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doi:10.1016/j.so

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(H.R. Masoum

Soil Dynamics and Earthquake Engineering 27 (2007) 126–143

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Prediction of free field vibrations due to pile driving using a dynamicsoil–structure interaction formulation

H.R. Masoumi�, G. Degrande, G. Lombaert

Department of Civil Engineering, K.U. Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium

Received 3 May 2006; accepted 20 May 2006

Abstract

This paper presents a numerical model for the prediction of free field vibrations due to vibratory and impact pile driving. As the focus

is on the response in the far field where deformations are relatively small, a linear elastic constitutive behaviour is assumed for the soil.

The free field vibrations are calculated by means of a coupled FE–BE model based on a subdomain formulation. First, the case of

vibratory pile driving is considered, where the contributions of different types of waves are investigated for several penetration depths. In

the near field, the soil response is dominated by a vertically polarized shear wave, whereas in the far field, body waves are importantly

attenuated and Rayleigh waves dominate the ground vibration. Second, the case of impact pile driving is considered. A linear wave

equation model is used to estimate the impact force during the driving process. Apart from the response of a homogeneous halfspace, it is

also investigated how the soil stratification influences the ground vibration for the case of a soft layer on a stiffer halfspace. When the

penetration depth is smaller than the layer thickness, the layered medium has no significant influence on ground vibrations. However,

when the penetration depth is larger than the layer thickness, the influence of the layered medium becomes more significant. The

computed ground vibrations are finally compared with field measurements reported in the literature.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Dynamic soil–pile interaction; Vibratory pile driving; Impact pile driving; Ground vibrations; Coupled finite element–boundary element

formulation

1. Introduction

In an urban environment, vibrations generated byconstruction activities often affect surrounding buildings.These vibrations may disturb sensitive equipment or peopleand cause cracks in the walls or in the facade. Permanentsettlements, densification and liquefaction may also occurin the soil due to construction activities.

The present study aims to develop a model to predict freefield vibrations in the environment due to pile driving.Ground motions due to pile driving generally depend on(1) the source parameters (driving method, released energyand pile depth), (2) the interaction between the drivingmachine, the pile and the soil and (3) the propagation ofthe waves through the soil. Therefore, the pile driving

e front matter r 2006 Elsevier Ltd. All rights reserved.

ildyn.2006.05.005

ing author. Tel.: +3216 32 16 68; fax: +3216 32 19 88.

ess: [email protected]

i).

problem can be divided into three main subproblems: (1)excitation mechanisms and dynamic pile-driver interaction,(2) dynamic soil–pile interaction and (3) wave propagationin the soil.Three main driving techniques to install foundation and

sheet piles can be distinguished: impact pile driving,vibratory pile driving and jacking.Impact pile driving is known as the oldest technique of

driving and produces transient vibrations in the ground.The driving energy is provided by a ram mass that dropsfrom a specific height and strikes the pile head with adownward impact velocity. Impact driving generates a highenergy and can drive piles even in difficult soil conditions.As impact driving generates high noise and vibration levels,it is not preferred in an urban area and its use is more andmore limited.During the last decades, vibratory pile driving has been

preferred to other driving techniques due to economical,practical and environmental concerns. Vibratory pile

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p

s

s

p

sext

Fig. 1. Geometry of the subdomains.

H.R. Masoumi et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 126–143 127

driving produces a harmonic vibration in the ground and isused to install piles in granular soils and, with lowoperating frequency, also in cohesive soils. It causes asignificant degradation in the soil: the cyclic movementscause a reduction of the soil strength, which allows the pileto penetrate with a much lower surcharge than required fora non-degraded soil. The vibratory motion is generated bycounter-rotation of eccentric masses actuated within anexciter block.

Jacking does not induce environmental disturbance butis very expensive. This technique is only used wheresensitive environmental conditions are encountered.

The transmission of vibrations through the soil duringpile driving is complex, as the soil behaviour around thepile is difficult to describe. Research on pile drivingproblems has concentrated on two main directions. Settingup numerous field measurements, several authors havefocussed on environmental effects (far field or externaleffects) [1–3]. Some efforts have also been made to developmodels to assess the driving efficiency, investigating thedriveability and the bearing capacity of driven piles (nearfield or internal effects) [4–6].

Modelling of the dynamic soil–pile interaction problemduring pile driving requires to account for the non-linearconstitutive behaviour of the soil. Such model should beable to consider not only the behaviour of the soil undercyclic loading (which occurs during pile driving) but alsothe geometrical non-linearities due to large deformations inthe soil around the pile. Accounting for the non-linearity ofthe soil, some significant advances have been made usingthe non-linear finite element (FE) method applied togranular and cohesive materials [7,8]. The driving responseof open-ended piles subjected to multiple hammer blowshas been studied by Liyanapathirana et al. [7]. Anaxisymmetric FE model has been used, and the soil domainsurrounding the pile shaft is truncated using the axisym-metric shear and dilatational wave transmitting boundariesdeveloped by Deeks and Randolph [9]. In this work, thesoil has been modelled as a perfect elastoplastic Von Misesmaterial.

During pile driving, the transmitted energy through thesoil is high and causes plastic deformations in the nearfield. In the far field, however, reported data show that thevibrations cause deformations in the elastic range [10]. Asthe focus here is on the response in the far field, wheredeformations are relatively small, a linear elastic constitu-tive behaviour is assumed for the soil.

In the present study, a model for both vibratory andimpact driving is proposed to predict free field vibrations.Using a subdomain formulation for dynamic soil–structureinteraction, developed by Aubry et al. [11,12], the dynamicsoil–pile interaction problem is investigated. A coupledfinite element–boundary element (FE–BE) model is pre-sented, in which the pile (bounded domain) and the soil(unbounded domain) are modelled using the FE methodand the BE technique, respectively. The subdomainformulation has been implemented in a computer program

MISS (Modele d’Interaction Sol-Structure) [13]. Theprogram MISS is used to compute in the frequency domainthe impedance functions as well as scattered wave fields inthe soil using a BE formulation based on the Green’sfunctions of a horizontally stratified soil.Free field vibrations in the soil are evaluated due to

vibratory and impact driving of a concrete pile in ahomogeneous halfspace and a layered medium, consistingof a soft layer on a stiffer halfspace. Results are discussedfor different penetration depths of the pile.

