Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 304856 9 pageshttpdxdoiorg1011552013304856

Research ArticleVertical Dynamic Impedance of Tapered Pile consideringCompacting Effect

Wenbing Wu1 Guosheng Jiang1 Bin Dou1 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 Penrith SydneyNSW 2751 Australia

Correspondence should be addressed to Wenbing Wu zjuwwb1126163com

Received 7 July 2013 Revised 12 October 2013 Accepted 20 October 2013

Academic Editor Mohammad Younis

Copyright copy 2013 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

Based on complex stiffness transfermodel the vertical vibration of tapered pile embedded in layered soil is theoretically investigatedby considering the compacting effect of the soil layer surrounding the tapered pile in the piling process Allowing for thestratification of the surrounding soil and variable crosssection of the tapered pile the pile-soil system is discretized into finitesegments By virtue of the complex stiffness transfer model to simulate the compacting effect the complex stiffness of different soilsegments surrounding the tapered pile is obtained Then substituting the complex stiffness into the vertical dynamic governingequation of tapered pile the analytical solution of vertical dynamic impedance of tapered pile under vertical exciting force is derivedby means of the Laplace technique and impedance function transfer method Based on the presented solutions the influence ofcompacting effect of surrounding soil on vertical dynamic impedance at the pile head is investigated within the low frequency rangeconcerned in the design of dynamic foundation

1 Introduction

The influence of compacting effect on the bearing capacityof jacked pile is always a focus problem in the field ofgeotechnical engineering and many researchers have paidtheir attention to investigate this problem systematically fromtwo aspects The one is to study the development processof displacement field affected by compacting effect and theinfluence of soil parameters on this development process Forinstance Randolph et al [1 2] observed the deformation ofsurrounding soil caused by pile driving process Luo et al[3] pointed out that the compacting effect of jacked pile wasclosely related to the interaction of pile and soil Lu et al[4] further indicated that the compacting effect was actuallyaffected by both the vertical pressure applied by pile driverand driving process of jacked pile Lu et al [5] obtained thevariation law of soil displacements in the horizontal directionand depth direction by means of model test to simulate thepiling process Gong and Li [6] analyzed the mechanicalproblems of piling process in saturated soft clay and presenteda systemic method to investigate the compacting effect of

jacked pile Luo et al [7] discussed the influence of frictioncondition of pile and soil interface on the displacementfield in piling process The other one is to investigate theinfluence of compacting effect on the bearing capacity andsediment of pile For example Peng et al [8] modified theexisting load transfer function method by considering thenonlinear properties of soil Zhang et al [9] pointed outthat the further consolidation of the surrounding soil maylead to larger settlement of pile foundation As can be seenfrom the above research results the compacting effect hasa significant influence on the surrounding environment andbearing capacity of pile Therefore it is of great realisticsignificance to reveal the mechanism of compacting effect

In recent years the tapered pile has been widely used inengineering due to its good bearing capacityThereforemanyresearchers have investigated the bearing capacity by meansof field tests and model tests [10 11] theoretical research [1213] and numerical analysis [14] Although the research on thebearing capacity of tapered pile has been very extensive andin-depth the investigation on the compacting effect caused

2 Mathematical Problems in Engineering

by pile driving is still insufficient especially the influenceof compacting effect on the vertical dynamic response oftapered pile In the existing literature there are only researchresults about the influence of compacting effect in pile drivingprocess on the bearing capacity and ground heave presentedby He et al [15] When investigating the vertical dynamicresponse of pile there are three kinds of modeling methodsfor pile that is circular cross-section pile with finite length[16] pile with variable section impedance [17] and circularuniform viscoelastic pile [18] The vertical dynamic responseof tapered pile is only studied by Wu et al [19] and Cai etal [20] by taking into account its variable section propertyFor the interaction models of pile and surrounding soilthere are four main models that is the dynamic Winklermodel [21] plane strain model [22 23] three-dimensionalcontinuum model not considering the radial displacement[24] and three-dimensional continuum model taking intoaccount true three-dimensional wave effect of soil [25] Inorder to consider the construction effect of pile the complexstiffness transfer model is proposed [26 27] It can be seenthat the research on the vertical coupling vibration of pile andsoil varies broadly in complexity However the research onvertical dynamic response of tapered pile is still insufficient

In light of this in order to extend the application oftapered pile this paper has introduced the complex stiffnesstransfer model [27] to simulate the radially inhomogeneousproperty of surrounding soil caused by compacting effectThen an analytical solution of vertical dynamic impedanceat tapered pile head is derived Based on this solutionthe influence of compacting effect on the vertical dynamicimpedance at the tapered pile head is studied in detail

2 Mathematical Model and Assumptions

21 Computational Model The problem studied in this paperis the vertical vibration of tapered pipe embedded in layeredsoil when the compacting effect is taken into account Theschematic of pile-soil interaction model is shown in Figure 1119876(119905) represents the harmonic force acting on the pile headThe modeling process of tapered pile can be seen in [19]where 119903119895 = 119903119901 + (119867119898)(119895 minus 1) tan 120579 denotes the radius ofthe 119895th tapered pile segment119867 120579 and 119903119901 denote the lengthcone angle and pile end radius of tapered pile respectively120588119901119895 119881119901119895 and 119860119901119895 represent the density elastic longitudinalwave velocity and cross-sectional area of the 119895th tapered pilesegment respectively The thickness of the 119895th soil layer isdenoted by 119897119895 and the depth of the top surface of the 119895thsoil layer is denoted by ℎ119895 The tapered pile segment can beassumed to be cylinder and the properties of pile are assumedto be homogeneous within each segment when the segmentlength is small enough [19] but may vary from segment tosegment

As can be seen from the existing results the pile sur-rounding soil is compacted in the piling process This com-paction may strengthen the shear modulus of surroundingsoil or change the density of surrounding soil which causesthe radial inhomogeneity of surrounding soil of taperedpile The specific modeling process can be shown as followsthe surrounding soil is divided into two annular zones an

outer semi-infinite undisturbed zone and an inner disturbedzone of width 119887 The outer zone medium is homogeneousisotropic and viscoelastic with frequency independent mate-rial damping In order to consider the compacting effectthe inner zone is further divided into 119899 thin concentricannular subzones numbered by 1 2 119896 119899 and theouter radius of the 119896th subzone within the 119895th soil layer isdenoted by 119887119895119896 The soil medium of each subzone is assumedto be homogeneous The dynamic shear interaction on theinterface of the 119895th surrounding soil layer and tapered pilesegment shaft is described by vertical shear complex stiffness1198701198951 which can be obtained by means of complex stiffnesstransfer model

22 Underlying Assumptions The pile-soil model is devel-oped on the basis of the following assumptions

(1) The tapered pile is viscoelastic and vertical with a vari-able circular cross-section and the radius decreaseslinearly along the pile length direction The supportat the bottom of the tapered pile is viscoelastic

(2) The surrounding soil is isotropic linear viscoelasticmedium with a hysteretic-type damping regardlessof the frequency and it has a perfect contact withthe tapered pile during vibration The support at thebottom of the surrounding soil is also viscoelastic

(3) The surrounding soil is infinite in the radial directionand its free top surface has no normal or shear stress

(4) The dynamic stress of surrounding soil transfers tothe outer shaft of tapered pile through the complexstiffness on the contact interface of pile-soil system

3 Governing Equations and Their Solutions

31 Dynamic Equation of Soil and Its Solution The dynamicequilibrium equation of the 119896th subzone within the 119895th soillayer undergoing vertical deformation can be expressed as[22 23]

1199032d2119882119895119896d1199032

+ 119903d119882119895119896d119903

minus 12057311989511989621199032119882119895119896 = 0 (1)

where 120573119895119896 = 119894120596Vs119895119896radic1 + 119894119863s119895119896 Vs119895119896 denotes the shearwave velocity of the 119896th subzone within the 119895th soil layerand satisfies the equation Vs119895119896 = radic119866s119895119896120588s119895119896 120588s119895119896 119866s119895119896119863s119895119896 and 119882119895119896 = 119882119895119896(119903) denote the density shear modulusmaterial damping and vertical displacement of the 119896thsubzone within the 119895th soil layer respectively 120596 denotes thecircular frequency which satisfies the equation 120596 = 2120587119891 and119891 is the general frequency

The general solution of (1) can be expressed as

119882119895119896 (119903) = 1198601198951198961198700 (120573119895119896119903) + 1198611198951198961198680 (120573119895119896119903) (2)

where 1198680(sdot) and 1198700(sdot) denote the modified Bessel functionsof order zero of the first and second kinds respectively 119860119895119896and 119861119895119896 are undetermined constants which can be obtainedby means of boundary conditions

Mathematical Problems in Engineering 3

H

Tapered pile

o

z

r

Disturbed area

Undisturbed area

Q(t)

120579

kb120578b

2rp

(a) Schematic of pile-soil system

1

2

m

j

Q(t)

hj

lj

m minus 1

kb120578b

uarrfj(z t) uarr

(b) Schematic of pile element

Figure 1 Schematic of pile-soil interaction model

(1) Vertical Shear Complex Stiffness at the Interface of Inner-Outer Zones In the outer semi-infinite undisturbed zone thevertical displacement diminishes as 119903 approaches infinity sothat 119861119899119896 = 0 Thus using the same method as proposedin [27] the vertical shear complex stiffness at the interfacebetween the inner and outer zones can be derived as follows

119870119895119899 =minus2120587119887119895119899120591119895119899 (119887119895119899)

119882119895119899 (119887119895119899)

=2120587119887119895119899119866119904119895119899

lowast1205731198951198991198701 (120573119895119899119887119895119899)

1198700 (120573119895119899119887119895119899)

(3)

where 120573119895119899 = 119894120596V119904119895119899radic1 + 119894119863119904119895119899 Vs119895119899 denotes the shear wavevelocity of the outer semi-infinite undisturbed zone withinthe 119895th soil layer 119866s119895119899

lowast= 119866s119895119899(1 + 119894119863s119895119899) 1198701(sdot) denotes the

modified Bessel function of order one of the second kind(2) Vertical Shear Complex Stiffness of the Inner Zones Sincethe vertical shear complex stiffness of surrounding soil isdefined as the ratio of the vertical shear force to the verticaldisplacement at the same position for the 119896th subzone withinthe 119895th inner soil layer the vertical shear complex stiffness atthe outer boundary of the 119896th subzone (119903 = 119887119895119896) namely119870119895119896can be easily derived from (2) and can be expressed as

119870119895119896 =2120587119887119895119896120573119895119896119866

lowasts119895119896 [1198601198951198961198701 (120573119895119896119887119895119896) minus 1198611198951198961198681 (120573119895119896119887119895119896)]

1198601198951198961198700 (120573119895119896119887119895119896) + 1198611198951198961198680 (120573119895119896119887119895119896)

(4)

where 119860119895119896 and 119861119895119896 are undetermined constants of the 119896thsubzone within the 119895th inner soil layer 1198681(sdot) denotes the

modified Bessel function of order one of the one kind Then(4) can be rewritten as follows

119860119895119896

119861119895119896=

2120587119887119895119896120573119895119896119866lowast1199041198951198961198681 (120573119895119896119887119895119896) + 1198701198951198961198680 (120573119895119896119887119895119896)

2120587119887119895119896120573119895119896119866lowast119904119895119896

1198701 (120573119895119896119887119895119896) minus 1198701198951198961198700 (120573119895119896119887119895119896) (5)

In the same manner the vertical shear complex stiffnessat the outer boundary of the (119896minus1)th subzone (119903 = 119887119895119896minus1) canbe derived as

119870119895119896minus1 =2120587119887119895119896minus1120573119895119896119866

lowast119904119895119896 [1198601198951198961198701 (120573119895119896119887119895119896minus1) minus 1198611198951198961198681 (120573119895119896119887119895119896minus1)]

1198601198951198961198700 (120573119895119896119887119895119896minus1) + 1198611198951198961198680 (120573119895119896119887119895119896minus1)

(6)

Substituting (5) into (6) 119870119895119896minus1 can be further rewrittenas follows

