Torsional Vibrations of Single Piles Embedded in Saturated Medium

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Torsional vibrations of single piles embedded in saturated medium

Guocai Wang a,*, Wei Ge b, Xiaodong Pan a, Zhe Wang a

a School of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou 310014, Chinab College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310027, China

Received 10 September 2006; received in revised form 31 December 2006; accepted 1 February 2007Available online 23 March 2007

Abstract

From the axisymmetrical point of view, the dynamic response of single piles vertically embedded in the saturated half space and sub-jected to harmonic torsional loadings is investigated by means of integral transform and Mukis methods for the first time. By treatingthe pile as a one-dimensional bar and the half space as a three-dimensional poroelastic continuum, the dynamic interaction between thepile and poroelastic medium is formulated in a Fredholm integral equation of the second kind. Numerical results are presented to illus-trate the influence of pile flexibility, geometrical properties of pilesaturated soil system and frequency of excitation on the piles torque,torsional angle and coupled impedances. The conclusions obtained can serve as guidelines for the design and dynamic pile testing ofpractical engineering. 2007 Elsevier Ltd. All rights reserved.

Keywords: Saturated half space; Integral equation; Single pile; Torsional vibration; Dynamic response

1. Introduction

The dynamic response of pile foundations has beenstudied extensively during the past few decades. This wasprompted by the design of unclear plants, machine founda-tions and other facilities. Various methods have been usedto take the soilpile interaction in the dynamic analysis ofpile foundations. For example, Nogami and Konagai[11,12] presented the linear transient analysis of single piles.In the analysis, the soil response to the pile motion wasintegrated through a simplified model based on Winklers

assumption, the parameters of which were determined fromthe consideration of plane strain wave propagation. Rajap-akse and Shah [13] used fictitious bar-extended half spacemodel originally proposed by Muki and Sternberg [9,10]and investigated the motion of a long cylindrical elasticbar which is partially embedded in a homogeneous elastichalf-space and subjected to harmonic axial loadings.Mamoon et al. [7] used hybrid boundary element formula-

tions and got the dynamic impedance and compliance func-tions of piles and inclined pile groups. EI Naggar andNovak [4] presented a non-linear model and investigatedthe effect of non-linear soil behavior, energy dissipationthrough radiation damping, soil hysteresis and the loadingrate dependency of the soil resistance on the piles axialresponse to transient and harmonic loadings. Militanoand Rajapakse [8] used analytical solutions and investi-gated the elastic pile subjected to transient torsional andaxial loadings. Khan and Pise [6] presented an analyticalmodel and developed an associated computer program to

investigate the vertical vibration of curved piles embeddedin a homogeneous elastic half-space. Das and Sargand [3]used variational principle and presented a two-parametermodel to study the response of a single circular cylindricalpile subjected to lateral dynamic loadings.

However, the assumption that the medium is an idealelastic solid is not satisfactory for most cases where themedium contains a fluid, such as in case of soil systemsor in some biomechanic applications. For such cases, itis more realistic to assume that the medium is a porouselastic solid filled with fluid and behaves according to

0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.02.001

* Corresponding author. Tel.: +86 571 8832 0460.E-mail address: [email protected] (G. Wang).

www.elsevier.com/locate/compgeo

Available online at www.sciencedirect.com

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the theory developed a half century ago by Biot [1]. UsingBiots theory, Zeng and Rajapakse [18] studied the verti-cal dynamic response of a pile in a poroelastic half spaceunder vertical loadings. By means of the method pro-posed by Pak and Jennings, Jin et al. [5] studied thetime-harmonic response of a pile in a poroelastic half

space under lateral loadings. Wang et al. [17] extendedthe Mukis method to dynamic response of pile groupsembedded in a poroelastic medium and computed theaxial forces of pile groups and pore pressures along thepile body. Recently, Wang et al. [16] used finiteinfiniteelement method and investigated the vertical vibrationsof composite saturated foundations.