2. Numerical modelling

2.1. Problem outline

The proposed model is based on the following hypoth-eses: (1) the soil medium is elastic with frequencyindependent material damping (hysteresis damping), (2)no separation is allowed between the pile and the soilmedium, (3) all displacements and strains remain suffi-ciently small and (4) the pile is embedded in a horizontallylayered soil.The domain is decomposed into two subdomains: the

unbounded semi-infinite layered soil denoted by Oexts and

the bounded structure (the pile) denoted by Op. Theinterface between the soil and the pile is denoted by S(Fig. 1).

2.2. Governing equations

The deformations are assumed to be small enough toallow for a linear approximation of the constitutiveequations, so that all equations can be elaborated in thefrequency domain. The displacement wave fields in eachsubdomain during the dynamic excitation are denoted byuaðx;oÞ where a ¼ s denotes the soil and a ¼ p refers to thepile. The stress tensors raðuaÞ can be expressed as linearfunctions of the strain tensors eaðuaÞ using Hooke’s law:

saij ¼ la�akkdij þ ma�aij, (1)

ARTICLE IN PRESS

s

s

sext

Fig. 2. The displacement field uscðupÞ radiated in the soil.

H.R. Masoumi et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 126–143128

where the components of the small strain tensor arecalculated from the displacements as follows:

�aij ¼12ðuai;j þ uaj;iÞ, (2)

where la and ma are the Lame parameters. In order toaccount for the dissipation of the internal energy in the soil(material damping), complex Lame coefficients are used:

m�s ¼ msð1þ 2ibsÞ, (3)

l�s ¼ lsð1þ 2ibsÞ, (4)

where bs is the material damping ratio.First, the pile Op is considered. The boundary Gp ¼

Gps [ S of the pile is decomposed into a boundary Gps

where tractions tp are imposed and the soil–pile interface S.The displacement vector up of the pile satisfies thefollowing Navier equation and boundary conditions:

divrpðupÞ þ rpb ¼ �rpo2up in Op, (5)

tpðupÞ ¼ tp on Gps, (6)

where rpb denotes the body force on the pile and tðuÞ ¼

rðuÞ � n is the traction vector on a boundary with a unitoutward normal vector n. Body forces will not beaccounted for in the following.

Second, the exterior soil domain Oexts is taken into

consideration. The boundary Gexts ¼ Gss [ Gs1 [ S of the

soil domain Oexts is decomposed into the boundary Gss

where tractions are imposed, the outer boundary Gs1

where radiation conditions are imposed and the soil–struc-ture interface S. A free boundary condition or zerotractions are assumed on Gss. The displacement vector usof the soil satisfies the Navier equation and the followingboundary conditions:

divrsðusÞ ¼ �rso2us in Oext

s , (7)

tsðusÞ ¼ 0 on Gss. (8)

The displacement compatibility and stress equilibriumconditions are imposed on the interface S as follows:

us ¼ up on S, (9)

tpðupÞ þ tsðusÞ ¼ 0 on S. (10)

According to the compatibility condition (9), the displace-ment vector us is the wave field uscðupÞ that is radiated in thesoil due to the pile motion up on the interface S (Fig. 2):

us ¼ uscðupÞ in Oexts . (11)

The wave field uscðupÞ radiated in the soil satisfies theelastodynamic equation in Oext

s and the following boundaryconditions:

divrsðuscðupÞÞ ¼ �rso2uscðupÞ in Oext

s , (12)

tsðuscðupÞÞ ¼ 0 on Gss, (13)

uscðupÞ ¼ up on S. (14)

2.3. Variational formulation

The equation of motion of the dynamic soil–pileinteraction problem can be formulated in a variationalform for any virtual displacement field v imposed on thepile:Z

Op

eðvÞ : rpðupÞdO� o2

ZOp

v � rpup dO

¼

ZGps

v � tp dGþZSv � tpðupÞdS, ð15Þ

where : denotes the contraction of two tensors. Accountingfor the equilibrium on the soil–pile interface S in Eq. (10),the variational equation (15) becomesZ

Op

eðvÞ : rpðupÞdO� o2

ZOp

v � rpup dO

þ

ZSv � tsðuscðupÞÞdS ¼

ZGps

v � tp dG. ð16Þ

A modal reduction technique is applied where anapproximate solution is sought in a subspace of finitedimension. The pile displacement vector up is decomposedas a linear combination of vibration modeswm ðm ¼ 1; . . . ; qÞ:

up ’Xq

m¼1

wmam ¼ Wa, (17)

where the modes wm ðm ¼ 1; . . . ; qÞ are collected in amatrix W and the modal coordinates am ðm ¼ 1; . . . ; qÞare collected in a vector a. Several alternatives are possiblefor the selection of the modes wm to describe the kinematicsof the pile, as will be demonstrated later. In a Galerkinformulation, a similar modal decomposition is used for thevector v of virtual displacements:

v ¼Xq

m¼1

wmdam ¼ Wda, (18)

where da is a vector with virtual modal coordinates.Introducing the decompositions (17) and (18) in thevirtual work expression (16) and expressing that theresulting equation must hold for any set of virtual modal

ARTICLE IN PRESSH.R. Masoumi et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 126–143 129

coordinates da results into the following q-dimensionalsystem of equations:Z

Op

eðWÞ : rpðWÞdO� o2

ZOp

WTrpWdO

"

þ

ZS

WTtsðuscðWÞÞdS

#a ¼

ZGps

WT tp dG. ð19Þ

2.4. Discretization of the variational formulation

In an FE formulation, the displacement field up isapproximated as up ’ Npup, where Np are the globallydefined shape functions and up is the displacement vector atall nodal points. Analogously, the strain vector ep isapproximated as ep ¼ LNpup ¼ Bpup, with L the matrixwith derivative operators. The stress vector is defined asrp ¼ Dpep, with Dp the constitutive matrix for an isotropiclinear elastic material.