119870119895119896minus1 =2120587119887119895119896minus1120573119895119896119866

lowast119904119895119896 [1198621198951198961198701 (120573119895119896119887119895119896minus1) minus 1198631198951198961198681 (120573119895119896119887119895119896minus1)]

1198621198951198961198700 (120573119895119896119887119895119896minus1) + 1198631198951198961198680 (120573119895119896119887119895119896minus1)

(7)

where

119862119895119896 = 2120587119887119895119896120573119895119896119866lowast1199041198951198961198681 (120573119895119896119887119895119896) + 1198680 (120573119895119896119887119895119896)119870119895119896

119863119895119896 = 2120587119887119895119896120573119895119896119866lowast1199041198951198961198701 (120573119895119896119887119895119896) minus 1198700 (120573119895119896119887119895119896)119870119895119896

(8)

By now the recursion formula of the vertical shearcomplex stiffness of the inner zone has been derived in aclosed formThen according to the stiffness of the outer zoneas presented in (3) and the recursion formula (7) it is easyto obtain the vertical shear complex stiffness of the innerdisturbed zones

4 Mathematical Problems in Engineering

32 Dynamic Equation of Tapered Pile and Its SolutionDenoting 119906119895(119911 119905) to be the vertical displacement of the 119895thpile segment the frictional force of the 119895th soil layer acting onthe surface of the 119895th pile segment shaft can be expressed as119891119895(119911 119905) = 1198701198951119906119895(119911 119905) Then the dynamic equation of the 119895thpile segment can be derived as

119864119901119895119860119901119895

1205972119906119895 (119911 119905)

1205971199112+ 119860119901119895120575119901119895

1205973119906119895 (119911 119905)

1205971199112120597119905

minus 119898119901119895

1205972119906119895 (119911 119905)

1205971199052minus 119891119895 (119911 119905) = 0

(9)

where 119864119901119895 119860119901119895 = 1205871199031198952 119898119901119895 and 120575119901119895 denote the elastic

modulus cross-section area mass per unit length andviscous damping of the 119895th pile segment respectively

Boundary conditions at the top and the bottom of taperedpile can be expressed as

[119864119901119898119860119901119898120597119906119898 (119911 119905)

120597119911+ 119860119901119898120575119901119898

1205972119906119898 (119911 119905)

120597119911120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=0

= minus119876 (119905)

[11986411990111205971199061 (119911 119905)

120597119911+1205751199011

12059721199061 (119911 119905)

120597119911120597119905+1198961198871199061 (119911 119905)+120578119887

1205971199061 (119911 119905)

120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=119867

= 0

(10)

where 119896119887 and 120578119887 represent the distributed spring and dampingcoefficient in unit area at the bottom of the tapered pilerespectively The value of 119896119887 and 120578119887 can be obtained using thefollowing equations [28]

119896119887 =4120588119887119904119881

2119887119904119903119901

(1 minus 120592)

120578119887 =34120588119887119904119881119887119904119903

2119901

(1 minus 120592)

(11)

where 120588119887119904 119881119887119904 120592 and 119903119901 denote the density shear wavevelocity Poissonrsquos ratio and pile end radius of tapered pilerespectively

Continuity conditions of the interface of the adjacenttapered pile segments can be founded as

119906119895 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

= 119906119895+1 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

[119864119901119895119860119901119895

120597119906119895 (119911 119905)

120597119911+ 119860119901119895120575119901119895

1205972119906119895 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= [119864119901(119895+1)119860119901(119895+1)

120597119906119895+1 (119911 119905)

120597119911+ 119860119901(119895+1)120575119901(119895+1)

1205972119906119895+1 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(12)

Initial conditions of the tapered pile segments can befounded as

119906119895 (119911 119905)10038161003816100381610038161003816119905=0

= 0

120597119906119895 (119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(13)

Denote 119880119895(119911 119904) = intinfin

0119906119895(119911 119905)119890

minus119904119905dt to be the Laplacetransform of 119906119895(119911 119905) with respect to 119905 Combining (9) withthe initial conditions (13) and applying the Laplace transformyield

1198811199011198952(1 +

120575119901119895

119864119901119895119904)

1205972119880119895 (119911 119904)

1205972119911minus (1199042+

1

1205881199011198951198601199011198951198701198951)119880119895 (119911 119904) = 0

(14)

where119881119901119895 120588119901119895 and 119864119901119895 denote the longitudinal wave velocitydensity and elastic modulus of the 119895th tapered pile segmentrespectively and 119864119901119895 = 120588119901119895119881

2119901119895 119904 is the Laplace transform

constant which should satisfy 119904 = 119894120596 during the numericalanalysis

It is not difficult to obtain that the general solution of (14)which can be expressed as

119880119895 (119911 119904) = 119864119895 cos(120582119895119911

119897119895) + 119865119895 sin(

120582119895119911

119897119895) (15)

where 120582119895 = radicminus(1199042 + 119870119895120588119901119895119860119901119895)1199052119895(1 + (120575119901119895119864119901119895)119904) is dimen-

sionless eigenvalue 119864119895 and 119865119895 are complex constants whichcan be obtained from boundary conditions 119905119895 = 119897119895119881119901119895denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment

Taking into consideration the boundary conditions thedisplacement impedance function at the head of the 119895th pilesegment can be expressed as

11988511990111989510038161003816100381610038161003816119911=ℎ119895

= minus120588119901119895119860119901119895119881119901119895 (1 + (120575119901119895119864119901119895) 119904) 120582119895 tan (120582119895 minus 120593119895)

119905119895

(16)

where 120593119895 = arctan (119885119901(119895minus1)119905119895120588119901119895119860119901119895119881119901119895120582119895(1 + (120575119901119895119864119901119895)119904))119885119901(119895minus1) is the displacement impedance function at the headof the (119895 minus 1)th pile segment which can be obtained by virtueof the boundary conditions

Then following themethod of recursion typically utilizedin the transfer function technique [18] the displacementimpedance function at the head of tapered pile can be derivedas

11988511990111989810038161003816100381610038161003816119911=ℎ119898

= minus120588119901119898119860119901119898119881119901119898 (1 + (120575119901119898119864119901119898) 119904) 120582119898 tan (120582119898 minus 120593119898)

119905119898

(17)

where 120593119898 = arctan (119885119901(119898minus1)119905119898120588119901119898119860119901119898119881119901119898120582119898(1 + (120575119901119898

119864119901119898)119904))As shown in (18) the displacement impedance function

can be further rewritten in terms of its real part and imaginarypart which represent the dynamic stiffness and dampingrespectively And the dynamic stiffness and damping reflectthe ability to resist the vertical deformation and the energydissipation

119885119901119898 = 119870119903 + 119894119862119894 (18)

Mathematical Problems in Engineering 5

4 Analysis of Vibration Characteristics

In order to focus on the influence of compacting effect on thedynamic response of tapered pile the surrounding soil is setto be homogeneous in the following analysis For practicalengineering it only needs to discretize the surroundingsoil into finite layers according to the actual conditionsParameters used here are as follows length density elasticlongitudinal wave velocity pile end radius and materialdamping of tapered pile are 10m 2500 kgm3 4000ms03m and 0 respectively As shown in [19] it is worth notingthat the calculation results may be convergent if the numberof the pile segment is more than 119898 = 200 Therefore inthe following section the tapered pile is divided into 200segments with the same length

41 Influence of the Number of Inner Disturbed Zone on theVertical Dynamic Impedance at the Pile Head Because thecore idea of this paper is using the complex stiffness transfermodel to simulate the radial inhomogeneity of surroundingsoil caused by compacting effect it needs to discuss theinfluence of the number of inner disturbed zone on thecalculation results Parameters used here are as follows Theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityof surrounding soil from outer undisturbed zone to innerdisturbed zone increases linearly from 150ms to 200msThe density of surrounding soil is 2000 kgm3 and the widthof inner disturbed zone is 119887 = 119903119901 respectively The numberof inner disturbed zone are 119899 = 2 5 10 15 20 50 100

Figure 2 shows the influence of the number of innerdisturbed zone on the vertical dynamic impedance at the pilehead As shown in Figure 2(a) it can be observed that thedynamic stiffness increases as the frequency increases at firstbut when the frequency exceeds a certain value the dynamicstiffness may decrease with the increase of frequency Asshown in Figure 2(b) it can be noted that the dynamicdampingmay increase linearly with the increase of frequencywhen it is beyond 5Hz It can also be seen from Figure 2that the computational results may be convergent whenthe number of inner disturbed zone is big enough for thecomputational curves tend to be convergent By repeatedlycalculating for different cases all the computational resultsmay be convergent when 119899 ge 20 Therefore unless otherwisespecified the number of inner disturbed zone is set to be119899 = 20 in the following analysis

42 Influence of the Coneangle of Tapered Pile on the VerticalDynamic Impedance at the Pile Head In this section theinfluence of coneangle on the vertical dynamic impedanceat the pile head is studied in detail by taking into accountthe compacting effect Parameters used here are as followsthe shear wave velocity density of surrounding soil andthe width of inner disturbed zone are the same as thoseshown in Section 41 The coneangle of tapered pile are 120579 =

0∘ 03∘ 05∘ 1∘ 15∘ and 2

∘Figure 3 shows the influence of coneangle of tapered

pile on the vertical dynamic impedance at the pile headwithin the low frequency range concerned in dynamicfoundation design As shown in Figure 3(a) the dynamic

stiffness increases with the increase of frequency when theconeangle is small (ie 120579 le 1

∘ as shown in this example)However when the coneangle is big enough (ie 120579 gt 1

∘

as shown in this example) it can be seen that the dynamicstiffness increases with the increase of frequency at firstbut the dynamic stiffness may decrease as the frequencyincreases when the frequency exceeds a certain value Itcan also be noted that the dynamic stiffness increases asthe coneangle increases at low frequencies whereas at highfrequencies the dynamic stiffness decreases with the increaseof coneangle This means that increasing the coneangle canimprove the ability to resist vertical deformation of taperedpile within a certain low frequency range when other designparameters of tapered pile remain unchanged Neverthelessif the frequency is big enough the ability to resist verticaldeformation of tapered pile cannot be improved by simplyincreasing the coneangle As shown in Figure 3(b) it can beobserved that the dynamic damping may increase linearlywith the increase of frequency when it is beyond 5Hz Itcan also be noted the dynamic damping increases as theconeangle increases for the same frequency which shows thatthe stress wave energy propagating in the tapered pile maydecay more quickly as the coneangle increases

43 Influence of the Compacting Range on the Vertical Dy-namic Impedance at the Pile Head The compacting rangeof surrounding soil is affected by the design parameters oftapered pile construction technology and properties of sur-rounding soil and so on So it needs to discuss the influenceof compacting range on the vertical dynamic impedanceat the pile head Parameters used here are as follows theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityand density of surrounding soil are the same as those shownin Section 41 The width of inner disturbed zone are 119887 =

0 05119903119901 1119903119901 15119903119901 2119903119901 and 25119903119901Figure 4 shows the influence of compacting range on the

vertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 4(a) when the compacting range is small(ie 119887 le 2119903119901 as shown in this example) the dynamicstiffness increases as the frequency increases within lowrange whereas the dynamic stiffness deceases if the frequencyis beyond a certain value However it can be seen that thedynamic stiffness increases with the increase of frequencywhen the compacting range is big enough (ie 119887 gt 2119903119901 asshown in this example) It can also be seen that the dynamicstiffness increases with the increase of compacting range atlow frequenciesWhen the frequency exceeds a certain valuethe dynamic stiffness increases with the increase of the com-pacting range at first and then decreases as the compactingrange further increases As shown in Figure 4(b) it can benoted the dynamic damping increases as the compactingrange increases for the same frequency which means that thestress wave energy propagating in the tapered pile may decaymore quickly as the compacting range increases

In order to investigate the influence of compacting rangeon the vertical dynamic impedance at the pile head moredetailedly it needs to discuss the influence of compactingrange changing within smaller scale on the vertical dynamic

6 Mathematical Problems in Engineering

0 10 20 30 40

Kr

(Nmiddotm

minus1)

f (Hz)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

0

6 times108

12 times109

18 times109

24 times109

3 times109

36 times109

0 10 20 30 40f (Hz)

0

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(b) Dynamic damping curves

Figure 2 Influence of the number of inner disturbed zone on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(b) Dynamic damping curves