It is worth mentioning that, although many problemsinvolving dynamic response of piles and pile groups inelastic/saturated medium have been studied, the solutionsconcerning the dynamic torsional response of piles embed-ded in poroelastic half space are not reported in the liter-ature. It is, therefore, important to examine the torsional

dynamic response of piles within the framework of Biotstheory, and to determine the influence of poroelastic med-ium. The main objective of this paper is to investigate rig-orously the torsional vibrations of an elastic circularcylindrical pile, which is embedded in an isotropic homo-geneous and completely saturated half space. The problemis formulated by using an approximative scheme similar tothat used by Muki and Sternberg for the case of an elas-tostatic load-transfer to a half space from an axiallyloaded bar. The governing equation of the problem isfound to be a Fredholm integral equation of the secondkind and can be solved by an appropriate numerical

method. Selected numerical results for the torque transfer,torsional angle and dynamic compliance can be obtainedto portray the influence of pilesoil system on the dynamictorsional response of piles. The results obtained can beused in practice as a guideline in obtaining the dynamicstiffness and damping of piles, machine foundations anddynamic torsional wave testing of composite pilefoundations.

2. Governing equations and general solutions of poroelastic

medium

2.1. Dynamic equations of saturated medium

Biots theory of linear, isotropic poroelasticity isemployed in this study. This formulation assumes thatthe porous material is constructed so that the solid phaseforms a structure that contains statistically distributedsmall pores filled with a Newtonian-viscous compressiblefluid. The bulk material is assumed to be homogeneouson a macroscopic scale, and the pores are assumed to beinterconnected. The solid skeleton is taken to be linear elas-tic and undergoing small deformations. The fluid flow isassumed to be of the Poiseuille type so that the fluid inertiaand friction are uniquely characterized by density, viscosity

and pore dimensions.

According to Biots theory, the governing differentialequations for a completely saturated poroelastic mediacan be written as [1]

lr2ur kc l oeor

l 2r2

ouh

oh l ur

r2 aMon

or

qur

qfwr

1a

lr2uh kc l oe

oh l 2

r2our

oh l uh

r2 aMon

oh

quh qfwh 1blr2uz kc l oe

oz aMon

oz quz qfwz 1c

aMoe

orMon

or qfur mwr b _wr 2a

aM1

r

oe

ohM1

r

on

oh qfuh mwh b _wh 2b

aMoe

ozMon

oz qfuz mwz b _wz 2c

In the above equations, ui and wi (i= r,h, z) are the dis-placement components of the solid and of the fluid relativeto the solid in the cylindrical polar coordinate system,respectively; n is the porosity; k and l are the Lames con-stants; kc = k + a

2M, where a and Mare the Biots param-eters accounting for compressibility in the two-phasematerial; q = nqf + (1 n)qs is the mass density of bulkmaterial, where qs and qf are the mass density of fluidand of grains, respectively; m = qf/n is a Biots density-likeparameter; b = g/k, where g is the pore fluid viscosity and kthe permeability.

r2

o

2

or2 1

r

o

or1

r2

o2

oh2 o

2

oz2is the Laplace operator, while

e ouror

urr

1r

ouh

oh ouz

ozand

n owror

wrr

1r

owh

oh owz

oz

are the dilatations of the solid and of the fluid relative tothe solid, respectively. A dot over a symbol indicates thedifferentiation with respect to time variable t.

The fluid pressure pf can be expressed as

pf aMe Mn 3Consider a semi-infinite saturated half-space as shown in

Fig. 1 in which (r,h, z) is the cylindrical polar coordinatesystem with z-axis normal to the free surface. A harmonicuniform torque eixt (x is the angular frequency) is appliedat the depth z 0 of the half space. a is the radius of theapplied loading region. Considering the symmetry of theproblem, the only non-vanishing displacements of the solidand of the fluid relative to the solid for a homogeneous,isotropic, saturated soil are uh (r, z)e

ixt and wh(r, z)eixt,

respectively. So Eqs. (1) and (2) can be further written as

lr2 1r2

uh qx2uh qfx2wh 4a

qfxuh mxwh ibwh 4b

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For brevity, the time harmonic factor eixt is suppressedfrom all the expressions in the sequal.

The only non-vanishing bulk stress components of thesoil are

szhlouh

oz 5a

srh l ouh

or uh

r

5b

It is convenient to introduce the dimensionless constantsand variables: r r=a, z z=a, uh uh=a, wh wh=a,qf qf=q, m m=q, d ax

ffiffiffiffiffiffiffiffiq=l

p, b ab= ffiffiffiffiffiffiqlp ,

szh szh=l, srh srh=l.After introducing the dimensionless constants and vari-

ables, Eqs. (4) and (5) can be written as

r2 1r2

uh d2uh qfd2wh 6a

qfduh mdwh ibwh 6b

szh ouh

oz7a

srh ouhor

uhr

7bWe denote the Hankel transform of order t of a functionfr;z by~ftp;z

Z10

rfr;zJtprdr

and its inverse transform by

fp;z Z

1

0 p

~f

t

p;zJtprdpwhere Jt(pr) is the Bessel function of the first kind of ordert. p is the transform parameter.