Applying the same FE approximation to the modes ofthe pile wm ’ Npwm

, the modal decomposition (17) of thepile displacement vector can be written as follows:

up ’ Npup ’Xq

m¼1

Npwmam ¼ Np W a. (20)

Introducing this discretization into the system of equations(19) results in the following system of discretized equations,neglecting the effect of material damping in the pile:

WT½Kp � o2Mp þ Ks�W a ¼ WTfp, (21)

where the stiffness matrix Kp of the pile is defined as:

Kp ¼

ZOp

BTpDpBp dO, (22)

the mass matrix Mp of the pile is

Mp ¼

ZOp

NTprpNp dO (23)

and the vector of external forces fp on the pile is defined as

fp ¼

ZGps

NTp tp dG. (24)

The dynamic stiffness matrix or impedance matrix Ks ofthe semi-infinite layered soil domain Oext

s around the pile isequal to

Ks ¼

ZSNT

p tsðuscðNpÞÞdS (25)

although it is preferable to compute directly its projectionon the imposed displacement fields W of the pile:

WTKs W ¼ZSðNp W ÞTtsðuscðNp WÞÞdS. (26)

This impedance matrix is calculated using a BE technique.As the BE method is based on the Green’s functions of ahorizontally layered halfspace, only a discretization of theinterface S between the soil and the pile is required and thenumber of the unknowns is drastically reduced.

A computation procedure is developed to solve thegoverning system of equations (21). This procedure can bepartitioned into two parts. First, using the StructuralDynamics Toolbox in MATLAB, the FE model of the pileis made. In the second part, using the program MISS 6.3,the soil impedance WTKs W as well as the modal responsesof the soil uscðWÞ are computed. The solution of thedynamic soil–pile interaction problem in terms of themodal coordinates allows to compute the soil tractions onthe interface and, subsequently, in the free field.The BE analysis applied for the computation of the soil’s

impedance of structures embedded in the soil may sufferfrom the appearance of fictitious eigenfrequencies. Thisnumerical deficiency predominantly occurs when theexcitation frequency is equal to an eigenfrequency of theinterior soil domain with Dirichlet boundary conditionsalong the interface S and free boundary conditions alongthe free surface Gps, leading to spurious results. Thephenomenon of fictitious eigenfrequencies generally occursin the high frequency range, depending on the geometry ofthe pile and the stiffness of the excavated soil. For vibratorypile driving, the frequency of excitation is low, between 10and 50Hz, and fictitious eigenfrequencies are usually not amatter of concern. For impact driving, however, thefrequency content of the loading is higher and fictitiousfrequencies need to be mitigated. A solution technique forelastodynamic problems has been proposed by Pyl et al.[14], by analogy with Burton and Miller’s approach foracoustic problems. The method is based on a combinationof the boundary integral equation for the displacement andthe displacement gradient along the normal direction on theboundary, using an imaginary coupling parameter a. In thefollowing examples, this method has been applied wheneverfictitious eigenfrequencies are an issue.

3. Validation of the dynamic soil–pile interaction model

In order to validate the numerical soil–pile interactionmodel, the vertical impedance of a single pile is comparedwith results reported by Kaynia and Kausel [15]. The pileimpedance functions are generally defined as the harmonicforces or moments that must be applied on the pile head inorder to generate a unit harmonic motion in a specifieddirection.Several alternatives are available to write the FE

displacement vector up of the pile as a superposition of q

modes wmaccording to Eq. (20):

up ’Xq

m¼1

wmam ¼ W a. (27)

When the distinction is made between the six rigid bodymodes Lm and q� 6 flexible modes /m, this decompositioncan alternatively be written as

up ’X6m¼1

Lma1m þXq�6m¼1

/ma2m ¼ L a1 þU a2, (28)

ARTICLE IN PRESS

(a) (b) (c) (d)

Fig. 3. (a) Rigid body mode and first three flexible modes of the pile fixed

at the head at (b) 100.2Hz, (c) 300.6Hz and (d) 501.5Hz.


where a1m and a2m are the modal coordinates correspond-ing to the rigid body modes and the flexible modes of thepile, respectively. Based on this modal decomposition, thesystem of equations (21) can alternatively be written as

LTð�o2Mp þ KsÞL LTð�o2Mp þ KsÞU

UTð�o2Mp þ KsÞL UTðKp � o2Mp þ KsÞU

24

35 a1

a2

( )

¼LTfp

UTfp

8<:

9=;. ð29Þ

Different alternatives can now be considered, depending onthe choice of the modes of the pile: (1) the modes of thepile, fixed at the head Gps, (2) the modes of the free pile and(3) the modes of the pile on an elastic foundation. In thefirst two cases, it is preferred to explicitly differentiatebetween the rigid body modes and the flexible modes of thepile, using the modal decomposition (28) and the equili-brium equation (29), whereas in the third case it is aspractical to use the decomposition (27) together with theequilibrium equation (21) in its original form.

The first alternative is particularly appealing for thecalculation of the impedance matrix of the pile, as the forcevector fp on the pile head is orthogonal to the modes U,which is reflected by a zero vector UTfp on the right-handside of Eq. (29). Therefore, this system of equations caneasily be reduced to

SðoÞa1 ¼ LTfp, (30)

where SðoÞ is the six-by-six condensed impedance matrix,obtained by evaluating the Schur complement of theequation of motion (29) in the modal coordinates a1. Thismatrix is complex and its imaginary part representsthe effect of geometric and material damping in the soil.The dynamic stiffness coefficients of the pile Smnða0Þ aregenerally described in terms of a dimensionless frequencya0 ¼ odp=Cs, with dp the pile diameter and Cs arepresentative value for the shear wave velocity in the soil,and are normalized by the static stiffness coefficients K s

mn:

Smnða0Þ ¼ K smn½kmnða0Þ þ ia0cmnða0Þ�, (31)

where the subscripts mn ¼ vv; hh; rr and tt denote theterms on the diagonal of the impedance matrix and thesubscripts mn ¼ hr indicate the coupled horizontal-rockingterm. kmnða0Þ and cmnða0Þ are the dimensionless dynamicstiffness and damping coefficients of the pile.

The vertical impedance of a pile embedded in ahomogeneous soil domain is now considered. The pilehas a length Lp ¼ 10m, a diameter dp ¼ 0:50m, a Young’smodulus Ep ¼ 40 000MPa, a Poisson’s ratio np ¼ 0:25 anda density rp ¼ 2500 kg/m3. The longitudinal wave velocityin the pile is equal to Cp ¼ 4000m/s.