Figure 3 Influence of coneangle of tapered pile on the vertical dynamic impedance at the pile head

impedance Parameters used here are as follows the width ofinner disturbed zone are 119887 = 0 005119903119901 01119903119901 02119903119901 03119903119901and 04119903119901 The other parameters are the same as those shownin the above analysis

Figure 5 shows the influence of compacting range chang-ing within smaller scale on the vertical dynamic impedanceat the pile head within the low frequency range concerned indynamic foundation design As shown in Figure 5(a) it canbe seen that the dynamic stiffness increases with the increaseof compacting range at low frequencies but the increaseratio is small When the frequency is beyond a certain valuethe dynamic stiffness decreases as the compacting rangeincreases As shown in Figure 5(b) it can be seen that theinfluence of compacting range changing within smaller scale

on the vertical dynamic impedance is the same as that whenthe compacting range changes within bigger scale

44 Influence of the Compacting Degree on the Vertical Dy-namic Impedance at the Pile Head Parameters used hereare as follows the coneangle of tapered pile is 120579 = 2

∘ andthe width of inner disturbed zone is 119887 = 119903119901 Denote shearwave velocity ratio to describe the compacting degree thatis 119902 represents the ratio of the shear wave velocity of innerdisturbed zone to that of outer undisturbed zone The shearwave velocity of outer undisturbed zone is 150ms and theshear wave velocity ratio are 119902 = 1 12 14 16 18 and 2 Theother parameters are the same as those shown in Section 41

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

by pile driving is still insufficient especially the influenceof compacting effect on the vertical dynamic response oftapered pile In the existing literature there are only researchresults about the influence of compacting effect in pile drivingprocess on the bearing capacity and ground heave presentedby He et al [15] When investigating the vertical dynamicresponse of pile there are three kinds of modeling methodsfor pile that is circular cross-section pile with finite length[16] pile with variable section impedance [17] and circularuniform viscoelastic pile [18] The vertical dynamic responseof tapered pile is only studied by Wu et al [19] and Cai etal [20] by taking into account its variable section propertyFor the interaction models of pile and surrounding soilthere are four main models that is the dynamic Winklermodel [21] plane strain model [22 23] three-dimensionalcontinuum model not considering the radial displacement[24] and three-dimensional continuum model taking intoaccount true three-dimensional wave effect of soil [25] Inorder to consider the construction effect of pile the complexstiffness transfer model is proposed [26 27] It can be seenthat the research on the vertical coupling vibration of pile andsoil varies broadly in complexity However the research onvertical dynamic response of tapered pile is still insufficient

In light of this in order to extend the application oftapered pile this paper has introduced the complex stiffnesstransfer model [27] to simulate the radially inhomogeneousproperty of surrounding soil caused by compacting effectThen an analytical solution of vertical dynamic impedanceat tapered pile head is derived Based on this solutionthe influence of compacting effect on the vertical dynamicimpedance at the tapered pile head is studied in detail

2 Mathematical Model and Assumptions

21 Computational Model The problem studied in this paperis the vertical vibration of tapered pipe embedded in layeredsoil when the compacting effect is taken into account Theschematic of pile-soil interaction model is shown in Figure 1119876(119905) represents the harmonic force acting on the pile headThe modeling process of tapered pile can be seen in [19]where 119903119895 = 119903119901 + (119867119898)(119895 minus 1) tan 120579 denotes the radius ofthe 119895th tapered pile segment119867 120579 and 119903119901 denote the lengthcone angle and pile end radius of tapered pile respectively120588119901119895 119881119901119895 and 119860119901119895 represent the density elastic longitudinalwave velocity and cross-sectional area of the 119895th tapered pilesegment respectively The thickness of the 119895th soil layer isdenoted by 119897119895 and the depth of the top surface of the 119895thsoil layer is denoted by ℎ119895 The tapered pile segment can beassumed to be cylinder and the properties of pile are assumedto be homogeneous within each segment when the segmentlength is small enough [19] but may vary from segment tosegment

As can be seen from the existing results the pile sur-rounding soil is compacted in the piling process This com-paction may strengthen the shear modulus of surroundingsoil or change the density of surrounding soil which causesthe radial inhomogeneity of surrounding soil of taperedpile The specific modeling process can be shown as followsthe surrounding soil is divided into two annular zones an

outer semi-infinite undisturbed zone and an inner disturbedzone of width 119887 The outer zone medium is homogeneousisotropic and viscoelastic with frequency independent mate-rial damping In order to consider the compacting effectthe inner zone is further divided into 119899 thin concentricannular subzones numbered by 1 2 119896 119899 and theouter radius of the 119896th subzone within the 119895th soil layer isdenoted by 119887119895119896 The soil medium of each subzone is assumedto be homogeneous The dynamic shear interaction on theinterface of the 119895th surrounding soil layer and tapered pilesegment shaft is described by vertical shear complex stiffness1198701198951 which can be obtained by means of complex stiffnesstransfer model

22 Underlying Assumptions The pile-soil model is devel-oped on the basis of the following assumptions

(1) The tapered pile is viscoelastic and vertical with a vari-able circular cross-section and the radius decreaseslinearly along the pile length direction The supportat the bottom of the tapered pile is viscoelastic

(2) The surrounding soil is isotropic linear viscoelasticmedium with a hysteretic-type damping regardlessof the frequency and it has a perfect contact withthe tapered pile during vibration The support at thebottom of the surrounding soil is also viscoelastic

(3) The surrounding soil is infinite in the radial directionand its free top surface has no normal or shear stress

(4) The dynamic stress of surrounding soil transfers tothe outer shaft of tapered pile through the complexstiffness on the contact interface of pile-soil system

3 Governing Equations and Their Solutions

31 Dynamic Equation of Soil and Its Solution The dynamicequilibrium equation of the 119896th subzone within the 119895th soillayer undergoing vertical deformation can be expressed as[22 23]

1199032d2119882119895119896d1199032

+ 119903d119882119895119896d119903

minus 12057311989511989621199032119882119895119896 = 0 (1)

where 120573119895119896 = 119894120596Vs119895119896radic1 + 119894119863s119895119896 Vs119895119896 denotes the shearwave velocity of the 119896th subzone within the 119895th soil layerand satisfies the equation Vs119895119896 = radic119866s119895119896120588s119895119896 120588s119895119896 119866s119895119896119863s119895119896 and 119882119895119896 = 119882119895119896(119903) denote the density shear modulusmaterial damping and vertical displacement of the 119896thsubzone within the 119895th soil layer respectively 120596 denotes thecircular frequency which satisfies the equation 120596 = 2120587119891 and119891 is the general frequency

The general solution of (1) can be expressed as

119882119895119896 (119903) = 1198601198951198961198700 (120573119895119896119903) + 1198611198951198961198680 (120573119895119896119903) (2)

where 1198680(sdot) and 1198700(sdot) denote the modified Bessel functionsof order zero of the first and second kinds respectively 119860119895119896and 119861119895119896 are undetermined constants which can be obtainedby means of boundary conditions

Mathematical Problems in Engineering 3

H

Tapered pile

o

z

r

Disturbed area

Undisturbed area

Q(t)

120579

kb120578b

2rp

(a) Schematic of pile-soil system

1

2

m

j

Q(t)

hj

lj

m minus 1

kb120578b

uarrfj(z t) uarr

(b) Schematic of pile element

Figure 1 Schematic of pile-soil interaction model

(1) Vertical Shear Complex Stiffness at the Interface of Inner-Outer Zones In the outer semi-infinite undisturbed zone thevertical displacement diminishes as 119903 approaches infinity sothat 119861119899119896 = 0 Thus using the same method as proposedin [27] the vertical shear complex stiffness at the interfacebetween the inner and outer zones can be derived as follows

119870119895119899 =minus2120587119887119895119899120591119895119899 (119887119895119899)

119882119895119899 (119887119895119899)

=2120587119887119895119899119866119904119895119899

lowast1205731198951198991198701 (120573119895119899119887119895119899)

1198700 (120573119895119899119887119895119899)

(3)

where 120573119895119899 = 119894120596V119904119895119899radic1 + 119894119863119904119895119899 Vs119895119899 denotes the shear wavevelocity of the outer semi-infinite undisturbed zone withinthe 119895th soil layer 119866s119895119899

lowast= 119866s119895119899(1 + 119894119863s119895119899) 1198701(sdot) denotes the

modified Bessel function of order one of the second kind(2) Vertical Shear Complex Stiffness of the Inner Zones Sincethe vertical shear complex stiffness of surrounding soil isdefined as the ratio of the vertical shear force to the verticaldisplacement at the same position for the 119896th subzone withinthe 119895th inner soil layer the vertical shear complex stiffness atthe outer boundary of the 119896th subzone (119903 = 119887119895119896) namely119870119895119896can be easily derived from (2) and can be expressed as

119870119895119896 =2120587119887119895119896120573119895119896119866

lowasts119895119896 [1198601198951198961198701 (120573119895119896119887119895119896) minus 1198611198951198961198681 (120573119895119896119887119895119896)]

1198601198951198961198700 (120573119895119896119887119895119896) + 1198611198951198961198680 (120573119895119896119887119895119896)

(4)

where 119860119895119896 and 119861119895119896 are undetermined constants of the 119896thsubzone within the 119895th inner soil layer 1198681(sdot) denotes the

modified Bessel function of order one of the one kind Then(4) can be rewritten as follows

119860119895119896

119861119895119896=

2120587119887119895119896120573119895119896119866lowast1199041198951198961198681 (120573119895119896119887119895119896) + 1198701198951198961198680 (120573119895119896119887119895119896)

2120587119887119895119896120573119895119896119866lowast119904119895119896

1198701 (120573119895119896119887119895119896) minus 1198701198951198961198700 (120573119895119896119887119895119896) (5)

In the same manner the vertical shear complex stiffnessat the outer boundary of the (119896minus1)th subzone (119903 = 119887119895119896minus1) canbe derived as

119870119895119896minus1 =2120587119887119895119896minus1120573119895119896119866

lowast119904119895119896 [1198601198951198961198701 (120573119895119896119887119895119896minus1) minus 1198611198951198961198681 (120573119895119896119887119895119896minus1)]

1198601198951198961198700 (120573119895119896119887119895119896minus1) + 1198611198951198961198680 (120573119895119896119887119895119896minus1)

(6)

Substituting (5) into (6) 119870119895119896minus1 can be further rewrittenas follows

119870119895119896minus1 =2120587119887119895119896minus1120573119895119896119866

lowast119904119895119896 [1198621198951198961198701 (120573119895119896119887119895119896minus1) minus 1198631198951198961198681 (120573119895119896119887119895119896minus1)]

1198621198951198961198700 (120573119895119896119887119895119896minus1) + 1198631198951198961198680 (120573119895119896119887119895119896minus1)

(7)

where

119862119895119896 = 2120587119887119895119896120573119895119896119866lowast1199041198951198961198681 (120573119895119896119887119895119896) + 1198680 (120573119895119896119887119895119896)119870119895119896

119863119895119896 = 2120587119887119895119896120573119895119896119866lowast1199041198951198961198701 (120573119895119896119887119895119896) minus 1198700 (120573119895119896119887119895119896)119870119895119896

(8)

By now the recursion formula of the vertical shearcomplex stiffness of the inner zone has been derived in aclosed formThen according to the stiffness of the outer zoneas presented in (3) and the recursion formula (7) it is easyto obtain the vertical shear complex stiffness of the innerdisturbed zones

4 Mathematical Problems in Engineering

32 Dynamic Equation of Tapered Pile and Its SolutionDenoting 119906119895(119911 119905) to be the vertical displacement of the 119895thpile segment the frictional force of the 119895th soil layer acting onthe surface of the 119895th pile segment shaft can be expressed as119891119895(119911 119905) = 1198701198951119906119895(119911 119905) Then the dynamic equation of the 119895thpile segment can be derived as

119864119901119895119860119901119895

1205972119906119895 (119911 119905)

1205971199112+ 119860119901119895120575119901119895

1205973119906119895 (119911 119905)

1205971199112120597119905

minus 119898119901119895

1205972119906119895 (119911 119905)

