (

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) minus 120582119898 sin 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[cos 120582119898 +infin

sum

119899=1

1205941015840119898119899 sin (120573119898119899 + 120601119898119899)])

minus1

1205941015840119898119899 = 120594119898119899 [

cos (120573119898119899 + 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 + 120582119898

+

cos (120573119898119899 minus 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 minus 120582119898

]

12059410158401015840119898119899 = 120594119898119899 [

sin (120573119898119899 + 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 + 120582119898

minus

sin (120573119898119899 minus 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 minus 120582119898

]

120594119898119899 =

(119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

120588119901119898119897119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898] 120593119898119899119871119898119899

120593119898119899 = 1198700 (120585119898119899119903119901119898)

+

2 (119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

1205881199011198981198972119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898]

119871119898119899 = int

119897119898

0

sin2 (1205731198981198991199111015840+ 120601119898119899) d119911

1015840

(25)

where 120582119898 = radicminus11990421199052119898(1 + (120578

119901119898119864119901119898) sdot 119904) 120573119898119899 = 120573119898119899119897119898 120585119898119899 =

120585119898119899119897119898 and 119903119901119898 = 119903119901119898119897119898 are all dimensionless parameters 119905119898 =

119897119898119881119901119898 denotes the propagation time of elastic longitudinal

Mathematical Problems in Engineering 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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 5

(7b) and (7c) are transformed by using the Laplace transformtechnique and can be rewritten as follows

[

[

(119896119904119895 + 120575119904119895 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) minus

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=0

= 0 (16a)

[

[

(119896119904119895minus1 + 120575

119904119895minus1 sdot 119904)

119864119904V119895

119882119895 (119903 1199111015840 119904) +

120597119882119895 (119903 1199111015840 119904)

1205971199111015840

]

]

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161199111015840=119897119895

= 0

(16b)

120590119895 (infin 1199111015840) = 0 119882119895 (infin 119911

1015840) = 0 (16c)

According to the properties of the modified Bessel func-tions that is if 119903 rarr infin then 119868119899(sdot) rarr infin and 119870119899(sdot) rarr 0it can be obtained that 119861119895 = 0 from boundary conditions(16c) By means of boundary conditions (16a) and (16b) thefollowing equation can be obtained

tan (120573119895119897119895)

= (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895 +

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895)120573119895119897119895

times ((120573119895119897119895)2minus (

119896119904119895 + 120575119904119895 sdot 119904

119864119904V119895

119897119895)(

119896119904119895minus1 + 120575

119904119895minus1 sdot 119904

119864119904V119895

119897119895))

minus1

=

(119870119895 + 119870

1015840

119895) 120573119895119897119895

(120573119895119897119895)2minus 119870119895119870

1015840

119895

(17)

where 119870119895 = ((119896119904119895 + 120575

119904119895 sdot 119904)119864

119904V119895)119897119895 and 119870

1015840

119895 = ((119896119904119895minus1 +

120575119904119895minus1 sdot 119904)119864

119904V119895)119897119895 denote the dimensionless complex stiffness

of the upper surface and lower surface of the 119895th soil layerSubstituting 119904 = 119894120596 into (17) and solving it by using bisectionmethod in the frequency domain a series of eigenvalues 120573119895119899can be derived 120585119895119899 can also be derived by substituting120573119895119899 into(14)

Then the vertical displacement of the 119895th soil layer can berewritten as

119882119895 (119903 1199111015840 119904) =

infin

sum

119899=1

1198601198951198991198700 (120585119895119899119903) sin (1205731198951198991199111015840+ 120601119895119899) (18)

where120601119895119899 = arctan(120573119895119899119897119895119870119895) and119860119895119899 are a series of constantsdetermined by the boundary conditions which reflect thedynamic interaction of soil and pile

32 Vibrations of the Pile Denoting119880119895(119911 119904) to be the Laplacetransform with respect to time of 119906119895(119911 119905) (5) is transformed

by using the Laplace transform technique and can be rewrit-ten combining with (6) and (18)

(119881119901

119895 )2(1 +

120578119901

119895

119864119901

119895

sdot 119904)