The pile is modelled using 8-node isoparametric brickelements. Fig. 3 shows one rigid body and three axialmodes of the pile fixed at the head. The first moderepresents the rigid body mode of translation in the z-direction and the second mode at 100.2Hz corresponds to

the first axial mode. The dashed-dotted lines display theundeformed shape of the pile.The soil medium consists of a homogeneous halfspace

with a Young’s modulus Es ¼ 400MPa, a Poisson’s rations ¼ 0:4, a material damping ratio bs ¼ 0:05 and a densityrs ¼ 1750 kg/m2. The shear and the dilatational wavevelocity in the soil are equal to 285 and 700m/s,respectively.The BE method is applied to compute the impedance

matrix of the soil. The size of a BE on the soil–pile interfaceshould be sufficiently small with respect to the minimumwavelength lmin ¼ Cs=f max, where Cs is the shear wavevelocity in the soil and f max is the highest frequency of thedynamic excitation. It is proposed that minimum eight BEsare used over one wavelength lmin.Fig. 4 shows the vertical dimensionless dynamic stiffness

coefficient kvvða0Þ and damping coefficient cvvða0Þ as afunction of the dimensionless frequency a0 for a single pileembedded in a homogeneous halfspace. The numericalresults are put in dimensionless form using the verticalstatic stiffness K s

vv, which according to Kaynia and Kausel[15] is equal to 1:58EpAp=Lp for the soil characteristicsconsidered. Results for one rigid body and two axial modesof the pile fixed at the head, as well as one rigid body andseven axial modes of the pile fixed at the head, arecompared with Kaynia and Kausel’s solution. The resultsshow a very good agreement with the reference solution inthe full range of dimensionless frequencies, even when onlyone rigid and two axial modes are considered.A convergence analysis has been carried out to

determine the required number of modes in the frequencyrange upto a dimensionless frequency amax

0 ¼ 1. Thisanalysis is based on the evaluation of the error norm e,defined as the mean square of the difference between thenumerical solution and Kaynia and Kausel’s referencesolution over the frequency range of interest:

e ¼1

amax0

Z amax0

0

jSvvða0Þ � Srefvv ða0Þj

2 da0. (32)

ARTICLE IN PRESS

(a)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Dimensionless frequency a0

k vv(

a 0)

(b)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Dimensionless frequency a0

Cvv

(a0)

Fig. 4. Vertical (a) stiffness and (b) damping coefficients of a single pile

with Lp=dp ¼ 20 embedded in a homogeneous halfspace with

Es ¼ Ep=100, ns ¼ 0:40, bs ¼ 0:05 and rs=rp ¼ 0:70. Results for one rigid

body and two axial modes of the pile fixed at the head (dashed line) and

one rigid body and seven axial modes of the pile fixed at the head (solid

line) are compared with Kaynia and Kausel’s solution (dashed-dotted

line).

0 2 4 6 8

0

2

-2

-4

-6

-8

4

Number of modes

log

e

Fig. 5. Logarithm of the error e as a function of the number of modes q,

using one rigid body mode and the axial modes of the pile fixed at the head

(-�-), the free axial modes of the pile (-�-), or the modes of the pile on an

elastic foundation (-�-).

(a) (b) (c) (d)

Fig. 6. (a) Rigid body mode and first three flexible axial modes of the pile

with free boundary conditions at (b) 200.0Hz, (c) 400.3Hz and

(d) 601.1Hz.

(a) (b) (c) (d)

Fig. 7. Four lowest modes of the pile on an elastic foundation at

(a) 150.8Hz, (b) 253.1Hz, (c) 429.4Hz and (d) 620.9Hz.

Fp

Lpep

dp

Fig. 8. Geometry of an embedded pile in a layered medium.


Fig. 5 shows the logarithm of the error e as a function ofthe number of modes q of the pile. The error rapidlydecreases with increasing number of modes. This erroranalysis confirms that very accurate results in the frequencyrange upto amax

0 ¼ 1 are obtained when one rigid body andtwo axial modes fixed at the head are used.

A second alternative for the modal basis to describe thekinematics of the pile is to use the free modes of the pile

(Fig. 6). The projection of the force vector fp on theflexible modes U on the right-hand side of Eq. (29) isnon-zero now, which makes this modal basis less ap-propriate for the computation of the impedance, although

ARTICLE IN PRESS

(a)

0 2 4 6 80

2

4

6

8x 10-8

Number of modes

Vel

ocity

[m/s

]

(b)

0 2 4 6 80

2

4

6

8x 10-8

Number of modes

Vel

ocity

[m/s

]

(c)

0 2 4 6 80

1

2

3

4x 10-8

Number of modes

Vel

ocity

[m/s

]

(d)

0 2 4 6 80

1

2

3

4x 10-8

Number of modes

Vel

ocity

[m/s

]

Fig. 9. Norm of the velocity in the free field at (a) ð5m; 0Þ, (b) ð10m; 0Þ, (c) ð5m; 10mÞ and (d) ð10m; 10mÞ for harmonic excitation at 20Hz at a

penetration depth ep ¼ 10m as a function of the number of modes: one rigid body and axial modes fixed at the head (-�-) and one rigid body and free axial

modes (-�-).

Fig. 10. The norm of the particle velocity in a homogeneous halfspace due

to vibratory pile driving at 20Hz for penetration depths (a) ep ¼ 2m,

(b) ep ¼ 5m and (c) ep ¼ 10m.

1 10 1001

10

100

1000

PP

V [m

m/s

]

r [m]

Fig. 11. PPV versus the distance from the pile due to vibratory pile driving

at 20Hz. Results obtained for the penetration depths ep ¼ 2m (� � �),

ep ¼ 5m (� � �) and ep ¼ 10m (� � �) are compared with results of field

measurements (dashed-dotted line) reported by Wiss [2].


not impossible. Fig. 5 shows that the required numberof modes q that result in a specified error value is largerthan for a modal basis based on flexible modes fixed at thehead.

The third alternative consists of using the modes of thepile on an elastic foundation, represented by springs anddampers along the pile shaft and at its bottom (Fig. 7); thecoefficients of these springs and dampers can be estimatedusing Novak’s solution (along the shaft) [16] and Lysmer’sanalogue (at the bottom) [17], respectively. Fig. 5, however,proves that this modal basis results in a convergence that issimilar as obtained with the free modes of the pile and isnot better than obtained with the modes of the pile fixed atthe head.The convergence study demonstrates that, in the

frequency range upto amax0 ¼ 1, very good convergence is

obtained for the vertical impedance of the pile when one

ARTICLE IN PRESS

(a)

0 100 200 300 400 500

0

5

10

15

Particle velocity [mm/s]

Dep

th [m

]

(b)

0 10 20 30

0

5

10

15


Dep

th [m

]

Fig. 12. PPV versus the depth at (a) r ¼ 0:1 and (b) r ¼ 5 due to vibratory

pile driving at 20Hz for the penetration depths ep ¼ 2m (� ��), ep ¼ 5m

(� � �) and ep ¼ 10m (� � �).


rigid body mode and two flexible axial modes of the pilefixed at the head are used.