1205971199052minus 119891119895 (119911 119905) = 0

(9)

where 119864119901119895 119860119901119895 = 1205871199031198952 119898119901119895 and 120575119901119895 denote the elastic

modulus cross-section area mass per unit length andviscous damping of the 119895th pile segment respectively

Boundary conditions at the top and the bottom of taperedpile can be expressed as

[119864119901119898119860119901119898120597119906119898 (119911 119905)

120597119911+ 119860119901119898120575119901119898

1205972119906119898 (119911 119905)

120597119911120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=0

= minus119876 (119905)

[11986411990111205971199061 (119911 119905)

120597119911+1205751199011

12059721199061 (119911 119905)

120597119911120597119905+1198961198871199061 (119911 119905)+120578119887

1205971199061 (119911 119905)

120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=119867

= 0

(10)

where 119896119887 and 120578119887 represent the distributed spring and dampingcoefficient in unit area at the bottom of the tapered pilerespectively The value of 119896119887 and 120578119887 can be obtained using thefollowing equations [28]

119896119887 =4120588119887119904119881

2119887119904119903119901

(1 minus 120592)

120578119887 =34120588119887119904119881119887119904119903

2119901

(1 minus 120592)

(11)

where 120588119887119904 119881119887119904 120592 and 119903119901 denote the density shear wavevelocity Poissonrsquos ratio and pile end radius of tapered pilerespectively

Continuity conditions of the interface of the adjacenttapered pile segments can be founded as

119906119895 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

= 119906119895+1 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

[119864119901119895119860119901119895

120597119906119895 (119911 119905)

120597119911+ 119860119901119895120575119901119895

1205972119906119895 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= [119864119901(119895+1)119860119901(119895+1)

120597119906119895+1 (119911 119905)

120597119911+ 119860119901(119895+1)120575119901(119895+1)

1205972119906119895+1 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(12)

Initial conditions of the tapered pile segments can befounded as

119906119895 (119911 119905)10038161003816100381610038161003816119905=0

= 0

120597119906119895 (119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(13)

Denote 119880119895(119911 119904) = intinfin

0119906119895(119911 119905)119890

minus119904119905dt to be the Laplacetransform of 119906119895(119911 119905) with respect to 119905 Combining (9) withthe initial conditions (13) and applying the Laplace transformyield

1198811199011198952(1 +

120575119901119895

119864119901119895119904)

1205972119880119895 (119911 119904)

1205972119911minus (1199042+

1

1205881199011198951198601199011198951198701198951)119880119895 (119911 119904) = 0

(14)

where119881119901119895 120588119901119895 and 119864119901119895 denote the longitudinal wave velocitydensity and elastic modulus of the 119895th tapered pile segmentrespectively and 119864119901119895 = 120588119901119895119881

2119901119895 119904 is the Laplace transform

constant which should satisfy 119904 = 119894120596 during the numericalanalysis

It is not difficult to obtain that the general solution of (14)which can be expressed as

119880119895 (119911 119904) = 119864119895 cos(120582119895119911

119897119895) + 119865119895 sin(

120582119895119911

119897119895) (15)

where 120582119895 = radicminus(1199042 + 119870119895120588119901119895119860119901119895)1199052119895(1 + (120575119901119895119864119901119895)119904) is dimen-

sionless eigenvalue 119864119895 and 119865119895 are complex constants whichcan be obtained from boundary conditions 119905119895 = 119897119895119881119901119895denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment

Taking into consideration the boundary conditions thedisplacement impedance function at the head of the 119895th pilesegment can be expressed as

11988511990111989510038161003816100381610038161003816119911=ℎ119895

= minus120588119901119895119860119901119895119881119901119895 (1 + (120575119901119895119864119901119895) 119904) 120582119895 tan (120582119895 minus 120593119895)

119905119895

(16)

where 120593119895 = arctan (119885119901(119895minus1)119905119895120588119901119895119860119901119895119881119901119895120582119895(1 + (120575119901119895119864119901119895)119904))119885119901(119895minus1) is the displacement impedance function at the headof the (119895 minus 1)th pile segment which can be obtained by virtueof the boundary conditions

Then following themethod of recursion typically utilizedin the transfer function technique [18] the displacementimpedance function at the head of tapered pile can be derivedas

11988511990111989810038161003816100381610038161003816119911=ℎ119898

= minus120588119901119898119860119901119898119881119901119898 (1 + (120575119901119898119864119901119898) 119904) 120582119898 tan (120582119898 minus 120593119898)

119905119898

(17)

where 120593119898 = arctan (119885119901(119898minus1)119905119898120588119901119898119860119901119898119881119901119898120582119898(1 + (120575119901119898

119864119901119898)119904))As shown in (18) the displacement impedance function

can be further rewritten in terms of its real part and imaginarypart which represent the dynamic stiffness and dampingrespectively And the dynamic stiffness and damping reflectthe ability to resist the vertical deformation and the energydissipation

119885119901119898 = 119870119903 + 119894119862119894 (18)

Mathematical Problems in Engineering 5

4 Analysis of Vibration Characteristics

In order to focus on the influence of compacting effect on thedynamic response of tapered pile the surrounding soil is setto be homogeneous in the following analysis For practicalengineering it only needs to discretize the surroundingsoil into finite layers according to the actual conditionsParameters used here are as follows length density elasticlongitudinal wave velocity pile end radius and materialdamping of tapered pile are 10m 2500 kgm3 4000ms03m and 0 respectively As shown in [19] it is worth notingthat the calculation results may be convergent if the numberof the pile segment is more than 119898 = 200 Therefore inthe following section the tapered pile is divided into 200segments with the same length

41 Influence of the Number of Inner Disturbed Zone on theVertical Dynamic Impedance at the Pile Head Because thecore idea of this paper is using the complex stiffness transfermodel to simulate the radial inhomogeneity of surroundingsoil caused by compacting effect it needs to discuss theinfluence of the number of inner disturbed zone on thecalculation results Parameters used here are as follows Theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityof surrounding soil from outer undisturbed zone to innerdisturbed zone increases linearly from 150ms to 200msThe density of surrounding soil is 2000 kgm3 and the widthof inner disturbed zone is 119887 = 119903119901 respectively The numberof inner disturbed zone are 119899 = 2 5 10 15 20 50 100

Figure 2 shows the influence of the number of innerdisturbed zone on the vertical dynamic impedance at the pilehead As shown in Figure 2(a) it can be observed that thedynamic stiffness increases as the frequency increases at firstbut when the frequency exceeds a certain value the dynamicstiffness may decrease with the increase of frequency Asshown in Figure 2(b) it can be noted that the dynamicdampingmay increase linearly with the increase of frequencywhen it is beyond 5Hz It can also be seen from Figure 2that the computational results may be convergent whenthe number of inner disturbed zone is big enough for thecomputational curves tend to be convergent By repeatedlycalculating for different cases all the computational resultsmay be convergent when 119899 ge 20 Therefore unless otherwisespecified the number of inner disturbed zone is set to be119899 = 20 in the following analysis

42 Influence of the Coneangle of Tapered Pile on the VerticalDynamic Impedance at the Pile Head In this section theinfluence of coneangle on the vertical dynamic impedanceat the pile head is studied in detail by taking into accountthe compacting effect Parameters used here are as followsthe shear wave velocity density of surrounding soil andthe width of inner disturbed zone are the same as thoseshown in Section 41 The coneangle of tapered pile are 120579 =

0∘ 03∘ 05∘ 1∘ 15∘ and 2

∘Figure 3 shows the influence of coneangle of tapered

pile on the vertical dynamic impedance at the pile headwithin the low frequency range concerned in dynamicfoundation design As shown in Figure 3(a) the dynamic

stiffness increases with the increase of frequency when theconeangle is small (ie 120579 le 1

∘ as shown in this example)However when the coneangle is big enough (ie 120579 gt 1

∘

as shown in this example) it can be seen that the dynamicstiffness increases with the increase of frequency at firstbut the dynamic stiffness may decrease as the frequencyincreases when the frequency exceeds a certain value Itcan also be noted that the dynamic stiffness increases asthe coneangle increases at low frequencies whereas at highfrequencies the dynamic stiffness decreases with the increaseof coneangle This means that increasing the coneangle canimprove the ability to resist vertical deformation of taperedpile within a certain low frequency range when other designparameters of tapered pile remain unchanged Neverthelessif the frequency is big enough the ability to resist verticaldeformation of tapered pile cannot be improved by simplyincreasing the coneangle As shown in Figure 3(b) it can beobserved that the dynamic damping may increase linearlywith the increase of frequency when it is beyond 5Hz Itcan also be noted the dynamic damping increases as theconeangle increases for the same frequency which shows thatthe stress wave energy propagating in the tapered pile maydecay more quickly as the coneangle increases

43 Influence of the Compacting Range on the Vertical Dy-namic Impedance at the Pile Head The compacting rangeof surrounding soil is affected by the design parameters oftapered pile construction technology and properties of sur-rounding soil and so on So it needs to discuss the influenceof compacting range on the vertical dynamic impedanceat the pile head Parameters used here are as follows theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityand density of surrounding soil are the same as those shownin Section 41 The width of inner disturbed zone are 119887 =

0 05119903119901 1119903119901 15119903119901 2119903119901 and 25119903119901Figure 4 shows the influence of compacting range on the

vertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 4(a) when the compacting range is small(ie 119887 le 2119903119901 as shown in this example) the dynamicstiffness increases as the frequency increases within lowrange whereas the dynamic stiffness deceases if the frequencyis beyond a certain value However it can be seen that thedynamic stiffness increases with the increase of frequencywhen the compacting range is big enough (ie 119887 gt 2119903119901 asshown in this example) It can also be seen that the dynamicstiffness increases with the increase of compacting range atlow frequenciesWhen the frequency exceeds a certain valuethe dynamic stiffness increases with the increase of the com-pacting range at first and then decreases as the compactingrange further increases As shown in Figure 4(b) it can benoted the dynamic damping increases as the compactingrange increases for the same frequency which means that thestress wave energy propagating in the tapered pile may decaymore quickly as the compacting range increases

In order to investigate the influence of compacting rangeon the vertical dynamic impedance at the pile head moredetailedly it needs to discuss the influence of compactingrange changing within smaller scale on the vertical dynamic

6 Mathematical Problems in Engineering

0 10 20 30 40

Kr

(Nmiddotm

minus1)

f (Hz)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

0

6 times108

12 times109

18 times109

24 times109

3 times109

36 times109

0 10 20 30 40f (Hz)

0

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(b) Dynamic damping curves

Figure 2 Influence of the number of inner disturbed zone on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(b) Dynamic damping curves

Figure 3 Influence of coneangle of tapered pile on the vertical dynamic impedance at the pile head

impedance Parameters used here are as follows the width ofinner disturbed zone are 119887 = 0 005119903119901 01119903119901 02119903119901 03119903119901and 04119903119901 The other parameters are the same as those shownin the above analysis

Figure 5 shows the influence of compacting range chang-ing within smaller scale on the vertical dynamic impedanceat the pile head within the low frequency range concerned indynamic foundation design As shown in Figure 5(a) it canbe seen that the dynamic stiffness increases with the increaseof compacting range at low frequencies but the increaseratio is small When the frequency is beyond a certain valuethe dynamic stiffness decreases as the compacting rangeincreases As shown in Figure 5(b) it can be seen that theinfluence of compacting range changing within smaller scale

on the vertical dynamic impedance is the same as that whenthe compacting range changes within bigger scale

44 Influence of the Compacting Degree on the Vertical Dy-namic Impedance at the Pile Head Parameters used hereare as follows the coneangle of tapered pile is 120579 = 2

∘ andthe width of inner disturbed zone is 119887 = 119903119901 Denote shearwave velocity ratio to describe the compacting degree thatis 119902 represents the ratio of the shear wave velocity of innerdisturbed zone to that of outer undisturbed zone The shearwave velocity of outer undisturbed zone is 150ms and theshear wave velocity ratio are 119902 = 1 12 14 16 18 and 2 Theother parameters are the same as those shown in Section 41

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

H

Tapered pile

o

z

r

Disturbed area

Undisturbed area

Q(t)