1205972119880119895

12059711991110158402

minus 1199042119880119895

minus

2120587119903119901

119895

120588119901

119895 119860119901

119895

(119866119904V119895 + 120578119904V119895 sdot 119904)

infin

sum

119899=1

1198601198951198991205851198951198991198701 (120585119895119899119903119901

119895 )

times sin (1205731198951198991199111015840+ 120601119895119899) = 0

(19)

It is not difficult to obtain that the general solution of (19)can be expressed as

119880119895 = 119872119895 [cos(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

+ 119873119895 [sin(1205821198951199111015840

119897119895

) +

infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899119911

1015840+ 120601119895119899)]

(20)

where

1205941015840119895119899 = 120594119895119899

[

[

cos (120573119895119899 + 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 + 120582119895

+

cos (120573119895119899 minus 120582119895 + 120601119895119899) minus cos120601119895119899120573119895119899 minus 120582119895

]

]

12059410158401015840119895119899 = 120594119895119899

[

[

sin (120573119895119899 + 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 + 120582119895

minus

sin (120573119895119899 minus 120582119895 + 120601119895119899) minus sin120601119895119899120573119895119899 minus 120582119895

]

]

120594119895119899 =

(119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 119897119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895] 120593119895119899119871119895119899

120593119895119899 = 1198700 (120585119895119899119903119901

119895 ) +

2 (119866119904V119895 + 120578119904119895 sdot 119904) 1205851198951198991198701 (120585119895119899119903

119901

119895 ) 1199052119895

120588119901

119895 1198972119895119903119901

119895 [1205732

119895119899 (1 + (120578119901

119895 119864119901

119895 ) sdot 119904) + 11990421199052119895]

119871119895119899 = int

119897119895

0

sin2 (1205731198951198991199111015840+ 120601119895119899) d119911

1015840

(21)

where 120582119895 = radicminus11990421199052119895(1 + (120578

119901

119895 119864119901

119895 ) sdot 119904) 120573119895119899 = 120573119895119899119897119895 120585119895119899 = 120585119895119899119897119895and 119903119901

119895 = 119903119901

119895 119897119895 are all dimensionless parameters 119905119895 = 119897119895119881119901

119895

denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment1198701(sdot) is the modified Bessel functions oforder one of the second kind

6 Mathematical Problems in Engineering

Combining with the boundary conditions (8a) and (8b)the displacement impedance function at the head of the 119895thpile segment can be derived in the local coordinates as follows

119885119895 (119904) =

minus119864119901

119895119860119901

119895 (1205971198801198951205971199111015840)

100381610038161003816100381610038161199111015840=0

119880119895

100381610038161003816100381610038161199111015840=0

= minus

119864119901

119895119860119901

119895

119897119895

times (

119872119895

119873119895

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos120601119895119899

+ 120582119895 +

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos120601119895119899)

times (

119872119895

119873119895

(1 +

infin

sum

119899=1

1205941015840119895119899 sin120601119895119899)

+

infin

sum

119899=1

12059410158401015840119895119899 sin120601119895119899)

minus1

(22)

where

119872119895

119873119895

= (

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) + 120582119895 cos 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[sin 120582119895 +infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899 + 120601119895119899)])

times (

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) minus 120582119895 sin 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[cos 120582119895 +infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899 + 120601119895119899)])

minus1

(23)

where 119885119895minus1(119904) denotes the displacement impedance functionat the head of the (119895 minus 1)th pile segmentwhich can be obtainedby using boundary conditionsThen following themethod ofrecursion typically used in the transfer function techniquethe displacement impedance function at the head of pile canbe derived as

119885119898 (119904) =

minus119864119901119898119860119901119898 (120597119880119898120597119911

1015840)

100381610038161003816100381610038161199111015840=0

11988011989810038161003816100381610038161199111015840=0

= minus

119864119901119898119860119901119898

119897119898

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

minus1

(24)

where

119872119898

119873119898

= minus(

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) + 120582119898 cos 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[sin 120582119898 +infin

sum

119899=1

12059410158401015840119898119899 sin (120573119898119899 + 120601119898119899)])

times (

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) minus 120582119898 sin 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[cos 120582119898 +infin

sum

119899=1

1205941015840119898119899 sin (120573119898119899 + 120601119898119899)])

minus1

1205941015840119898119899 = 120594119898119899 [

cos (120573119898119899 + 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 + 120582119898