In the case of vibratory and impact pile driving, as willbe considered in the following sections, the convergencestudy will be based on the response in the free field.

4. Ground vibrations due to vibratory pile driving


Ground vibrations due to vibratory driving are con-sidered. A standard hydraulic vibratory driver ICE 44-30Vis selected. It operates at the frequency f ¼ 20Hz with aneccentric moment me ¼ 50:7 kgm, resulting in a centrifugalforce Fp ¼ 800 kN.

The free field vibrations due to vibratory driving of aconcrete pile in a semi-infinite layered medium areinvestigated. The pile has a length Lp ¼ 10m, a diameterdp ¼ 0:50m, a Young’s modulus Ep ¼ 40 000MPa, aPoisson’s ratio np ¼ 0:25 and a density rp ¼ 2500 kg/m3.The longitudinal wave velocity of the pile Cp ¼ 4000m/s.The contributions of different types of waves are investi-gated for several penetration depths ep ¼ 2, 5 and 10m(Fig. 8).

4.2. A homogeneous halfspace

First, results are discussed for vibratory pile driving in ahomogeneous halfspace with a Young’s modulusEs ¼ 80MPa, a Poisson’s ratio ns ¼ 0:4, a materialdamping ratio bs ¼ 0:025 and a density rs ¼ 2000 kg/m3.The shear and the dilatational wave velocity are Cs ¼

120m/s and Cp ¼ 293m/s, respectively.The excitation frequency of 20Hz corresponds to a

dimensionless frequency a0 ¼ 0:52, which is well below themaximum frequency amax

0 ¼ 1 considered in the conver-gence study for the vertical impedance of the pile. It istherefore expected that one rigid body mode and twoflexible axial modes of the pile represent well thekinematical basis of the pile. As no analytical results havebeen found in the literature, a convergence study is limitedto an assessment of the free field response when anincreasing number of modes is incorporated in thekinematical basis. Two bases are considered, consisting of(1) a vertical rigid body mode and the axial modes of thepile, fixed at the head, and (2) a vertical rigid body modeand the axial modes of the free pile.

Fig. 9 shows the norm of the velocity at four locations inthe free field for a harmonic excitation at 20Hz at apenetration depth ep ¼ 10m as a function of the number ofmodes. There is no significant difference for both modalbases and convergence is obtained if at least one rigid bodymode and two axial modes are used. Whereas for the studyof the vertical impedance of the pile, it turned out to bemathematically more convenient to use the axial modes ofthe pile fixed at the head, the choice of free axial modes isas good when free field vibrations due to vibratory or

impact pile driving are studied. In the following, the latterkinematical basis will be used.Fig. 10 shows the norm of the particle velocity in a

homogeneous halfspace due to vibratory pile driving at20Hz for different penetration depths ep ¼ 2, 5 and 10m.Results are presented in the ðr; zÞ plane, where the verticalcoordinate varies from z ¼ 0:0m on the ground surface toz ¼ 20m; the horizontal coordinate varies from the pileshaft at r ¼ 0:5m up to a distance r ¼ 20:5m from the pilecentre. The characteristics of the propagating wavesinduced by vibratory pile driving can be classified asfollows: (1) because of the soil–shaft contact, verticallypolarized shear waves are generated which propagateradially from the shaft on a cylindrical surface, (2) at thepile toe, shear and compression waves propagate in alldirections from the toe on a spherical wave front and (3)Rayleigh waves propagate radially on a cylindrical wavefront along the surface. When the pile toe is near thesurface, ground motions are influenced only by the toeresistance and the response of the soil around the pile shaftis dominated by Rayleigh waves. As the pile is driven, thecontact area along the pile shaft increases and verticallypolarized shear waves dominate the response around thepile shaft. In an elastic halfspace, both body waves andRayleigh waves decrease in amplitude with increasingdistance from the pile due to the geometrical damping.Theoretically, ground vibrations in the far field attenuateinversely proportional to the square of the area AðrÞ of thewave front [18] or according to r�n, with r the distance and

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(a)

0 3

0

3

-3-3

-2-2

u z [m

m]

u z [m

m]

ur [mm]

(b)0 3

3

-3-3

u z [m

m]

ur [mm]

(c)0 3

3

-3-3

u z [m

m]

ur [mm]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

00

0

2

Fig. 13. Particle trajectories in the points at r ¼ 0:1 and (a) z ¼ 0, (b) z ¼ 13and (c) z ¼ 2

3due to vibratory pile driving at 20Hz for the penetration depths

ep ¼ 2m (left), ep ¼ 5m (centre) and ep ¼ 10m (right).


n the geometrical attenuation coefficient. The latter is equalto 0.5 for surface waves propagating on a cylindrical wavefront and equal to 1 for body waves propagating on aspherical wave front in the interior of the halfspace; forbody waves propagating along the surface, the coefficient n

is equal to 2. Fig. 10 shows how Rayleigh waves attenuateslower than body waves and propagate in a zone close tothe surface of the halfspace.

Fig. 11 illustrates the decrease of the peak particlevelocity (PPV) at the surface with the distance r from thepile for three penetration depths ep ¼ 2, 5 and 10m.The dashed-dotted line on the figure represents dataobtained by several field measurements during vibratorypile driving [2].

The attenuation is due to both geometrical and materialdamping in the soil. The material damping ratio bs in thesoil is assumed to be constant and minimum below theelastic threshold shear strain. The slope of the attenuationcurve can be interpreted as an attenuation coefficient thatrepresents the effect of both damping mechanisms. Thenumerical predictions reveal a larger attenuation coefficientfor distances larger than 15m, which may be attributed tothe increased relative importance of material damping.