120579

kb120578b

2rp

(a) Schematic of pile-soil system

1

2

m

j

Q(t)

hj

lj

m minus 1

kb120578b

uarrfj(z t) uarr

(b) Schematic of pile element

Figure 1 Schematic of pile-soil interaction model

(1) Vertical Shear Complex Stiffness at the Interface of Inner-Outer Zones In the outer semi-infinite undisturbed zone thevertical displacement diminishes as 119903 approaches infinity sothat 119861119899119896 = 0 Thus using the same method as proposedin [27] the vertical shear complex stiffness at the interfacebetween the inner and outer zones can be derived as follows

119870119895119899 =minus2120587119887119895119899120591119895119899 (119887119895119899)

119882119895119899 (119887119895119899)

=2120587119887119895119899119866119904119895119899

lowast1205731198951198991198701 (120573119895119899119887119895119899)

1198700 (120573119895119899119887119895119899)

(3)

where 120573119895119899 = 119894120596V119904119895119899radic1 + 119894119863119904119895119899 Vs119895119899 denotes the shear wavevelocity of the outer semi-infinite undisturbed zone withinthe 119895th soil layer 119866s119895119899

lowast= 119866s119895119899(1 + 119894119863s119895119899) 1198701(sdot) denotes the

modified Bessel function of order one of the second kind(2) Vertical Shear Complex Stiffness of the Inner Zones Sincethe vertical shear complex stiffness of surrounding soil isdefined as the ratio of the vertical shear force to the verticaldisplacement at the same position for the 119896th subzone withinthe 119895th inner soil layer the vertical shear complex stiffness atthe outer boundary of the 119896th subzone (119903 = 119887119895119896) namely119870119895119896can be easily derived from (2) and can be expressed as

119870119895119896 =2120587119887119895119896120573119895119896119866

lowasts119895119896 [1198601198951198961198701 (120573119895119896119887119895119896) minus 1198611198951198961198681 (120573119895119896119887119895119896)]

1198601198951198961198700 (120573119895119896119887119895119896) + 1198611198951198961198680 (120573119895119896119887119895119896)

(4)

where 119860119895119896 and 119861119895119896 are undetermined constants of the 119896thsubzone within the 119895th inner soil layer 1198681(sdot) denotes the

modified Bessel function of order one of the one kind Then(4) can be rewritten as follows

119860119895119896

119861119895119896=

2120587119887119895119896120573119895119896119866lowast1199041198951198961198681 (120573119895119896119887119895119896) + 1198701198951198961198680 (120573119895119896119887119895119896)

2120587119887119895119896120573119895119896119866lowast119904119895119896

1198701 (120573119895119896119887119895119896) minus 1198701198951198961198700 (120573119895119896119887119895119896) (5)

In the same manner the vertical shear complex stiffnessat the outer boundary of the (119896minus1)th subzone (119903 = 119887119895119896minus1) canbe derived as

119870119895119896minus1 =2120587119887119895119896minus1120573119895119896119866

lowast119904119895119896 [1198601198951198961198701 (120573119895119896119887119895119896minus1) minus 1198611198951198961198681 (120573119895119896119887119895119896minus1)]

1198601198951198961198700 (120573119895119896119887119895119896minus1) + 1198611198951198961198680 (120573119895119896119887119895119896minus1)

(6)

Substituting (5) into (6) 119870119895119896minus1 can be further rewrittenas follows

119870119895119896minus1 =2120587119887119895119896minus1120573119895119896119866

lowast119904119895119896 [1198621198951198961198701 (120573119895119896119887119895119896minus1) minus 1198631198951198961198681 (120573119895119896119887119895119896minus1)]

1198621198951198961198700 (120573119895119896119887119895119896minus1) + 1198631198951198961198680 (120573119895119896119887119895119896minus1)

(7)

where

119862119895119896 = 2120587119887119895119896120573119895119896119866lowast1199041198951198961198681 (120573119895119896119887119895119896) + 1198680 (120573119895119896119887119895119896)119870119895119896

119863119895119896 = 2120587119887119895119896120573119895119896119866lowast1199041198951198961198701 (120573119895119896119887119895119896) minus 1198700 (120573119895119896119887119895119896)119870119895119896

(8)

By now the recursion formula of the vertical shearcomplex stiffness of the inner zone has been derived in aclosed formThen according to the stiffness of the outer zoneas presented in (3) and the recursion formula (7) it is easyto obtain the vertical shear complex stiffness of the innerdisturbed zones

4 Mathematical Problems in Engineering

32 Dynamic Equation of Tapered Pile and Its SolutionDenoting 119906119895(119911 119905) to be the vertical displacement of the 119895thpile segment the frictional force of the 119895th soil layer acting onthe surface of the 119895th pile segment shaft can be expressed as119891119895(119911 119905) = 1198701198951119906119895(119911 119905) Then the dynamic equation of the 119895thpile segment can be derived as

119864119901119895119860119901119895

1205972119906119895 (119911 119905)

1205971199112+ 119860119901119895120575119901119895

1205973119906119895 (119911 119905)

1205971199112120597119905

minus 119898119901119895

1205972119906119895 (119911 119905)

1205971199052minus 119891119895 (119911 119905) = 0

(9)

where 119864119901119895 119860119901119895 = 1205871199031198952 119898119901119895 and 120575119901119895 denote the elastic

modulus cross-section area mass per unit length andviscous damping of the 119895th pile segment respectively

Boundary conditions at the top and the bottom of taperedpile can be expressed as

[119864119901119898119860119901119898120597119906119898 (119911 119905)

120597119911+ 119860119901119898120575119901119898

1205972119906119898 (119911 119905)

120597119911120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=0

= minus119876 (119905)

[11986411990111205971199061 (119911 119905)

120597119911+1205751199011

12059721199061 (119911 119905)

120597119911120597119905+1198961198871199061 (119911 119905)+120578119887

1205971199061 (119911 119905)

120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=119867

= 0

(10)

where 119896119887 and 120578119887 represent the distributed spring and dampingcoefficient in unit area at the bottom of the tapered pilerespectively The value of 119896119887 and 120578119887 can be obtained using thefollowing equations [28]

119896119887 =4120588119887119904119881

2119887119904119903119901

(1 minus 120592)

120578119887 =34120588119887119904119881119887119904119903

2119901

(1 minus 120592)

(11)

where 120588119887119904 119881119887119904 120592 and 119903119901 denote the density shear wavevelocity Poissonrsquos ratio and pile end radius of tapered pilerespectively

Continuity conditions of the interface of the adjacenttapered pile segments can be founded as

119906119895 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

= 119906119895+1 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

[119864119901119895119860119901119895

120597119906119895 (119911 119905)

120597119911+ 119860119901119895120575119901119895

1205972119906119895 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= [119864119901(119895+1)119860119901(119895+1)

120597119906119895+1 (119911 119905)

120597119911+ 119860119901(119895+1)120575119901(119895+1)

1205972119906119895+1 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(12)

Initial conditions of the tapered pile segments can befounded as

119906119895 (119911 119905)10038161003816100381610038161003816119905=0

= 0

120597119906119895 (119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(13)

Denote 119880119895(119911 119904) = intinfin

0119906119895(119911 119905)119890

minus119904119905dt to be the Laplacetransform of 119906119895(119911 119905) with respect to 119905 Combining (9) withthe initial conditions (13) and applying the Laplace transformyield

1198811199011198952(1 +

120575119901119895

119864119901119895119904)

1205972119880119895 (119911 119904)

1205972119911minus (1199042+

1

1205881199011198951198601199011198951198701198951)119880119895 (119911 119904) = 0

(14)

where119881119901119895 120588119901119895 and 119864119901119895 denote the longitudinal wave velocitydensity and elastic modulus of the 119895th tapered pile segmentrespectively and 119864119901119895 = 120588119901119895119881

2119901119895 119904 is the Laplace transform

constant which should satisfy 119904 = 119894120596 during the numericalanalysis

It is not difficult to obtain that the general solution of (14)which can be expressed as

119880119895 (119911 119904) = 119864119895 cos(120582119895119911

119897119895) + 119865119895 sin(

120582119895119911

119897119895) (15)

where 120582119895 = radicminus(1199042 + 119870119895120588119901119895119860119901119895)1199052119895(1 + (120575119901119895119864119901119895)119904) is dimen-

sionless eigenvalue 119864119895 and 119865119895 are complex constants whichcan be obtained from boundary conditions 119905119895 = 119897119895119881119901119895denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment

Taking into consideration the boundary conditions thedisplacement impedance function at the head of the 119895th pilesegment can be expressed as

11988511990111989510038161003816100381610038161003816119911=ℎ119895

= minus120588119901119895119860119901119895119881119901119895 (1 + (120575119901119895119864119901119895) 119904) 120582119895 tan (120582119895 minus 120593119895)

119905119895

(16)

where 120593119895 = arctan (119885119901(119895minus1)119905119895120588119901119895119860119901119895119881119901119895120582119895(1 + (120575119901119895119864119901119895)119904))119885119901(119895minus1) is the displacement impedance function at the headof the (119895 minus 1)th pile segment which can be obtained by virtueof the boundary conditions

Then following themethod of recursion typically utilizedin the transfer function technique [18] the displacementimpedance function at the head of tapered pile can be derivedas

11988511990111989810038161003816100381610038161003816119911=ℎ119898

= minus120588119901119898119860119901119898119881119901119898 (1 + (120575119901119898119864119901119898) 119904) 120582119898 tan (120582119898 minus 120593119898)

119905119898

(17)

where 120593119898 = arctan (119885119901(119898minus1)119905119898120588119901119898119860119901119898119881119901119898120582119898(1 + (120575119901119898

119864119901119898)119904))As shown in (18) the displacement impedance function

can be further rewritten in terms of its real part and imaginarypart which represent the dynamic stiffness and dampingrespectively And the dynamic stiffness and damping reflectthe ability to resist the vertical deformation and the energydissipation

119885119901119898 = 119870119903 + 119894119862119894 (18)

Mathematical Problems in Engineering 5

4 Analysis of Vibration Characteristics

In order to focus on the influence of compacting effect on thedynamic response of tapered pile the surrounding soil is setto be homogeneous in the following analysis For practicalengineering it only needs to discretize the surroundingsoil into finite layers according to the actual conditionsParameters used here are as follows length density elasticlongitudinal wave velocity pile end radius and materialdamping of tapered pile are 10m 2500 kgm3 4000ms03m and 0 respectively As shown in [19] it is worth notingthat the calculation results may be convergent if the numberof the pile segment is more than 119898 = 200 Therefore inthe following section the tapered pile is divided into 200segments with the same length

41 Influence of the Number of Inner Disturbed Zone on theVertical Dynamic Impedance at the Pile Head Because thecore idea of this paper is using the complex stiffness transfermodel to simulate the radial inhomogeneity of surroundingsoil caused by compacting effect it needs to discuss theinfluence of the number of inner disturbed zone on thecalculation results Parameters used here are as follows Theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityof surrounding soil from outer undisturbed zone to innerdisturbed zone increases linearly from 150ms to 200msThe density of surrounding soil is 2000 kgm3 and the widthof inner disturbed zone is 119887 = 119903119901 respectively The numberof inner disturbed zone are 119899 = 2 5 10 15 20 50 100

Figure 2 shows the influence of the number of innerdisturbed zone on the vertical dynamic impedance at the pilehead As shown in Figure 2(a) it can be observed that thedynamic stiffness increases as the frequency increases at firstbut when the frequency exceeds a certain value the dynamicstiffness may decrease with the increase of frequency Asshown in Figure 2(b) it can be noted that the dynamicdampingmay increase linearly with the increase of frequencywhen it is beyond 5Hz It can also be seen from Figure 2that the computational results may be convergent whenthe number of inner disturbed zone is big enough for thecomputational curves tend to be convergent By repeatedlycalculating for different cases all the computational resultsmay be convergent when 119899 ge 20 Therefore unless otherwisespecified the number of inner disturbed zone is set to be119899 = 20 in the following analysis