+

cos (120573119898119899 minus 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 minus 120582119898

]

12059410158401015840119898119899 = 120594119898119899 [

sin (120573119898119899 + 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 + 120582119898

minus

sin (120573119898119899 minus 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 minus 120582119898

]

120594119898119899 =

(119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

120588119901119898119897119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898] 120593119898119899119871119898119899

120593119898119899 = 1198700 (120585119898119899119903119901119898)

+

2 (119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

1205881199011198981198972119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898]

119871119898119899 = int

119897119898

0

sin2 (1205731198981198991199111015840+ 120601119898119899) d119911

1015840

(25)

where 120582119898 = radicminus11990421199052119898(1 + (120578

119901119898119864119901119898) sdot 119904) 120573119898119899 = 120573119898119899119897119898 120585119898119899 =

120585119898119899119897119898 and 119903119901119898 = 119903119901119898119897119898 are all dimensionless parameters 119905119898 =

119897119898119881119901119898 denotes the propagation time of elastic longitudinal

Mathematical Problems in Engineering 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

6 Mathematical Problems in Engineering

Combining with the boundary conditions (8a) and (8b)the displacement impedance function at the head of the 119895thpile segment can be derived in the local coordinates as follows

119885119895 (119904) =

minus119864119901

119895119860119901

119895 (1205971198801198951205971199111015840)

100381610038161003816100381610038161199111015840=0

119880119895

100381610038161003816100381610038161199111015840=0

= minus

119864119901

119895119860119901

119895

119897119895

times (

119872119895

119873119895

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos120601119895119899

+ 120582119895 +

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos120601119895119899)

times (

119872119895

119873119895

(1 +

infin

sum

119899=1

1205941015840119895119899 sin120601119895119899)

+

infin

sum

119899=1

12059410158401015840119895119899 sin120601119895119899)

minus1

(22)

where

119872119895

119873119895

= (

infin

sum

119899=1

12059410158401015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) + 120582119895 cos 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[sin 120582119895 +infin

sum

119899=1

12059410158401015840119895119899 sin (120573119895119899 + 120601119895119899)])

times (

infin

sum

119899=1

1205941015840119895119899120573119895119899 cos (120573119895119899 + 120601119895119899) minus 120582119895 sin 120582119895

+

119885119895minus1 (119904) 119897119895

119864119901

119895119860119901

119895

[cos 120582119895 +infin

sum

119899=1

1205941015840119895119899 sin (120573119895119899 + 120601119895119899)])

minus1

(23)

where 119885119895minus1(119904) denotes the displacement impedance functionat the head of the (119895 minus 1)th pile segmentwhich can be obtainedby using boundary conditionsThen following themethod ofrecursion typically used in the transfer function techniquethe displacement impedance function at the head of pile canbe derived as

119885119898 (119904) =

minus119864119901119898119860119901119898 (120597119880119898120597119911

1015840)

100381610038161003816100381610038161199111015840=0

11988011989810038161003816100381610038161199111015840=0

= minus

119864119901119898119860119901119898

119897119898

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

minus1

(24)

where

119872119898

119873119898

= minus(

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) + 120582119898 cos 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[sin 120582119898 +infin

sum

119899=1

12059410158401015840119898119899 sin (120573119898119899 + 120601119898119899)])

times (

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos (120573119898119899 + 120601119898119899) minus 120582119898 sin 120582119898

+

119885119898minus1 (119904) 119897119898

119864119901119898119860119901119898

[cos 120582119898 +infin

sum

119899=1

1205941015840119898119899 sin (120573119898119899 + 120601119898119899)])

minus1

1205941015840119898119899 = 120594119898119899 [

cos (120573119898119899 + 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 + 120582119898

+

cos (120573119898119899 minus 120582119898 + 120601119898119899) minus cos120601119898119899120573119898119899 minus 120582119898

]