The maximum vibration amplitude in the near fieldoccurs when the pile toe is near the ground surface

(ep ¼ 2m), whereas reported data show that groundvibrations are mostly independent of the pile penetrationdepth [3,19]. It is also observed that the present modelpredicts higher ground vibrations in the far field, especiallywhen the penetration depth is small. This may reflect thefact that during actual vibratory pile driving, higher(plastic) strains are induced in the soil, leading to morematerial damping [3]. It must also be noted that vibrationamplitudes strongly depend on the soil conditions, the piletype, and the method of driving, which may also explainthe discrepancy between the field data and the numericalpredictions.Fig. 12 displays the variation of the PPV versus the

depth at different dimensionless distances r ¼ r=ls from thepile, with ls the wavelength of the shear wave, which isequal to 6m at 20Hz. As the strain level is proportional tothe particle velocity [1,2], the profile of the particle velocityaround the pile can also be interpreted as the variation ofthe shear deformation. Fig. 12a shows how this profile isuniformly distributed along the shaft, except when thepenetration depth is small.The particle trajectories due to vibratory pile driving at

20Hz for the penetration depths ep ¼ 2, 5 and 10m areplotted at r ¼ 0:1 and r ¼ 5:0 and at different dimension-less depths z ¼ z=ls in Figs. 13 and 14, respectively.

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(a)0 0.2

0

0

0 0

0

0

0.2

-0.2-0.2

u z [m

m]

ur [mm]0 0.2

0.2

-0.2-0.2

-0.1-0.1

u z [m

m]

u z [m

m]

ur [mm] ur [mm]

0 0.2

0.2

-0.2-0.2

u z [m

m]

ur [mm]

0 0.2

0.2

-0.2-0.2

u z [m

m]

ur [mm]

(b)0 0.2

0.2

-0.2-0.2

u z [m

m]

ur [mm]

(c)0 0.2

0.2

-0.2-0.2

u z [m

m]

ur [mm]

0 0.1

0

0.1

-0.1-0.1

u z [m

m]

ur [mm]0 0.1

0

0.1

-0.1-0.1

u z [m

m]

ur [mm]0 0.1

0

0.1


3due to vibratory pile driving at 20Hz for the penetration depths

ep ¼ 2m (left), ep ¼ 5m (centre) and ep ¼ 10m (right).


Fig. 13a shows that, for a penetration depth ep ¼ 2m, thedisplacement has vertical and horizontal componentsresulting in a retrograde elliptical motion, typical forRayleigh waves. As the penetration depth increases,ground vibrations around the shaft are dominated byvertically polarized shear waves (Figs. 13b and c). Fig. 14gives results at a distance r ¼ 5:0, where Rayleigh wavesdominate the response in a zone near the surface,independent of the penetration depth of the pile. Bodywaves are already attenuated and only Rayleigh wavesremain in the far field.

4.3. Soft layer on a stiffer halfspace

Second, results are discussed for vibratory pile driving ina layered soil medium, consisting of a soft layer with aYoung’s modulus EL

s ¼ 80MPa resting on a stifferhomogeneous halfspace with a Young’s modulusEH

s ¼ 320MPa. The thickness of the top layer is equal to7:5m. The soil in both the layer and the halfspace has aPoisson’s ratio ns ¼ 0:4, a material damping ratio bs ¼0:025 and a density rs ¼ 2000 kg/m3. The shear wavevelocity in the top layer and the underlying halfspace isequal to CL

s ¼ 120m/s and CHs ¼ 240m/s, respectively. It is

investigated how the soil stratification influences theintensity of the ground vibrations.Fig. 15 shows the norm of the particle velocity in a

layered medium due to vibratory pile driving at 20Hz forthe penetration depths ep ¼ 2, 5 and 10m. Figs. 15a and billustrate that the body waves which encounter the under-lying halfspace are mostly reflected into the top layerand a small part of the propagating wave is transmitted tothe stiffer underlying halfspace. When the penetra-tion depth is equal to ep ¼ 10m, the shear wave front isaffected by the reflected and refracted waves. Fig. 15cshows how the radial direction of the shear waves ischanged.Fig. 16 shows the variation of the PPV on the surface as

a function of the dimensionless distance r from the pile,where a comparison is made between the results of thelayered medium and the homogeneous halfspace. When thepenetration depth is less than the layer thickness, a smalldiscrepancy is observed (Figs. 16a and b). This discrepancybecomes more significant when the penetration depth isequal to 10m (Fig. 16c). Indeed, due to wave reflections,vibration amplitudes do not decay monotonically andshow an oscillatory behaviour. The minima are identi-fied at dimensionless distances r ¼ 1:1 and r ¼ 3:2 from

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Fig. 15. The norm of the particle velocity in a layered medium due to vibratory pile driving at 20Hz for the penetration depths (a) ep ¼ 2m, (b) ep ¼ 5m

and (c) ep ¼ 10m.

(a)

0 1 2 3 4 50

100

200

300

PP

V [m

m/s

]

r/ (b)

0 1 2 3 4 50

50

100

150

PP

V [m

m/s

]

s r/ s

(c)0 1 2 3 4 5

0

50

150

PP

V [m

m/s

]

r/ s

Fig. 16. PPV at the free surface versus the distance from the pile due to

vibratory pile driving at 20Hz in a layered medium (� � �) and a

homogeneous halfspace (� ��) for the penetration depths (a) ep ¼ 2m,

(b) ep ¼ 5m and (c) ep ¼ 10m.

(a)0 100 200 300 400 500

0

5

10

15


Dep

th [m

]

(b)0 50 100 150 200

0 50 100 150 200

0

5

10

15


Dep

th [m

]

(c)

0

5

10

15


Dep

th [m

]

Fig. 17. PPV at r ¼ 0:1 versus the depth due to vibratory pile driving at

20Hz in a layered medium (� � �) and a homogeneous halfspace (� � �)

for the penetration depths (a) ep ¼ 2m, (b) ep ¼ 5m and (c) ep ¼ 10m.


the pile. The latter behaviour was also observed infield measurements reported by van Staalduinen andWaarts [20].

Figs. 17 and 18 display the variation of the PPV versusthe depth at dimensionless distances r ¼ 0:1 and r ¼ 5,

respectively. The results for the layered medium and thehomogeneous halfspace are compared. When the penetra-tion depth is less than the layer thickness, a small differenceis observed in the near field. When the pile penetrationdepth is ep ¼ 10m and larger than the layer thickness, thereflected and scattered waves change the PPV around the

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(a)0 10 20 30

0

5

10

15


Dep

th [m

]

(c)0 10 20 30

0

5

10

15


Dep

th [m

]

(b)0 10 20 30

0

5

10

15


Dep

th [m

]

Fig. 18. PPV at r ¼ 5 versus the depth due to vibratory pile driving at

20Hz in a layered medium (� � �) and a homogeneous halfspace (� ��)

for the penetration depths (a) ep ¼ 2m, (b) ep ¼ 5m and (c) ep ¼ 10m.