By means of Hankel transform of the first-order to Eqs.(6)(7a) and the second-order to Eq. (7b), we can get

p2~u1h d2

dz2~u1h qfd2~w1h d2~u1h 8a

ib~w1h qfd~u1h md~w1h 8b

~s1zh d

dz~u1h 9a

~s2rh

p~u1h

9b

Eqs. (8a) and (8b) are ordinary differential equations.Considering that there exists both incoming and out-com-ing waves in the system, the solutions of Eqs. (8a) and (8b)can be obtained directly as

~u1h AejzBejz 10a

~w1h qfd

ib md AejzBejz 10bin which j2 = p2 s2, s2 ibd2 q2fd3 md3=ib md, sis the dimensionless complex wave number associated withthe rotational wave. A and Bare the arbitrary functions ofp which can be determined from the boundary conditions.

Substitute Eqs. (10a) and (10b) into Eqs. (9a) and (9b)and yield

~s1zh AjejzBjejz 11a~s2rh pAejzBejz 11b

2.2. Mixed boundary-value problems

At this stage, it is appropriate to first derive the solu-tions for the case of a shearing traction equivalent to a uni-form harmonic torque eixt acting over a circular area ofradius a in the interior of a saturated half space at z = z0

as shown in Fig. 1. These solutions, which would serve asthe fundamental solutions necessary for solving the mainproblem, can be approached by considering the half spaceas divided into two domains, namely, domain D1(0 6 z6 z 0) and domain D2 (z 0 6 z < 1) as shown inFig. 1. The solutions of each domain are in the form of

Eqs. (10) and (11), containing two constants of integrationAi and Bi. The subscript iis used to denote a domain num-ber. However, the constants A2 should vanish to guaranteethe boundedness of the solutions as z approaches infinity.Thus there are three constants to be determined from thefollowing boundary and continuity conditions:

szh1r; 0 0 0 6 r< 1 12auh1r;z0 uh2r;z0 0 6 r< 1 12bszh2r;z0 szh1r;z0 2r

pr6 1 12c

szh2

r;z0

szh1

r;z0

0

r> 1

12d

Combining Eqs. (10)(12) yields

~u1h J2pppj

ejjz0zj ejz0z 13a

~w1h qfd

ib mdJ2pppj

ejjz0zj ejz0z 13b

~s1zh J2ppp

sgnz0 zejjz0zj ejz0z 13c~s2rh

J2ppj

ejjz0zj ejz0z 13d

where sgn[x] is a sign function, z0 z0=a.

Fig. 1. Geometry of a harmonic uniform torque embedded in saturatedsoil.

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3. Torsional vibrations of single piles in saturated half space

The problem of the axially symmetric torsional vibra-tions of an elastic pile embedded in the semi-infinite satu-rated half space is depicted in Fig. 2. In addition to thosepreviously defined, the following notations are introduced

in this system: h denotes the length of the pile; T0 denotesthe magnitude of the torque applied at the top end of thepile which is flush with the surface of the half space; x1,x2 and x3 are Cartesian components of position vector x;thus x1 is identical to r for h = 0 and x3 is identical to z.The pile is bonded to the half space in the area (r < a,z = h) and along the surface of the pile (r = a, 0 6 z 6 h).

Following the approximative scheme similarly used byMuki and Sternberg [10] in the elastostatic axial load-trans-fer, the system in Fig. 2 is decomposed into two systems: anextended saturated half space X as shown in Fig. 3(a) and afictitious pile Xv as shown in Fig. 3(b). Assuming that thepermeability and porosity of the pile are same as those of

the saturated soil, the shear moduli lv and mass densityqv of the fictitious pile are as follows:

lv l2 l 14aqv q2 q 14bwhere l and q denote the shear modulus and mass densityof the saturated half space, while l2 and q2 are the shearmodulus and mass density of the real pile.