12059410158401015840119898119899 = 120594119898119899 [

sin (120573119898119899 + 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 + 120582119898

minus

sin (120573119898119899 minus 120582119898 + 120601119898119899) minus sin120601119898119899120573119898119899 minus 120582119898

]

120594119898119899 =

(119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

120588119901119898119897119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898] 120593119898119899119871119898119899

120593119898119899 = 1198700 (120585119898119899119903119901119898)

+

2 (119866119904V119898 + 120578

119904119898 sdot 119904) 1205851198981198991198701 (120585119898119899119903

119901119898) 1199052119898

1205881199011198981198972119898119903119901119898 [1205732

119898119899 (1 + (120578119901119898119864119901119898) sdot 119904) + 119904

21199052119898]

119871119898119899 = int

119897119898

0

sin2 (1205731198981198991199111015840+ 120601119898119899) d119911

1015840

(25)

where 120582119898 = radicminus11990421199052119898(1 + (120578

119901119898119864119901119898) sdot 119904) 120573119898119899 = 120573119898119899119897119898 120585119898119899 =

120585119898119899119897119898 and 119903119901119898 = 119903119901119898119897119898 are all dimensionless parameters 119905119898 =

119897119898119881119901119898 denotes the propagation time of elastic longitudinal

Mathematical Problems in Engineering 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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 7

wave in the 119898th pile segment 120601119898119899 and 120573119898119899 can be obtainedfrom the following equations

120601119898119899 = arctan(120573119898119899119897119898

119870119898

)

tan (120573119898119897119898) =(119870119898 + 119870

1015840

119898) 120573119898119897119898

(120573119898119897119898)2minus 119870119898119870

1015840

119898

(26)

where 119870119898 = ((119896119904119898 + 120575119904119898 sdot 119904)119864

119904V119898)119897119898 and 119870

1015840

119898 = ((119896119904119898minus1 + 120575

119904119898minus1 sdot

119904)119864119904V119898)119897119898 denote the dimensionless complex stiffness of the

upper surface and lower surface of the119898th soil layerThen the velocity transfer function at the head of pile can

be obtained as

119866V (119904) =119904

119885119898 (119904)

= minus

119897119898 sdot 119904

119864119901119898119860119901119898

(

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+ 120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(27)

Letting 119904 = 119894120596 and substituting it into (27) the velocityresponse in the frequency domain at the head of pile can beobtained as

119867V (119894120596) =119894120596

1198852 (119894120596)= minus

1

120588119901119898119860119901119898119881119901119898

1198671015840V (28)

where1198671015840V is the dimensionless velocity admittance at the pilehead which can be expressed as

1198671015840V = 119894120596119905119898 (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899)

+

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899

+120582119898 +

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

(29)

By means of the inverse Fourier transform and convolu-tion theorem the velocity response in the time domain at thehead of pile can be expressed as 119881(119905) = IFT[119876(120596)119867V(119894120596)]where 119876(120596) denotes the Fourier transform of 119902(119905) which isthe vertical excitation acting on the pile head

In particular the excitation can be regarded as a half-sinepulse in the nondestructive detection of pile foundation asfollows

119902 (119905) =

119876max sin(120587

119879

119905) 119905 isin (0 119879)

0 119905 ge 119879

(30)

where 119879 and 119876max denote the duration of the impulse orimpulse width and the maximum amplitude of the verticalexcitation respectively Then the velocity response in thetime domain at the head of pile can be expressed as

119881 (119905) = 119902 (119905) lowast IFT [119867V (119894120596)]

= IFT [119876 (119894120596) sdot 119867V (119894120596)] = minus

119876max

120588119901119898119860119901119898119881119901119898

1198811015840V

(31)

where1198811015840V is the dimensionless velocity response which can beexpressed as