42 Influence of the Coneangle of Tapered Pile on the VerticalDynamic Impedance at the Pile Head In this section theinfluence of coneangle on the vertical dynamic impedanceat the pile head is studied in detail by taking into accountthe compacting effect Parameters used here are as followsthe shear wave velocity density of surrounding soil andthe width of inner disturbed zone are the same as thoseshown in Section 41 The coneangle of tapered pile are 120579 =

0∘ 03∘ 05∘ 1∘ 15∘ and 2

∘Figure 3 shows the influence of coneangle of tapered

pile on the vertical dynamic impedance at the pile headwithin the low frequency range concerned in dynamicfoundation design As shown in Figure 3(a) the dynamic

stiffness increases with the increase of frequency when theconeangle is small (ie 120579 le 1

∘ as shown in this example)However when the coneangle is big enough (ie 120579 gt 1

∘

as shown in this example) it can be seen that the dynamicstiffness increases with the increase of frequency at firstbut the dynamic stiffness may decrease as the frequencyincreases when the frequency exceeds a certain value Itcan also be noted that the dynamic stiffness increases asthe coneangle increases at low frequencies whereas at highfrequencies the dynamic stiffness decreases with the increaseof coneangle This means that increasing the coneangle canimprove the ability to resist vertical deformation of taperedpile within a certain low frequency range when other designparameters of tapered pile remain unchanged Neverthelessif the frequency is big enough the ability to resist verticaldeformation of tapered pile cannot be improved by simplyincreasing the coneangle As shown in Figure 3(b) it can beobserved that the dynamic damping may increase linearlywith the increase of frequency when it is beyond 5Hz Itcan also be noted the dynamic damping increases as theconeangle increases for the same frequency which shows thatthe stress wave energy propagating in the tapered pile maydecay more quickly as the coneangle increases

43 Influence of the Compacting Range on the Vertical Dy-namic Impedance at the Pile Head The compacting rangeof surrounding soil is affected by the design parameters oftapered pile construction technology and properties of sur-rounding soil and so on So it needs to discuss the influenceof compacting range on the vertical dynamic impedanceat the pile head Parameters used here are as follows theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityand density of surrounding soil are the same as those shownin Section 41 The width of inner disturbed zone are 119887 =

0 05119903119901 1119903119901 15119903119901 2119903119901 and 25119903119901Figure 4 shows the influence of compacting range on the

vertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 4(a) when the compacting range is small(ie 119887 le 2119903119901 as shown in this example) the dynamicstiffness increases as the frequency increases within lowrange whereas the dynamic stiffness deceases if the frequencyis beyond a certain value However it can be seen that thedynamic stiffness increases with the increase of frequencywhen the compacting range is big enough (ie 119887 gt 2119903119901 asshown in this example) It can also be seen that the dynamicstiffness increases with the increase of compacting range atlow frequenciesWhen the frequency exceeds a certain valuethe dynamic stiffness increases with the increase of the com-pacting range at first and then decreases as the compactingrange further increases As shown in Figure 4(b) it can benoted the dynamic damping increases as the compactingrange increases for the same frequency which means that thestress wave energy propagating in the tapered pile may decaymore quickly as the compacting range increases

In order to investigate the influence of compacting rangeon the vertical dynamic impedance at the pile head moredetailedly it needs to discuss the influence of compactingrange changing within smaller scale on the vertical dynamic

6 Mathematical Problems in Engineering

0 10 20 30 40

Kr

(Nmiddotm

minus1)

f (Hz)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

0

6 times108

12 times109

18 times109

24 times109

3 times109

36 times109

0 10 20 30 40f (Hz)

0

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(b) Dynamic damping curves

Figure 2 Influence of the number of inner disturbed zone on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(b) Dynamic damping curves

Figure 3 Influence of coneangle of tapered pile on the vertical dynamic impedance at the pile head

impedance Parameters used here are as follows the width ofinner disturbed zone are 119887 = 0 005119903119901 01119903119901 02119903119901 03119903119901and 04119903119901 The other parameters are the same as those shownin the above analysis

Figure 5 shows the influence of compacting range chang-ing within smaller scale on the vertical dynamic impedanceat the pile head within the low frequency range concerned indynamic foundation design As shown in Figure 5(a) it canbe seen that the dynamic stiffness increases with the increaseof compacting range at low frequencies but the increaseratio is small When the frequency is beyond a certain valuethe dynamic stiffness decreases as the compacting rangeincreases As shown in Figure 5(b) it can be seen that theinfluence of compacting range changing within smaller scale

on the vertical dynamic impedance is the same as that whenthe compacting range changes within bigger scale

44 Influence of the Compacting Degree on the Vertical Dy-namic Impedance at the Pile Head Parameters used hereare as follows the coneangle of tapered pile is 120579 = 2

∘ andthe width of inner disturbed zone is 119887 = 119903119901 Denote shearwave velocity ratio to describe the compacting degree thatis 119902 represents the ratio of the shear wave velocity of innerdisturbed zone to that of outer undisturbed zone The shearwave velocity of outer undisturbed zone is 150ms and theshear wave velocity ratio are 119902 = 1 12 14 16 18 and 2 Theother parameters are the same as those shown in Section 41

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

32 Dynamic Equation of Tapered Pile and Its SolutionDenoting 119906119895(119911 119905) to be the vertical displacement of the 119895thpile segment the frictional force of the 119895th soil layer acting onthe surface of the 119895th pile segment shaft can be expressed as119891119895(119911 119905) = 1198701198951119906119895(119911 119905) Then the dynamic equation of the 119895thpile segment can be derived as

119864119901119895119860119901119895

1205972119906119895 (119911 119905)

1205971199112+ 119860119901119895120575119901119895

1205973119906119895 (119911 119905)

1205971199112120597119905

minus 119898119901119895

1205972119906119895 (119911 119905)

1205971199052minus 119891119895 (119911 119905) = 0

(9)

where 119864119901119895 119860119901119895 = 1205871199031198952 119898119901119895 and 120575119901119895 denote the elastic

modulus cross-section area mass per unit length andviscous damping of the 119895th pile segment respectively

Boundary conditions at the top and the bottom of taperedpile can be expressed as

[119864119901119898119860119901119898120597119906119898 (119911 119905)

120597119911+ 119860119901119898120575119901119898

1205972119906119898 (119911 119905)

120597119911120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=0

= minus119876 (119905)

[11986411990111205971199061 (119911 119905)

120597119911+1205751199011

12059721199061 (119911 119905)

120597119911120597119905+1198961198871199061 (119911 119905)+120578119887

1205971199061 (119911 119905)

120597119905]

100381610038161003816100381610038161003816100381610038161003816119911=119867

= 0

(10)

where 119896119887 and 120578119887 represent the distributed spring and dampingcoefficient in unit area at the bottom of the tapered pilerespectively The value of 119896119887 and 120578119887 can be obtained using thefollowing equations [28]

119896119887 =4120588119887119904119881

2119887119904119903119901

(1 minus 120592)

120578119887 =34120588119887119904119881119887119904119903

2119901

(1 minus 120592)

(11)

where 120588119887119904 119881119887119904 120592 and 119903119901 denote the density shear wavevelocity Poissonrsquos ratio and pile end radius of tapered pilerespectively

Continuity conditions of the interface of the adjacenttapered pile segments can be founded as

119906119895 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

= 119906119895+1 (119911 119905)10038161003816100381610038161003816119911=ℎ119895

[119864119901119895119860119901119895

120597119906119895 (119911 119905)

120597119911+ 119860119901119895120575119901119895

1205972119906119895 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

= [119864119901(119895+1)119860119901(119895+1)

120597119906119895+1 (119911 119905)

120597119911+ 119860119901(119895+1)120575119901(119895+1)

1205972119906119895+1 (119911 119905)

120597119911120597119905]

1003816100381610038161003816100381610038161003816100381610038161003816119911=ℎ119895

(12)

Initial conditions of the tapered pile segments can befounded as

119906119895 (119911 119905)10038161003816100381610038161003816119905=0

= 0

120597119906119895 (119911 119905)

120597119905

100381610038161003816100381610038161003816100381610038161003816119905=0= 0

(13)

Denote 119880119895(119911 119904) = intinfin

0119906119895(119911 119905)119890

minus119904119905dt to be the Laplacetransform of 119906119895(119911 119905) with respect to 119905 Combining (9) withthe initial conditions (13) and applying the Laplace transformyield

1198811199011198952(1 +

120575119901119895

119864119901119895119904)

1205972119880119895 (119911 119904)

1205972119911minus (1199042+

1

1205881199011198951198601199011198951198701198951)119880119895 (119911 119904) = 0

(14)

where119881119901119895 120588119901119895 and 119864119901119895 denote the longitudinal wave velocitydensity and elastic modulus of the 119895th tapered pile segmentrespectively and 119864119901119895 = 120588119901119895119881

2119901119895 119904 is the Laplace transform

constant which should satisfy 119904 = 119894120596 during the numericalanalysis

It is not difficult to obtain that the general solution of (14)which can be expressed as

119880119895 (119911 119904) = 119864119895 cos(120582119895119911

119897119895) + 119865119895 sin(

120582119895119911

119897119895) (15)

where 120582119895 = radicminus(1199042 + 119870119895120588119901119895119860119901119895)1199052119895(1 + (120575119901119895119864119901119895)119904) is dimen-

sionless eigenvalue 119864119895 and 119865119895 are complex constants whichcan be obtained from boundary conditions 119905119895 = 119897119895119881119901119895denotes the propagation time of elastic longitudinal wave inthe 119895th pile segment

Taking into consideration the boundary conditions thedisplacement impedance function at the head of the 119895th pilesegment can be expressed as

11988511990111989510038161003816100381610038161003816119911=ℎ119895

= minus120588119901119895119860119901119895119881119901119895 (1 + (120575119901119895119864119901119895) 119904) 120582119895 tan (120582119895 minus 120593119895)

119905119895

(16)

where 120593119895 = arctan (119885119901(119895minus1)119905119895120588119901119895119860119901119895119881119901119895120582119895(1 + (120575119901119895119864119901119895)119904))119885119901(119895minus1) is the displacement impedance function at the headof the (119895 minus 1)th pile segment which can be obtained by virtueof the boundary conditions

Then following themethod of recursion typically utilizedin the transfer function technique [18] the displacementimpedance function at the head of tapered pile can be derivedas

11988511990111989810038161003816100381610038161003816119911=ℎ119898

= minus120588119901119898119860119901119898119881119901119898 (1 + (120575119901119898119864119901119898) 119904) 120582119898 tan (120582119898 minus 120593119898)

119905119898

(17)

where 120593119898 = arctan (119885119901(119898minus1)119905119898120588119901119898119860119901119898119881119901119898120582119898(1 + (120575119901119898

119864119901119898)119904))As shown in (18) the displacement impedance function

can be further rewritten in terms of its real part and imaginarypart which represent the dynamic stiffness and dampingrespectively And the dynamic stiffness and damping reflectthe ability to resist the vertical deformation and the energydissipation

119885119901119898 = 119870119903 + 119894119862119894 (18)

Mathematical Problems in Engineering 5

4 Analysis of Vibration Characteristics

In order to focus on the influence of compacting effect on thedynamic response of tapered pile the surrounding soil is setto be homogeneous in the following analysis For practicalengineering it only needs to discretize the surroundingsoil into finite layers according to the actual conditionsParameters used here are as follows length density elasticlongitudinal wave velocity pile end radius and materialdamping of tapered pile are 10m 2500 kgm3 4000ms03m and 0 respectively As shown in [19] it is worth notingthat the calculation results may be convergent if the numberof the pile segment is more than 119898 = 200 Therefore inthe following section the tapered pile is divided into 200segments with the same length

41 Influence of the Number of Inner Disturbed Zone on theVertical Dynamic Impedance at the Pile Head Because thecore idea of this paper is using the complex stiffness transfermodel to simulate the radial inhomogeneity of surroundingsoil caused by compacting effect it needs to discuss theinfluence of the number of inner disturbed zone on thecalculation results Parameters used here are as follows Theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityof surrounding soil from outer undisturbed zone to innerdisturbed zone increases linearly from 150ms to 200msThe density of surrounding soil is 2000 kgm3 and the widthof inner disturbed zone is 119887 = 119903119901 respectively The numberof inner disturbed zone are 119899 = 2 5 10 15 20 50 100