The extended half space is subjected to a distributedbond torque tv(z) which is exerted by Xv on X at x3 = zin the region D in place of the pile. In addition, X is alsosubjected to end torques T0 Tv(0) and Tv(h) applied atthe terminal cross sections as shown in Fig. 3(a). The bondtorque tv(z), the end torque T0 Tv(0) and Tv(h) areassumed to be distributed linearly in the form of Eq.(12c) over their respective cross sections pz 0 < z< h,p0 z 0 and ph z h.

Conversely, the bond torques tv(z), the end torques Tv(0)and Tv(h) are exerted by the extended half space X on thefictitious pile Xv, which may be treated as a one-dimen-sional elastic bar subjected to dynamic torsional loadings.

Considering the harmonic vibration, the equation of thedynamic equilibrium for the fictitious pile is [2]

lvJo

2/vzoz2

tvz qvJd2/vz 0 15

where /vz

is the torsional angle of the fictitious pile.

J p=2, tvz tvz=la2, lv lv=l, qv qv=q.The fictitious torquetorsional angle relationship for the

fictitious pile can be expressed as

Tvz lvJd/vzdz

0 6 z6 h 16

where Tvz Tvz=la3.The tangential displacement uhr;z of the extended sat-

urated half space can be written as

uhr;z T0 Tv0uhTr;z; 0 TvhuhTr;z; h

Zh

0

uhT

r;z;z0

tvz0

dz0

17

where T0 T0=la3, uhTr;z;z0 is the fundamental solu-tion derived in the previous section, i.e. the displacementin tangential direction at a point x(r,h, z) due to a uniformtorque applied at a depth z0 (Fig. 1).

Assuming that there is no disengagement between thepile and soil, i.e. the pile and soil is fully bonded when aharmonic torsional loading is applied on the pile, it has

uh1;z /vz 18In view of Eqs. (15)(17) and performing appropriate inte-grations in Eq. (18), it leads to

1

lvJ 2c1z;z

Tvz

cT1;z; 0 qvd2

lv

Zh0

cT1;z; fdfZhf

Tvf0df0

Zh

0

Tvf ocT1;z; fof

df DZh

0

cT1;z; fdf

/T1; h; 0 Zh

0

Tvf o/T1; h; fof

qvd2

lv/T1; h; f

"(

Zh

f

Tvf0df0#df) 19

Fig. 2. Geometry of pile embedded saturated half space.

Fig. 3. Decomposition of the problem.

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where D 1=qvJd2 Rh

0/T1; h;z0dz0

h i1, Tvz

Tvz=T0. /T, cT and c1 are influence functions, which aregiven in detail in Appendix.

Eq. (19) is a Fredholm integral equation of the secondkind with weak singularity, from which the dimensionlessfictitious torque Tvz can be calculated.

The torsional angle /z of the real pile is equal to /vzof the fictitious pile, which can be written as

/z T0DqvJd

2/T1;h;0

Zh0

Tvz0o/T1;h;z0

oz0

"(

qvd2

lv/T1;h;z0

Zhz0Tvfdf

#dz0 T0

lvJ

Zhz

Tvfdf)

20

while the real pile torque Tz is determined by combiningthe fictitious pile torque Tvz with the corresponding areaintegral of the shear stress shz in the region D ofX, i.e.

Tz Tvz T0 Tvhshzr;z; 0

Tvhshzr;z; h Z

h

0

oTvz0oz0

shzr;z;z0dz0

qvJd2Zh

0

/vz0shzr;z;z0dz0 21

where shzr;z;z0 is the influence function of the distributingtorque which is also listed in Appendix.

4. Numerical examples and discussions

The Fredholm integral equation of the second kindderived above can be easily solved numerically by discretiz-

ing it to a system of algebraic equations in the interval [0 ; h]using trapezoidal rule of integration with 35 subintervals[15]. Once the integral equation has been solved, the

dynamic response of the pile can be evaluated immediatelyfrom Eqs. (20) and (21).

The above questions can be reduced to the torsionalvibrations of single piles embedded in the one-phase elastichalf space [14,15] if we put qf = 0. The degenerated resultsfor the single pile embedded in the saturated half space are

compared with those of Rajapakse [14]. Fig. 4 shows thecomparison between the real and imaginary parts of thedegenerated dynamic torsional response of a single pilefor h/a = 10.0, q2/q = 1.2, l2/l = 21.0 and d = 1.0. FromFig. 4, excellent agreement is observed between the tworesults at all points and the accuracy of the present solu-tions is therefore confirmed.