(a)

0 10 20 300

0.5

1

1.5

2

For

ce [M

N]

Time [ms]

(b)

0 100 200 300 400 5000

2

4

6

8

For

ce [k

N/H

z]

Frequency [Hz]

Fig. 19. (a) Time history and (b) frequency content of the hammer impact

force.


pile shaft. Fig. 18 shows that, in the far field, effects ofreflected and scattered waves on ground motions arestronger than in the near field.

5. Ground vibrations induced by impact driving


Ground vibrations due to impact driving are subse-quently investigated. A BSP-357 hammer is used to drive aconcrete pile with a circular section. The pile characteristicsand properties are similar to the previous example.

Fig. 19 shows the time history and frequency content ofthe hammer impact force. The force is evaluated using a2DOF model developed by Deeks and Randolph [21]. Thehammer cushion is a steel plate with a stiffnesskc ¼ 1:6� 106 kN/m. The ram and anvil masses of thehammer are mr ¼ 6860 kg and ma ¼ 850 kg, respectively.The pile impedance is equal to Zp ¼ 1960 kNs/m.

In the numerical model, the rated energy Er ¼ mrgh isassumed to be equal to the energy Et ¼ mrv

20=2 that is

effectively transmitted to the pile, with v0 the impactvelocity. In a practical application, however, the trans-mitted energy Et ¼ ZEr is lower than the rated energy Er

due to energy loss related to precompression, frictionbetween the ram and the cylinder, preignition andmisalignment. In the literature, a transfer coefficient Z ofabout 0.40–0.45 is proposed for diesel hammers. In thefollowing, results are presented for an impact velocityequal to v0 ¼ 1m/s, resulting in a low impact with atransferred energy of about 3.4 kJ.

In the case of impact pile driving, the impact forces havea higher frequency content, necessitating a convergenceanalysis in a wider frequency range. As the frequencycontent of the force is mainly dominated by frequenciesbelow 300Hz, the soil stiffness and modal responses of the

soil are calculated in the frequency domain upto 300Hz.Because of the material damping in the soil, the resultingfrequency content of the free field vibrations is limited to150Hz.Figs. 20 and 21 show the norm of the velocity at four

locations in the free field for a harmonic excitation at 50and 150Hz and at a penetration depth ep ¼ 10m as afunction of the number of modes. There is no significantdifference between the results obtained with both modalbases. At 50Hz, convergence is obtained if one rigid bodymode and two axial modes are used, whereas at 150Hz,three axial modes are preferable. In the following calcula-tions, one rigid body mode and three flexible axial modesof the free pile are therefore included.In order to obtain an accurate inverse Fourier transform

of the response from the frequency domain to the timedomain, the frequency step Df must be sufficiently smallwith respect to CR=rmax, where rmax represents the largestconsidered distance from the pile axis.

5.2. A homogeneous halfspace

Results are discussed for impact pile driving in ahomogeneous halfspace with the same characteristics asfor the case of vibratory pile driving in Section 4.2.Fig. 22 shows the time history and the frequency content

of the vertical particle velocity in a homogeneous halfspaceat r ¼ 5m from the pile for the penetration depths ep ¼ 2, 5and 10m. The frequency content of the ground motionsis mainly situated below 150Hz and dominated by

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(a)0 2 4 6 8

0

2

4

6

8x 10-8

x 10-8

Number of modes

Vel

ocity

[m/s

]

(b)0 2 4 6 8

0

2

4

6

8x 10-8

Number of modes

Vel

ocity

[m/s

]

(c)0 2 4 6 8

0

1

2

3

4

Number of modes

Vel

ocity

[m/s

]

x 10-8

(d)0 2 4 6 8

0

1

2

3

4

Number of modes

Vel

ocity

[m/s

]



modes (-�-).

(a)0 2 4 6 8

0

2

4

x 10-8

x 10-8

Number of modes

Vel

ocity

[m/s

]

(b)0 2 4 6 8

2

4

x 10-8

Number of modes

Vel

ocity

[m/s

]

(c)

0 2 4 6 80

1

2

Number of modes

Vel

ocity

[m/s

]

x 10-8

(d)

0 2 4 6 80

1

2

Number of modes

Vel

ocity

[m/s

]



modes (-�-).


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(a)

0 50 100 150

0

100

-100

200P

artic

le v

eloc

ity [m

m/s

]

Time [ms]

(b)

0 100 200 3000

0.5

1

1.5

Par

ticle

vel

ocity

[mm

/s/H

z]

Frequency [Hz]

Fig. 22. (a) Time history and (b) frequency content of the vertical particle

velocity due to impact driving in a homogeneous halfspace at r ¼ 5m for

the penetration depths ep ¼ 2m (solid line), ep ¼ 5m (dashed line) and

ep ¼ 10m (dashed-dotted line).

0 100 200 3000

0.2

0.4

0.6

0.8

Par

ticle

vel

ocity

[mm

/s/H

z]

Frequency [Hz]

Fig. 23. Frequency content of the particle velocity due to impact driving

in a homogeneous halfspace at r ¼ 5m (solid line), r ¼ 15m (dashed line)

and r ¼ 30m (dashed-dotted line) for the penetration depth ep ¼ 5m.

Fig. 24. The norm of the particle velocity in a homogeneous halfspace at

(a) t ¼ 30ms, (b) t ¼ 80ms and (c) t ¼ 160ms due to impact pile driving

for the penetration depth ep ¼ 2m.





frequencies around 50Hz, independent of the penetrationdepth. A similar range of dominant frequencies has beenreported by Hwang et al. [22] and Thandavamoorthy [23].

Fig. 23 displays the frequency content of the particlevelocity at three distances from the pile. It can be observedthat the frequency content of ground vibrations decreasesas the distance from the pile increases. Such behaviour isexpected because of material damping, resulting in areduction in the frequency content of ground vibrationsin the far field. The ratio of the dominant frequency ofground vibrations to the natural frequency of adjacentstructures is an important parameter affecting the structur-

al response. For most sandy and clayey soils the dominantfrequency of vibration due to pile driving is generallyreported between 15 and 35Hz [2,10,19].Figs. 24–26 show the norm of the particle velocity in a

homogeneous halfspace due to impact driving for the

ARTICLE IN PRESS




1 10 1001

10

100

1000

PP

V [m

m/s

]

r [m]

Fig. 27. PPV versus the distance from the pile due to impact pile driving.