Figure 2 shows the influence of the number of innerdisturbed zone on the vertical dynamic impedance at the pilehead As shown in Figure 2(a) it can be observed that thedynamic stiffness increases as the frequency increases at firstbut when the frequency exceeds a certain value the dynamicstiffness may decrease with the increase of frequency Asshown in Figure 2(b) it can be noted that the dynamicdampingmay increase linearly with the increase of frequencywhen it is beyond 5Hz It can also be seen from Figure 2that the computational results may be convergent whenthe number of inner disturbed zone is big enough for thecomputational curves tend to be convergent By repeatedlycalculating for different cases all the computational resultsmay be convergent when 119899 ge 20 Therefore unless otherwisespecified the number of inner disturbed zone is set to be119899 = 20 in the following analysis

42 Influence of the Coneangle of Tapered Pile on the VerticalDynamic Impedance at the Pile Head In this section theinfluence of coneangle on the vertical dynamic impedanceat the pile head is studied in detail by taking into accountthe compacting effect Parameters used here are as followsthe shear wave velocity density of surrounding soil andthe width of inner disturbed zone are the same as thoseshown in Section 41 The coneangle of tapered pile are 120579 =

0∘ 03∘ 05∘ 1∘ 15∘ and 2

∘Figure 3 shows the influence of coneangle of tapered

pile on the vertical dynamic impedance at the pile headwithin the low frequency range concerned in dynamicfoundation design As shown in Figure 3(a) the dynamic

stiffness increases with the increase of frequency when theconeangle is small (ie 120579 le 1

∘ as shown in this example)However when the coneangle is big enough (ie 120579 gt 1

∘

as shown in this example) it can be seen that the dynamicstiffness increases with the increase of frequency at firstbut the dynamic stiffness may decrease as the frequencyincreases when the frequency exceeds a certain value Itcan also be noted that the dynamic stiffness increases asthe coneangle increases at low frequencies whereas at highfrequencies the dynamic stiffness decreases with the increaseof coneangle This means that increasing the coneangle canimprove the ability to resist vertical deformation of taperedpile within a certain low frequency range when other designparameters of tapered pile remain unchanged Neverthelessif the frequency is big enough the ability to resist verticaldeformation of tapered pile cannot be improved by simplyincreasing the coneangle As shown in Figure 3(b) it can beobserved that the dynamic damping may increase linearlywith the increase of frequency when it is beyond 5Hz Itcan also be noted the dynamic damping increases as theconeangle increases for the same frequency which shows thatthe stress wave energy propagating in the tapered pile maydecay more quickly as the coneangle increases

43 Influence of the Compacting Range on the Vertical Dy-namic Impedance at the Pile Head The compacting rangeof surrounding soil is affected by the design parameters oftapered pile construction technology and properties of sur-rounding soil and so on So it needs to discuss the influenceof compacting range on the vertical dynamic impedanceat the pile head Parameters used here are as follows theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityand density of surrounding soil are the same as those shownin Section 41 The width of inner disturbed zone are 119887 =

0 05119903119901 1119903119901 15119903119901 2119903119901 and 25119903119901Figure 4 shows the influence of compacting range on the

vertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 4(a) when the compacting range is small(ie 119887 le 2119903119901 as shown in this example) the dynamicstiffness increases as the frequency increases within lowrange whereas the dynamic stiffness deceases if the frequencyis beyond a certain value However it can be seen that thedynamic stiffness increases with the increase of frequencywhen the compacting range is big enough (ie 119887 gt 2119903119901 asshown in this example) It can also be seen that the dynamicstiffness increases with the increase of compacting range atlow frequenciesWhen the frequency exceeds a certain valuethe dynamic stiffness increases with the increase of the com-pacting range at first and then decreases as the compactingrange further increases As shown in Figure 4(b) it can benoted the dynamic damping increases as the compactingrange increases for the same frequency which means that thestress wave energy propagating in the tapered pile may decaymore quickly as the compacting range increases

In order to investigate the influence of compacting rangeon the vertical dynamic impedance at the pile head moredetailedly it needs to discuss the influence of compactingrange changing within smaller scale on the vertical dynamic

6 Mathematical Problems in Engineering

0 10 20 30 40

Kr

(Nmiddotm

minus1)

f (Hz)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

0

6 times108

12 times109

18 times109

24 times109

3 times109

36 times109

0 10 20 30 40f (Hz)

0

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(b) Dynamic damping curves

Figure 2 Influence of the number of inner disturbed zone on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(b) Dynamic damping curves

Figure 3 Influence of coneangle of tapered pile on the vertical dynamic impedance at the pile head

impedance Parameters used here are as follows the width ofinner disturbed zone are 119887 = 0 005119903119901 01119903119901 02119903119901 03119903119901and 04119903119901 The other parameters are the same as those shownin the above analysis

Figure 5 shows the influence of compacting range chang-ing within smaller scale on the vertical dynamic impedanceat the pile head within the low frequency range concerned indynamic foundation design As shown in Figure 5(a) it canbe seen that the dynamic stiffness increases with the increaseof compacting range at low frequencies but the increaseratio is small When the frequency is beyond a certain valuethe dynamic stiffness decreases as the compacting rangeincreases As shown in Figure 5(b) it can be seen that theinfluence of compacting range changing within smaller scale

on the vertical dynamic impedance is the same as that whenthe compacting range changes within bigger scale

44 Influence of the Compacting Degree on the Vertical Dy-namic Impedance at the Pile Head Parameters used hereare as follows the coneangle of tapered pile is 120579 = 2

∘ andthe width of inner disturbed zone is 119887 = 119903119901 Denote shearwave velocity ratio to describe the compacting degree thatis 119902 represents the ratio of the shear wave velocity of innerdisturbed zone to that of outer undisturbed zone The shearwave velocity of outer undisturbed zone is 150ms and theshear wave velocity ratio are 119902 = 1 12 14 16 18 and 2 Theother parameters are the same as those shown in Section 41

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

4 Analysis of Vibration Characteristics

In order to focus on the influence of compacting effect on thedynamic response of tapered pile the surrounding soil is setto be homogeneous in the following analysis For practicalengineering it only needs to discretize the surroundingsoil into finite layers according to the actual conditionsParameters used here are as follows length density elasticlongitudinal wave velocity pile end radius and materialdamping of tapered pile are 10m 2500 kgm3 4000ms03m and 0 respectively As shown in [19] it is worth notingthat the calculation results may be convergent if the numberof the pile segment is more than 119898 = 200 Therefore inthe following section the tapered pile is divided into 200segments with the same length

41 Influence of the Number of Inner Disturbed Zone on theVertical Dynamic Impedance at the Pile Head Because thecore idea of this paper is using the complex stiffness transfermodel to simulate the radial inhomogeneity of surroundingsoil caused by compacting effect it needs to discuss theinfluence of the number of inner disturbed zone on thecalculation results Parameters used here are as follows Theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityof surrounding soil from outer undisturbed zone to innerdisturbed zone increases linearly from 150ms to 200msThe density of surrounding soil is 2000 kgm3 and the widthof inner disturbed zone is 119887 = 119903119901 respectively The numberof inner disturbed zone are 119899 = 2 5 10 15 20 50 100

Figure 2 shows the influence of the number of innerdisturbed zone on the vertical dynamic impedance at the pilehead As shown in Figure 2(a) it can be observed that thedynamic stiffness increases as the frequency increases at firstbut when the frequency exceeds a certain value the dynamicstiffness may decrease with the increase of frequency Asshown in Figure 2(b) it can be noted that the dynamicdampingmay increase linearly with the increase of frequencywhen it is beyond 5Hz It can also be seen from Figure 2that the computational results may be convergent whenthe number of inner disturbed zone is big enough for thecomputational curves tend to be convergent By repeatedlycalculating for different cases all the computational resultsmay be convergent when 119899 ge 20 Therefore unless otherwisespecified the number of inner disturbed zone is set to be119899 = 20 in the following analysis

42 Influence of the Coneangle of Tapered Pile on the VerticalDynamic Impedance at the Pile Head In this section theinfluence of coneangle on the vertical dynamic impedanceat the pile head is studied in detail by taking into accountthe compacting effect Parameters used here are as followsthe shear wave velocity density of surrounding soil andthe width of inner disturbed zone are the same as thoseshown in Section 41 The coneangle of tapered pile are 120579 =

0∘ 03∘ 05∘ 1∘ 15∘ and 2

∘Figure 3 shows the influence of coneangle of tapered

pile on the vertical dynamic impedance at the pile headwithin the low frequency range concerned in dynamicfoundation design As shown in Figure 3(a) the dynamic

stiffness increases with the increase of frequency when theconeangle is small (ie 120579 le 1

∘ as shown in this example)However when the coneangle is big enough (ie 120579 gt 1

∘

as shown in this example) it can be seen that the dynamicstiffness increases with the increase of frequency at firstbut the dynamic stiffness may decrease as the frequencyincreases when the frequency exceeds a certain value Itcan also be noted that the dynamic stiffness increases asthe coneangle increases at low frequencies whereas at highfrequencies the dynamic stiffness decreases with the increaseof coneangle This means that increasing the coneangle canimprove the ability to resist vertical deformation of taperedpile within a certain low frequency range when other designparameters of tapered pile remain unchanged Neverthelessif the frequency is big enough the ability to resist verticaldeformation of tapered pile cannot be improved by simplyincreasing the coneangle As shown in Figure 3(b) it can beobserved that the dynamic damping may increase linearlywith the increase of frequency when it is beyond 5Hz Itcan also be noted the dynamic damping increases as theconeangle increases for the same frequency which shows thatthe stress wave energy propagating in the tapered pile maydecay more quickly as the coneangle increases

43 Influence of the Compacting Range on the Vertical Dy-namic Impedance at the Pile Head The compacting rangeof surrounding soil is affected by the design parameters oftapered pile construction technology and properties of sur-rounding soil and so on So it needs to discuss the influenceof compacting range on the vertical dynamic impedanceat the pile head Parameters used here are as follows theconeangle of tapered pile is 120579 = 2

∘ The shear wave velocityand density of surrounding soil are the same as those shownin Section 41 The width of inner disturbed zone are 119887 =

0 05119903119901 1119903119901 15119903119901 2119903119901 and 25119903119901Figure 4 shows the influence of compacting range on the

vertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 4(a) when the compacting range is small(ie 119887 le 2119903119901 as shown in this example) the dynamicstiffness increases as the frequency increases within lowrange whereas the dynamic stiffness deceases if the frequencyis beyond a certain value However it can be seen that thedynamic stiffness increases with the increase of frequencywhen the compacting range is big enough (ie 119887 gt 2119903119901 asshown in this example) It can also be seen that the dynamicstiffness increases with the increase of compacting range atlow frequenciesWhen the frequency exceeds a certain valuethe dynamic stiffness increases with the increase of the com-pacting range at first and then decreases as the compactingrange further increases As shown in Figure 4(b) it can benoted the dynamic damping increases as the compactingrange increases for the same frequency which means that thestress wave energy propagating in the tapered pile may decaymore quickly as the compacting range increases

In order to investigate the influence of compacting rangeon the vertical dynamic impedance at the pile head moredetailedly it needs to discuss the influence of compactingrange changing within smaller scale on the vertical dynamic

6 Mathematical Problems in Engineering

0 10 20 30 40

Kr

(Nmiddotm

minus1)

f (Hz)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

0

6 times108

12 times109

18 times109

24 times109

3 times109

36 times109

0 10 20 30 40f (Hz)

0

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(b) Dynamic damping curves

Figure 2 Influence of the number of inner disturbed zone on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(b) Dynamic damping curves

Figure 3 Influence of coneangle of tapered pile on the vertical dynamic impedance at the pile head

impedance Parameters used here are as follows the width ofinner disturbed zone are 119887 = 0 005119903119901 01119903119901 02119903119901 03119903119901and 04119903119901 The other parameters are the same as those shownin the above analysis