In the remainder of the paper, parametric studies aregiven to illustrate the influence of geometrical and physicalproperties of the saturated soilpile system and appliedloadings on the piles dynamic response. The followingset of dimensionless parameters are used throughout:b

0:1, qf

0:53, m

1:1, q2/q = 1.2, l2/l = 21.0, unless

otherwise stated.

4.1. Effect of pile length and vibration frequency

In order to explore the influence of pile length and vibra-tion frequency on the dynamic response of the pile, threevalues of h/a and four values ofd are considered, namely,h/a = 5.0,10.0,20.0 and d = 0.1,0.5,2.0,3.0. The numericalresults are shown in Figs. 510.

Figs. 510 show the typical results of the piles non-dimensionalized torque Tz=T0 and torsional angle/z=T0 along the pile body at different values of dand h/a. From these figures, it can be seen that the pileslengthradius ratio and vibration frequency have animportant influence on the piles dynamic response. At

Fig. 4. Comparison of the results of Rajapakse and present paper.

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a given value of h/a, when the vibration frequency d isvery low, the changes for the real and imaginary partsof the piles torque and torsional angle are gentle andthere is no fluctuation along the pile body. But whenthe vibration frequency is larger (for example, dP 2.0),the curves begin to fluctuate, especially when the vibra-tion frequency is very larger, this fluctuation phenomenais more evident.

Compared with Figs. 510, it can also be seen that: whenthe piles lengthradius ratio is smaller (for example, h/a 65.0), the dynamic response at the bottom of the pile is stilllarger. With the increase of the piles lengthradius ratio,

the real and imaginary parts of torque and torsional angleat the bottom of the pile decrease gradually. When h/areaches 10.0, the torque and torsional angle at the bottomof the pile are almost zero. This illustrates that, similar withthe static problem, there also exists the concept of effectivepile length for the dynamic torsional vibrations of piles.When the pile length increases to a certain value, thereexists a critical depth and the piles response is almost zerobelow this critical depth. It indicates that: if the pile lengthexceeds the effective pile length, the contribution of theextra part to the piles dynamic response is tiny enoughto be neglected.

Fig. 5. Effect of vibration frequency on pile torque for h/a = 5.0.

Fig. 6. Effect of vibration frequency on torque of pile for h/a = 10.0.



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4.2. Effect of pilesoil relative rigidity

In order to further investigate the effect of pile rigidityon the dynamic response, we also calculate the pilesdynamic compliance coefficient vs. the pilesoil relativeshear modulus ratio l2/l. The definition of the dimension-less dynamic compliance coefficient is

CT 16

3 /0=T0 22

where /(0) is the torsional angle of the pile head.

The physical and geometrical parameters of the pilesoilsystem used in this section are same as those given in sec-tion one. The results are shown in Figs. 1113.

As expected, the effect of pilesoil relative rigidity isnoticeable as l2/l changes. From Figs. 1113, it can beseen that, with the increase of l2/l, the real part of thepiles dynamic compliance coefficient decreases gradually,but the imaginary one is on the inverse. Compared withthese figures, it can also be seen that the piles length

radius ratio h/a has an obvious influence on the dynamiccompliance coefficient, especially on the imaginary partof CT. With the increase of h/a, the real and imaginary

Fig. 7. Effect of vibration frequency on the torque of pile for h/a = 20.0.

Fig. 8. Effect of vibration frequency on torsional angle for h/a = 5.0.



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Fig. 9. Effect of vibration frequency on torsional angle of pile for h/a = 10.0.

Fig. 10. Effect of vibration frequency on torsional angle of pile for h/a = 20.0.

Fig. 11. Variation of pile impedance vs. the ratio of pilesoil system for h/a = 5.0.



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parts of CT decrease gradually. Furthermore, these effectsare more pronounced at higher dimensionless frequenciesand shorter pile length.