1198811015840V =

1

2

int

infin

minusinfin

119894120596119905119898

times (

119872119898

119873119898

(1 +

infin

sum

119899=1

1205941015840119898119899 sin120601119898119899) +

infin

sum

119899=1

12059410158401015840119898119899 sin120601119898119899)

times (

119872119898

119873119898

infin

sum

119899=1

1205941015840119898119899120573119898119899 cos120601119898119899 + 120582119898

+

infin

sum

119899=1

12059410158401015840119898119899120573119898119899 cos120601119898119899)

minus1

times

119879

1205872minus 119879

21205962sdot (1 + 119890

minus119894120596119879) 119890119894120596 119905d120596

(32)

where 120596 = 119879119888120596 denotes the dimensionless frequency 119879denotes the dimensionless pulse width which should satisfy119879 = 119879119879119888 119905 denotes the dimensional time variable whichshould satisfy 119905 = 119905119879119888

4 Analysis of Vibration Characteristics

According to the derivation process shown in the previoussection it can be seen that the difference between the shearmodulus in the vertical plane and the shear modulus in thehorizontal plane reflects the soil anisotropy Therefore basedon the solutions the influence of these two kinds of shearmodulus of pile surrounding soil and pile end soil on thedynamic response of pile is studied in detail Unless otherwisespecified the length radius density and longitudinal wavevelocity of pile are 15m 05m 2500 kgm3 and 3800msrespectively The spring constant of the distributed Voigtmodel is equal to the elastic modulus of the lower soil layerand the damping coefficient of the distributed Voigt model is10000N sdotmminus3 sdot s

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

8 Mathematical Problems in Engineering

0 5 10 15 20

120596

minus15

minus10

minus05

00

05

10

15

20

05

10

15

20

H998400

2 3 4 5 6 7

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(a) Velocity admittance curves

20 22 24 26minus06

minus03

00

03

0 1 2 3 4 5 6 7

t

minus12

minus10

minus08

minus06

minus04

minus02

00

02

04

V998400

Gs2 = 20MPa

Gs2 = 40MPa

Gs2 = 60MPa

Gs2 = 80MPa

Gs2 = 100MPa

(b) Reflected wave signal curve

Figure 2 Influence of the shear modulus of pile surrounding soil in the vertical plane on the dynamic response of pile

41 Influence of the Anisotropy of Pile Surrounding Soil onthe Dynamic Response of Pile Firstly the influence of theshear modulus of pile surrounding soil in the vertical planeon the dynamic response of pile is investigated Parametersof pile end soil are as follows the thickness is three timesthat of pile diameter the soil density is 2000 kgm3 both theshear modulus in the vertical plane and the shear modulusin the horizontal plane are 120MPa both the Poissonrsquos ratioin the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035the damping coefficient is 1000N sdot mminus3 sdot s Parametersof pile surrounding soil are as follows the soil densityis 1800 kgm3 both the Poissonrsquos ratio in the horizontaldirection caused by the vertical stress and the Poissonrsquos ratioin the orthogonal direction of the horizontal strain causedby the horizontal stress are 04 the damping coefficient is1000N sdot mminus3 sdot s the shear modulus in the horizontal planeis 60MPa the shear modulus in the vertical plane is 119866119904V2 =20MPa 40MPa 60MPa 80MPa 100MPa respectively

Figure 2 shows the influence of the shear modulus ofpile surrounding soil in the vertical plane on the dynamicresponse of pile As shown in Figure 2(a) it can be notedthat the velocity admittance curves oscillate about a meanamplitude as the frequency increases As the shear modulusof pile surrounding soil in the vertical plane increases theamplitude of resonance peaks gradually decreases but theresonance frequency of velocity admittance curves almostremains unchanged As shown in Figure 2(b) it is observedthat the amplitude of the incident pulses and reflective wavesignals decreases with the increase of the shear modulus ofpile surrounding soil in the vertical plane As the shear mod-ulus of pile surrounding soil in the vertical plane increasesthe raising phenomenon between the incident pulses and the

primary reflective wave signals will be gradually aggravatedand the declining phenomenon between the primary reflec-tive wave signals and the secondary reflective wave signalswill also be gradually intensified

After that the influence of the shear modulus of pilesurrounding soil in the horizontal plane on the dynamicresponse of pile is studied Parameters of pile surroundingsoil are as follows the shear modulus in the vertical planeis 60MPa and the shear modulus in the horizontal planeis 119866119904ℎ2 = 20MPa 40MPa 60MPa 80MPa 100MPa respec-tively The other parameters of soil-pile system are the sameas those shown in the previous case