Results obtained for an envelope of situations with a transferred energy of

19.2 kJ at a penetration depth ep ¼ 2m and a transferred energy of 3.4 kJ

at a penetration depth ep ¼ 10m are compared with results of field

measurements (dashed-dotted line) reported by Wiss [2].

(a)

0 200 400 600 800

0

5

10

15


Dep

th [m

]

(b)

0 5 10 15 20 25

0

5

10

15


Dep

th [m

]

Fig. 28. PPV versus the depth at (a) r ¼ 0:25m and (b) r ¼ 30:0m due to

impact pile driving for the penetration depths ep ¼ 2m (� � �), ep ¼ 5m

(� � �) and ep ¼ 10m (� � �).


penetration depths ep ¼ 2, 5 and 10m, respectively, andat the time steps t ¼ 30, 80 and 160ms. The follow-ing observations can be made: (1) body waves domi-nate around the toe and propagate on a spherical wavefront, (2) vertically polarized shear waves dominatearound the pile and propagate radially on a cylindricalwave front and (3) Rayleigh waves propagate near thesurface with a velocity slightly less than the shear waves(Fig. 26).

Fig. 27 displays the envelope of the PPV with thedistance r from the pile due to impact pile driving with atransferred energy from 3.4 to 19.2 kJ. The upper boundrepresents results for an impact with an energy of 19.2 kJ ata penetration depth ep ¼ 2m, while the lower boundrepresents results for an impact with an energy of 3.4 kJ ata penetration depth ep ¼ 10m. The predicted vibrationsare compared with the results of field measurementsreported by Wiss [2] for impact pile driving with a dieselhammer at a rated energy of 48 kJ. Considering a transfercoefficient of about 0.40, this energy is equivalent with atransmitted energy of 19.2 kJ. In the near field, thepredicted vibrations attenuate slower than the experimentalresults. In the far field, however, almost the sameattenuation coefficient is observed, but conservative vibra-tion levels are predicted, even when a low impact isconsidered.

Fig. 28 illustrates the variation of the PPV as a functionof the depth at two distances r ¼ 0:25 and 30.0m from thepile. In the near field, where vertically polarized shearwaves propagate on a cylindrical surface, the distribution

of the PPV can be assumed to be uniform. In the far field,however, vertically polarized shear waves are attenuatedand the PPV decays exponentially along the depth, as istypical for Rayleigh waves.The particle trajectories at r ¼ 0:1 and r ¼ 5:0 and at

dimensionless depths z ¼ 0, z ¼ 13and z ¼ 2

3due to impact

pile driving for the penetration depths ep ¼ 2, 5 and 10mare plotted in Figs. 29 and 30, respectively. Fig. 29a showsresults for the case of ep ¼ 2m, which are mainly composed

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(a)0 3

0

3

-3-3

-2-2

u z [m

m]

ur [mm]

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

-2-2

u z [m

m]

ur [mm]0 2

0

2

(b)0 3

0

3

-3-3

u z [m

m]

ur [mm]

(c)0 3

0

3

-3-3

u z [m

m]

ur [mm]

Fig. 29. Particle trajectories in the points at r ¼ 0:1 and (a) z ¼ 0, (b) z ¼ 13 and (c) z ¼ 2

3 due to impact pile driving for the penetration depths ep ¼ 2m

(left), ep ¼ 5m (centre) and ep ¼ 10m (right).


of vertical and horizontal components with a retrogradeelliptical pattern, typical for Rayleigh waves. As thepenetration depth increases, ground vibrations aroundthe shaft are dominated by shear waves (Figs. 29b and c).Fig. 30 shows results at a dimensionless distance r ¼ 5:0,where Rayleigh waves dominate the response in a zonenear the surface, independent of the penetration depth ofthe pile. Body waves are already attenuated and onlyRayleigh waves remain in the far field.

6. Conclusion

In the present study, a coupled FE–BE model has beendeveloped to predict ground vibrations due to vibratoryand impact pile driving. The pile is modelled as a linearelastic material and the soil is modelled as a horizontallylayered elastic halfspace. The dynamic soil–pile interactionproblem is formulated using a subdomain formulation. ABE technique based on the Green’s functions of thehorizontally layered halfspace is applied to calculate thesoil tractions and soil impedances. An FE method is usedto calculate the stiffness and mass matrix of the pile.

The prediction of the free field response due to vibratoryand impact pile driving is illustrated in several examples. In

the case of vibratory driving, the induced waves can beclassified into: (1) the vertically polarized shear wavesaround the shaft, (2) the body waves around the pile toeand (3) the Rayleigh waves along the surface. The radiatedwaves due to impact driving are more complex.In addition, the profile of the particle velocity around the

pile shaft is dominated by the vertical shear deformation,whereas the normal and the radial deformations are smallcompared to shear deformations. When the pile iscompletely embedded, ground vibrations in the near fieldaround the pile shaft are dominated by vertically polarizedshear waves. In the far field, however, body waves areattenuated and Rayleigh waves dominate in the zone nearthe surface, independent of the penetration depth of thepile. Furthermore, it is observed that the frequency contentof ground vibrations shifts toward lower frequencies as thedistance from the pile increases.The results for vibratory pile driving in a layered

medium show that, when the penetration depth is smallerthan the layer thickness, the layering has a relatively smallinfluence on the ground vibrations. When the penetrationdepth is larger than the layer thickness, however, theseeffects become more significant, due to the influence ofreflected and diffracted waves.

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(a)0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]

0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]

0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm](c)0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]

(b)0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]0 0.1

0

-0.1-0.1

0.1

u z [m

m]

ur [mm]


3due to impact pile driving for the penetration depths ep ¼ 2m

(left), ep ¼ 5m (centre) and ep ¼ 10m (right).


Acknowledgements

The results presented in this paper have been obtainedwithin the frame of the SBO project IWT 03175 ‘‘Structuraldamage due to dynamic excitation: a multi-disciplinaryapproach’’. This project is funded by IWT Vlaanderen, theInstitute of the Promotion of Innovation by Science andTechnology in Flanders. Their financial support is grate-fully acknowledged.

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strain case. J Eng Mech Proc ASCE 1978;104(EM4):1024–41.

[17] Lysmer J, Richart F. Dynamic response of footings to

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[18] Wolf JP. Foundation vibration analysis using simple physical models.

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[19] Svinkin MR. Predicting and calculation of construction vibrations,

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