4.3. Effect of pore fluid

Finally, in order to demonstrate the effect of pore fluidon the dynamic response of the pile, we calculate the

dynamic response of the pilesoil system using h 10:0,d = 2.0, b 0:1; 10:0; 3000:0. The bigger the value of b is,the smaller the permeability and larger the viscosity ofthe soil is. If there is no fluid in the model, the half spacewill become a dry one, which had been considered by

Rajapakse [14]. The results are presented in Figs. 1416.Fig. 14 shows that b has significant influence on theresponse of the pile and tends to reduce the real and



Fig. 14. Effect of pore fluid on torque.



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imaginary parts of the torque, torsional angle and thedynamic compliance coefficients of the pile (compared tothe dry case). Furthermore, these effects are more pro-nounced, especially when b is smaller. The parameter b isassociated with the soil viscosity and permeability. Withthe increase ofb (i.e. the permeability of the soil decreasesor its viscosity increases gradually), the curves tends to thedry one. It indicates that the existing of the fluid canweaken the dynamic response of the pile, especially whenthe soil permeability is very larger.

All of the above phenomena demonstrate that theexisting of pore fluid, the lengthradius ratio of piles,vibration frequency and geometrical properties of pilesoil system have an important influence on the dynamictorsional response of piles. The differences shown inFigs. 516 can be attributed to the inter-granular energylosses in the solid phase, the viscous resistance to theflow of pore fluid and the deformation of piles. All ofthese influence factors cannot be neglected when analyz-ing and designing the pile foundations subjected to

dynamic torsional loadings.

5. Conclusions

The harmonic torsional vibrations of single piles embed-ded in the saturated half space have been considered for thefirst time by means of integral transform and Mukis meth-ods. The effect of geometrical properties of the pilesoil sys-tem and vibration frequency on the dynamic compliance,torque and torsional angle has been investigated. Fromthe analysis, some conclusions can be drawn as follows:

(1) Comparison with the existing solutions available forsingle piles embedded in a one-phase elastic mediumconfirms the accuracy of the proposed formulationsand also the numerical computation involved.

(2) The change trend of torsional vibrations of singlepiles embedded in saturated half space is similar tothat of single piles embedded in one-phase elastichalf-space.

(3) Selected numerical results illustrate that: the maininfluence factors affecting the piles dynamic response

are the physical and geometrical properties of pile

Fig. 15. Effect of pore fluid on torsional angle of pile.

Fig. 16. Effect of pore fluid on pile impedance.



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soil system and applied loadings. The piles dynamicresponse will be obviously different at different pilesoil rigidity, pile slenderness ratio, mechanical param-eters of saturated soil and vibration frequency ofloadings.

(4) Similar with that of static problem, there also exists

the concept of effective pile length for the dynamicproblems. If the pile length exceeds the effective pilelength, the contribution of the extra part to thepiles dynamic response is tiny enough to beneglected.

(5) The existing of the pore fluid can weaken thedynamic response of the pile, especially when thesoil permeability is very larger. But when the soilspermeability is very low, the dynamic response ofthe pile embedded in saturated media tends to thatin the dry one.

The parametric study presented by this paper can

provide guidelines for the analysis and design of pilefoundations subjected to dynamic torsional loadingsand dynamic pile testing of practical engineering. Itshould be mentioned that the solutions presented inthis paper are only applicable to single piles embeddedin the saturated half space. If the complex interactionfactors are introduced between the adjacent piles, themethods presented by this paper can be extended easilyto analyze the dynamic response of pile groups in sat-urated and layered saturated soils. The authors arecurrently carrying out this study and the results willbe available soon.

Acknowledgements

Support for this work was provided by Zhejiang NaturalScience Foundation (ZJNSF), Project number Y105480and Zhejiang Provincial Education Department Founda-tion, Project number 20051414. These supports are grate-fully acknowledged.

Appendix. Influence functions

^cT1;z;z0

1

pZ1

0 J1pJ2p z

0 zj jz0 z e

j z0zj j ejz0z

dp z6z0 23a

c11;z;z0 1p

Z10

J1pJ2pejjz0zjdp z6z0 23b

shzr;z;z02Z1

0

J22pp

jz0 zj

z0 z ejjz0zj ejz0z

dp z6z0 23c

/T1;z;z01p Z

1

0

J1pJ2pj

ejjz0zj ejz0zdp z6z0 23dwhere j2 = p2 s2, s2 ibd2 q2fd3 md3=ib md.

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Torsional Vibrations of Single Piles Embedded in Saturated Medium