Figure 3 shows the influence of the shear modulus ofpile surrounding soil in the horizontal plane on the dynamicresponse of pile As shown in Figure 3(a) it can be seen thatthe amplitude of resonance peaks gradually increases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the change of resonance frequency canbe ignored As shown in Figure 3(b) it can be seen that theamplitude of the reflective wave signals decreases with theincrease of the shear modulus of pile surrounding soil in thehorizontal plane but the decreasing ratio is smallThe raisingphenomenon between the primary reflective wave signalsand the secondary reflective wave signals will be graduallyintensified with the increase of the shear modulus of pilesurrounding soil in the horizontal plane

42 Influence of the Anisotropy of Pile End Soil on the DynamicResponse of Pile In this section the influence of the shearmodulus of pile end soil in the vertical plane on the dynamicresponse of pile is firstly investigated Parameters of pilesurrounding soil are as follows the soil density is 1800 kgm3both the shear modulus in the horizontal plane and the shearmodulus in the vertical plane are 60MPa both the Poissonrsquos

Mathematical Problems in Engineering 9

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

06

08

10

12

14

H998400

(a) Velocity admittance curves

Gsh2 = 20MPa

Gsh2 = 40MPa

Gsh2 = 60MPa

Gsh2 = 80MPa

Gsh2 = 100MPa

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus03

minus02

minus01

00

01

02

03

V998400

(b) Reflected wave signal curves

Figure 3 Influence of the shear modulus of pile surrounding soil in the horizontal plane on the dynamic response of pile

ratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 04 thedamping coefficient is 1000N sdotmminus3 sdot s Parameters of pile endsoil are as follows the thickness is three times that of pilediameter the soil density is 2000 kgm3 both the Poissonrsquosratio in the horizontal direction caused by the vertical stressand the Poissonrsquos ratio in the orthogonal direction of thehorizontal strain caused by the horizontal stress are 035 thedamping coefficient is 1000N sdot mminus3 sdot s the shear modulusin the horizontal plane is 120MPa the shear modulus in thevertical plane is 119866119904V1 = 80MPa 100MPa 120MPa 140MPa160MPa respectively

Figure 4 shows the influence of the shear modulus ofpile end soil in the vertical plane on the dynamic responseof pile As shown in Figure 4(a) as the shear modulus ofpile end soil in the vertical plane increases the amplitude ofresonance peaks gradually decreases with the same resonancefrequency but the decreasing ratio is small As shown inFigure 4(b) it can be seen that the amplitude of the reflectivewave signals decreases with the increase of the shearmodulusof pile surrounding soil in the vertical plane

Then the influence of the shear modulus of pile endsoil in the horizontal plane on the dynamic response of pileis studied Parameters of pile end soil are as follows theshear modulus in the vertical plane is 120MPa and the shearmodulus in the horizontal plane is 119866119904ℎ1 = 80MPa 100MPa120MPa 140MPa 160MPa respectively The other param-eters of soil-pile system are the same as those shown in theprevious case

Figure 5 shows the influence of the shear modulus of pileend soil in the horizontal plane on the dynamic response ofpile It can be seen that the influence of the shear modulus of

pile end soil in the horizontal plane on the dynamic responseof pile can be ignored

5 Conclusions

By considering a pile embedded in layered transverselyisotropic soil as a dynamic soil-pile interaction problem thegoverning equations of soil-pile system are established whenthere is arbitrary vertical harmonic force acting on the pileheadThen an analytical solution for the velocity response inthe frequency domain and its corresponding semianalyticalsolution for the velocity response in the time domain havebeen derived by virtue of the transform technique and theseparation of variables technique An extensive parameterstudy has been undertaken to investigate the influence of thesoil anisotropy on the vertical dynamic response of pile andthe following conclusions have been obtained

(1) Whether for the pile surrounding soil or for the pileend soil it can be seen that the influence of theshear modulus in the vertical plane on the dynamicresponse of pile is more notable than the influenceof the shear modulus in the horizontal plane onthe dynamic response of pile Therefore the shearmodulus of soil in the vertical plane plays a leadingrole in the dynamic response of pile when only thevertical wave effect of soil is taken into account