Figure 5 shows the influence of compacting range chang-ing within smaller scale on the vertical dynamic impedanceat the pile head within the low frequency range concerned indynamic foundation design As shown in Figure 5(a) it canbe seen that the dynamic stiffness increases with the increaseof compacting range at low frequencies but the increaseratio is small When the frequency is beyond a certain valuethe dynamic stiffness decreases as the compacting rangeincreases As shown in Figure 5(b) it can be seen that theinfluence of compacting range changing within smaller scale

on the vertical dynamic impedance is the same as that whenthe compacting range changes within bigger scale

44 Influence of the Compacting Degree on the Vertical Dy-namic Impedance at the Pile Head Parameters used hereare as follows the coneangle of tapered pile is 120579 = 2

∘ andthe width of inner disturbed zone is 119887 = 119903119901 Denote shearwave velocity ratio to describe the compacting degree thatis 119902 represents the ratio of the shear wave velocity of innerdisturbed zone to that of outer undisturbed zone The shearwave velocity of outer undisturbed zone is 150ms and theshear wave velocity ratio are 119902 = 1 12 14 16 18 and 2 Theother parameters are the same as those shown in Section 41

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 10 20 30 40

Kr

(Nmiddotm

minus1)

f (Hz)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

0

6 times108

12 times109

18 times109

24 times109

3 times109

36 times109

0 10 20 30 40f (Hz)

0

n = 2

n = 5

n = 10

n = 15

n = 20

n = 50

n = 100

(b) Dynamic damping curves

Figure 2 Influence of the number of inner disturbed zone on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

120579 = 0∘

120579 = 03∘

120579 = 05∘

120579 = 1∘

120579 = 15∘

120579 = 2∘

(b) Dynamic damping curves

Figure 3 Influence of coneangle of tapered pile on the vertical dynamic impedance at the pile head

impedance Parameters used here are as follows the width ofinner disturbed zone are 119887 = 0 005119903119901 01119903119901 02119903119901 03119903119901and 04119903119901 The other parameters are the same as those shownin the above analysis

Figure 5 shows the influence of compacting range chang-ing within smaller scale on the vertical dynamic impedanceat the pile head within the low frequency range concerned indynamic foundation design As shown in Figure 5(a) it canbe seen that the dynamic stiffness increases with the increaseof compacting range at low frequencies but the increaseratio is small When the frequency is beyond a certain valuethe dynamic stiffness decreases as the compacting rangeincreases As shown in Figure 5(b) it can be seen that theinfluence of compacting range changing within smaller scale

on the vertical dynamic impedance is the same as that whenthe compacting range changes within bigger scale

44 Influence of the Compacting Degree on the Vertical Dy-namic Impedance at the Pile Head Parameters used hereare as follows the coneangle of tapered pile is 120579 = 2

∘ andthe width of inner disturbed zone is 119887 = 119903119901 Denote shearwave velocity ratio to describe the compacting degree thatis 119902 represents the ratio of the shear wave velocity of innerdisturbed zone to that of outer undisturbed zone The shearwave velocity of outer undisturbed zone is 150ms and theshear wave velocity ratio are 119902 = 1 12 14 16 18 and 2 Theother parameters are the same as those shown in Section 41

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

14 times109

0 10 20 30 40f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(a) Dynamic stiffness curves

Ci

(Nmiddotsmiddot

mminus1)

3 times109

2 times109

1 times109

5 times108

25 times109

15 times109

00 10 20 30 40

f (Hz)

b = 25rp

Homogeneousb = 05rpb = 1rp

b = 15rpb = 2rp

(b) Dynamic damping curves

Figure 4 Influence of compacting range on the vertical dynamic impedance at the pile head

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40

Homogeneous b = 02rpb = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(a) Dynamic stiffness curves

00 10 20 30 40

Homogeneous b = 02rp

Ci

(Nmiddotsmiddot

mminus1)

2 times109

4 times108

8 times108

16 times109

24 times109

12 times109

b = 005rpb = 01rp

b = 03rpb = 04rp

f (Hz)

(b) Dynamic damping curve

Figure 5 Influence of compacting range on the vertical dynamic impedance at the pile head

Figure 6 shows the influence of compacting degree on thevertical dynamic impedance at the pile head within the lowfrequency range concerned in dynamic foundation designAs shown in Figure 6(a) it can be seen that the dynamicstiffness increases as the compacting degree increases atlow frequencies whereas at high frequencies the dynamicstiffness decreases with the increase of compacting degree Asshown in Figure 6(b) it can be seen that the dynamic damp-ing increases as the compacting degree increases but theincrease ratio decreases as the compacting degree increasesThe result shows that the stress wave energy propagating inthe tapered pile may decay more quickly as the compactingdegree increases

5 Conclusions

(1) When the frequency changes within a low range theability to resist vertical deformation of tapered pile can beimproved by increasing the coneangle of tapered pile How-ever if the frequency exceeds a certain value the ability toresist vertical deformation of tapered pile cannot be improvedby simply increasing the coneangle of tapered pile Withinthe low frequency range concerned in dynamic foundationdesign the ability to resist vertical vibration of tapered pilecan be improved by increasing the coneangle of tapered pile

(2) When the frequency varies within a low rangeincreasing the compacting range of surrounding soil can

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Kr

(Nmiddotm

minus1)

0

2 times108

4 times108

6 times108

8 times108

1 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14 q = 2

q = 16

q = 18

(a) Dynamic stiffness curves

0

Ci

(Nmiddotsmiddot

mminus1)

3 times109

6 times108

18 times109

24 times109

36 times109

12 times109

0 10 20 30 40(Hz)

Homogeneousq = 12

q = 14

q = 16

q = 2

q = 18

(b) Dynamic damping curves

Figure 6 Influence of compacting degree on the vertical dynamic impedance at the pile head

improve the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting range within smaller scale mayweaken the ability to resist vertical deformation of taperedpile but increasing the compacting range within bigger scalemay weaken the ability to resist vertical deformation oftapered pile at first and then improve the ability to resist ver-tical deformation of tapered pile when the compacting rangeis big enough Within the low frequency range concerned indynamic foundation design increasing the compacting rangeof surrounding soil can improve the ability to resist verticalvibration

(3) When the frequency varies within a low rangeincreasing the compacting degree of surrounding soil canimprove the ability to resist vertical deformation of taperedpile However if the frequency is beyond a certain valueincreasing the compacting degree of surrounding soil mayweaken the ability to resist vertical deformation of taperedpile Within the low frequency range concerned in dynamicfoundation design increasing the compacting degree ofsurrounding soil can improve the ability to resist verticalvibration

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China (Grant no 51309207) the ChinaPostdoctoral Science Foundation Funded Project (Grant no2012M521495 and no 2013T60759) and the FundamentalResearch Funds for the Central Universities (Grant noCUG120821)

References

[1] M F Randolph J S Steenfelt andC PWorth ldquoThe effect of piletype on design parameter for driven pilesrdquo in Proceedings of the

7th European Conference on Soil Mechanics and Foundation inEngineering pp 114ndash207 Brighton UK 1979

[2] J S Steenfelt M F Randolph and C P Worth ldquoInstru-mented model piles jacked into clayrdquo in Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation inEngineering pp 857ndash864 Stockholm Sweden 1981

[3] Z Y Luo X N Gong J L Wang and W T Wang ldquoNumericalsimulation and factor analysis of jacked pile compacting effectsrdquoJournal of Zhejiang University (Engineering Science) vol 39 no7 pp 992ndash996 2005

[4] Q Lu X N Gong M Ma and J L Wang ldquoSqueezing effectsof jacked pile considering static pile press machinersquos actionrdquoJournal of Zhejiang University (Engineering Science) vol 41 no7 pp 1132ndash1190 2007

[5] J P Xu J Zhou C Y Xu and C Y Zhang ldquoModel test researchon pile driving effect of squeezing against soilrdquo Rock and SoilMechanics vol 21 no 3 pp 235ndash238 2000

[6] X N Gong and X H Li ldquoSeveral mechanical problemsin compacting effects of static piling in soft clay groundrdquoEngineering Mechanics vol 17 no 4 pp 7ndash12 2000

[7] Z Y LuoW TWang andW Liu ldquoInfluence analysis of frictionbetween pile and soil on compacting effects of jacked pilerdquoChinese Journal of Rock Mechanics and Engineering vol 24 no18 pp 3299ndash3304 2005

[8] J Peng J Y Shi and G Huang ldquoAnalysis of bearing capacity ofpile foundation in consideration of compaction effectrdquo Journalof Hohai University vol 30 no 2 pp 105ndash108 2002

[9] X S Zhang Y M Tiu X N Gong and J Z Pan ldquoTime-dependency analysis of soil compaction pile in soft clay groundrdquoChinese Journal of Rock Mechanics and Engineering vol 23 no19 pp 3365ndash3369 2004

[10] J P Jiang and G Y Gao ldquoComparison of belled pile taperedpile and equal-diameter pilerdquo Chinese Journal of GeotechnicalEngineering vol 25 no 6 pp 764ndash766 2003

[11] J Liu and Z H Wang ldquoExperimental study on the bearingcapacity of wedge pilerdquo Journal of Tianjin University vol 35 no2 pp 257ndash260 2002

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[12] MH El Naggar and J QWei ldquoUplift behaviour of tapered pilesestablished from model testsrdquo Canadian Geotechnical Journalvol 37 no 1 pp 56ndash74 2000

[13] J Liu J He and C Q Min ldquoContrast research of bearingbehavior for composite foundation with tapered piles andcylindrical pilesrdquo Rock and Soil Mechanics vol 31 no 7 pp2202ndash2206 2010

[14] W A Take and A J Walsangkar ldquoThe elastic analysis ofcompressible tin-piles and pile groupsrdquo Geotechnique vol 31no 4 pp 456ndash474 2002

[15] J He J Liu K L Chen and C Q Min ldquoExperiment ofcompacting effects of static piling in clay ground with taperedpilerdquo Industrial Construction vol 38 no 1 pp 74ndash75 2008

[16] K H Wang K H Xie and G X Zeng ldquoAnalytical solutionto vibration of finite length pile under exciting force and itsapplicationrdquoChinese Journal of Geotechnical Engineering vol 19no 6 pp 27ndash35 1997

[17] K H Wang K H Xie and G X Zeng ldquoAn analytical solutionto forced vibration of foundation pile with variable sectionimpedance and its applicationrdquoChina Civil Engineering Journalvol 31 no 6 pp 56ndash67 1998

[18] 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

[19] W B Wu K H Wang D H Wu and B N Ma ldquoStudy ofdynamic longitudinal impedance of tapered pile consideringlateral inertial effectrdquo Chinese Journal of Rock Mechanics andEngineering vol 30 no 2 pp 3618ndash3625 2011

[20] Y Y Cai J Yu C T Zheng Z B Qi and B X Song ldquoAnalyt-ical solution for longitudinal dynamic complex impedance oftapered pilerdquo Chinese Journal of Geotechnical Engineering vol33 no 2 pp 392ndash398 2011

[21] D S Kong M T Luan and Q Yang ldquoReview of dynamicWinkler model applied in pile-soil interaction analysesrdquoWorldInformation on Earthquake Engineering vol 21 no 1 pp 12ndash172005

[22] M Novak and F Aboul-Ella ldquoImpedance functions of piles inlayered mediardquo Journal of the Engineering Mechanical Divisionvol 104 no 3 pp 643ndash661 1978

[23] MNovak T Nogami and F Aboul-ella ldquoDynamic soil reactionfor plane strain caserdquo Journal of the Engineering MechanicalDivision vol 104 no 4 pp 953ndash959 1978

[24] C B Hu and T Zhang ldquoTime domain torsional response ofdynamically loaded pile in hysteretic type damping soil layerrdquoChinese Journal of Rock Mechanics and Engineering vol 25 no1 pp 3190ndash3197 2006

[25] K H Wang R B Que and J Z Xia ldquoTheory of pile vibrationconsidering true three-dimensional wave effect of soil and itscheck on the approximate theoriesrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 8 pp 1362ndash1370 2005

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

[27] K H Wang D Y Yang and Z Q Zhang ldquoStudy on dynamicresponse of pile based on complex stiffness transfer modelof radial multizone plane strainrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 4 pp 825ndash831 2008

[28] J Lysmer and F E Richart ldquoDynamic response of footing tovertical loadrdquo Soil Mechanical and Foundation Division vol 2no 1 pp 65ndash91 1966

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of