(2) As the shear modulus of pile surrounding soil in thevertical plane increases both the amplitude of theresonance peaks of velocity admittance curves and thereflective wave signals of reflected wave signal curvesgradually decrease As the shear modulus of pilesurrounding soil in the horizontal plane increases the

10 Mathematical Problems in Engineering

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

H998400

07

08

09

10

11

12

13

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsv1 = 80MPa

Gsv1 = 160MPaGs

v1 = 100MPaGsv1 = 140MPa

Gsv1 = 120MPa

(b) Reflected wave signal curves

Figure 4 Influence of the shear modulus of pile end soil in the vertical plane on the dynamic response of pile

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

0 5 10 15 20

120596

2 3 4 5 6 700

02

04

06

08

10

12

14

07

08

09

10

11

12

13

H998400

(a) Velocity admittance curves

20 22 24 26

0 1 2 3 4 5 6 7

t

minus10

minus08

minus06

minus04

minus02

00

02

04

minus02

minus01

00

01

V998400

Gsh1 = 80MPa

Gsh1 = 160MPaGs

h1 = 100MPaGsh1 = 140MPa

Gsh1 = 120MPa

(b) Reflected wave signal curves

Figure 5 Influence of the shear modulus of pile end soil in the horizontal plane on the dynamic response of pile

amplitude of the resonance peaks of velocity admit-tance curves gradually increases but the reflectivewave signals of reflected wave signal curves graduallydecrease

(3) As the shear modulus of pile end soil in the verticalplane increases both the amplitude of the resonancepeaks of velocity admittance curves and the reflectivewave signals of reflected wave signal curves gradually

decrease The influence of the shear modulus of pileend soil in the horizontal plane on the dynamicresponse of pile can be ignored

Conflict of Interests

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

Mathematical Problems in Engineering 11

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grantnos 2012M521495 and 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M Novak and Y O Beredugo ldquoVertical vibration of embeddedfootingsrdquo Journal of the Soil Mechanics and Foundations Divi-sion vol 98 no 12 pp 1291ndash1310 1972

[2] T Nogami and K Konagai ldquoTime domain axial responseof dynamically loaded single pilesrdquo Journal of EngineeringMechanics ASCE vol 112 no 11 pp 1241ndash1252 1986

[3] 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

[4] 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

[5] O Michaelides G Gazetas G Bouckovalas and E ChrysikouldquoApproximate non-linear dynamic axial response of pilesrdquoGeotechnique vol 48 no 1 pp 33ndash53 1998

[6] D J Liu ldquoLongitudinal waves in piles with exponentially vary-ing cross sectionsrdquo Chinese Journal of Geotechnical Engineeringvol 30 no 7 pp 1066ndash1071 2008

[7] 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

[8] K A Kuo and H E M Hunt ldquoAn efficient model for thedynamic behaviour of a single pile in viscoelastic soilrdquo Journalof Sound and Vibration vol 332 no 10 pp 2549ndash2561 2013

[9] M Novak ldquoDynamic stiffness and damping of pilesrdquo CanadianGeotechnical Journal vol 11 no 4 pp 574ndash598 1974

[10] M Novak and F Aboul-Ella ldquoDynamic soil reaction for planestrain caserdquo Journal of the Engineering Mechanical Division vol104 no 4 pp 953ndash959 1978

[11] S M Mamoon and P K Banerjee ldquoTime-domain analysisof dynamically loaded single pilesrdquo Journal of EngineeringMechanics vol 118 no 1 pp 140ndash160 1992

[12] Y CHan ldquoDynamic vertical response of piles in nonlinear soilrdquoJournal of Geotechnical Engineering vol 123 no 8 pp 710ndash7161997

[13] M H El Naggar and M Novak ldquoNonlinear lateral interactionin pile dynamicsrdquo Soil Dynamics and Earthquake Engineeringvol 14 no 2 pp 141ndash157 1995

[14] M H El Naggar and M Novak ldquoNonlinear analysis fordynamic lateral pile responserdquo Soil Dynamics and EarthquakeEngineering vol 15 no 4 pp 233ndash244 1996

[15] 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

[16] W B Wu G S Jiang B Dou and C J Leo ldquoVertical dynamicimpedance of tapered pile considering compacting effectrdquoMathematical Problems in Engineering vol 2013 Article ID304856 p 9 2013

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

[18] R K N D Rajapakse Y Chen and T Senjuntichai ldquoElectroe-lastic field of a piezoelectric annular finite cylinderrdquo Interna-tional Journal of Solids and Structures vol 42 no 11-12 pp3487ndash3508 2005

[19] T Senjuntichai S Mani and R K N D Rajapakse ldquoVerticalvibration of an embedded rigid foundation in a poroelastic soilrdquoSoil Dynamics and Earthquake Engineering vol 26 no 6-7 pp626ndash636 2006

[20] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in saturated poroe-lastic mediumrdquo Computers and Geotechnics vol 35 no 3 pp450ndash458 2008

[21] C B Hu and X M Huang ldquoA quasi-analytical solution tosoil-pile interaction in longitudinal vibration in layered soilsconsidering vertical wave effect on soilsrdquo Journal of EarthquakeEngineering and Engineering Vibration vol 26 no 4 pp 205ndash211 2006

[22] L C Liu Q F Yan andX Yang ldquoVertical vibration of single pilein soil described by fractional derivative viscoelastic modelrdquoEngineering Mechanics vol 28 no 8 pp 177ndash182 2011

[23] A S Veletsos and K W Dotson ldquoVertical and torsionalvibration of foundations in inhomogeneous mediardquo Journal ofGeotechnical Engineering vol 114 no 9 pp 1002ndash1021 1988

[24] K W Dotson and A S Veletsos ldquoVertical and torsionalimpedances for radially inhomogeneous viscoelastic soil layersrdquoSoil Dynamics and Earthquake Engineering vol 9 no 3 pp 110ndash119 1990

[25] M H El Naggar ldquoVertical and torsional soil reactions forradially inhomogeneous soil layerrdquo Structural Engineering andMechanics vol 10 no 4 pp 299ndash312 2000

[26] H D Wang and S P Shang ldquoResearch on vertical dynamicresponse of single-pile in radially inhomogeneous soil duringthe passage of Rayleighwavesrdquo Journal of Vibration Engineeringvol 19 no 2 pp 258ndash264 2006

[27] 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

[28] S G Lekhnitskii Theory of Anisotropic Elastic Bodies Holden-day San Francisco Calif USA 1963

[29] Y M Tsai ldquoTorsional vibrations of a circular disk on an infinitetransversely isotropic mediumrdquo International Journal of Solidsand Structures vol 25 no 9 pp 1069ndash1076 1989

[30] M W Liu and M Novak ldquoDynamic response of single pilesembedded in transversely isotropic layered mediardquo EarthquakeEngineeringampStructuralDynamics vol 23 no 11 pp 1239ndash12571994

[31] R Chen C F Wan S T Xue and H S Tang ldquoDynamicimpedances of double piles in transversely isotropic layeredmediardquo Journal of Tongji University vol 31 no 2 pp 127ndash1312003

[32] G Chen Y Q Cai F Y Liu and H L Sun ldquoDynamic responseof a pile in a transversely isotropic saturated soil to transienttorsional loadingrdquoComputers and Geotechnics vol 35 no 2 pp165ndash172 2008

[33] K H Wang Z Q Zhang C J Leo and K H Xie ldquoDynamictorsional response of an end bearing pile in transverselyisotropic saturated soilrdquo Journal of Sound andVibration vol 327no 3ndash5 pp 440ndash453 2009

12 Mathematical Problems in Engineering

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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

[34] W B Wu K H Wang D Y Yang S J Ma and B NMa ldquoLongitudinal dynamic response to the pile embedded inlayered soil based on fictitious soil pile modelrdquo China Journal ofHighway and Transport vol 25 no 2 pp 72ndash80 2012

[35] H J Ding Transversely Isotropic Elastic Mechanics ZhejiangUniversity Publishing House Hangzhou China 1997

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