Numerical modeling of free field vibrations due to

pile driving using a dynamic soil-structure

interaction formulation

H.R. Masoumi∗, G. Degrande

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

Abstract

This paper presents a numerical model for the prediction of free field vibrations due to vi-bratory and impact pile driving. As the focus is on the response in the far field, wheredeformations are relatively small, a linear elastic constitutive behavior is assumed for thesoil. The free field vibrations are calculated by means of a coupled FE-BE model using asubdomain formulation. First, the case of vibratory pile driving is considered, where thecontributions of different types of waves are investigated for several penetration depths. Theresults show that, in the near field, the response of the soil is dominated by a vertically polar-ized shear wave, whereas in the far field, Rayleigh waves dominate the ground vibration andbody waves are importantly attenuated. Second, the case of impact pile driving is consid-ered. The results are discussed for different time steps and penetration depths. Finally, thecomputed ground vibrations are compared with the results of field measurements reportedin the literature.

Keywords: Dynamic soil-pile interaction, pile impedance, vibratory pile driving, impact piledriving, ground vibration, boundary element method, finite element method

1 Introduction

Construction vibrations often affect adjacent and surrounding buildings in urban environments.The induced waves propagate in the soil and interact with nearby structures, which inducesvibrations that may cause cracks in its walls or in the facade. Permanent settlement, densificationand liquefaction may also occur in the soil due to such vibrations. The present study aims todevelop a model to predict free field vibrations in the environment due to pile driving. Groundmotions due to pile driving generally depend on (1) the source parameters (method of driving,energy released, and pile depth), (2) the interaction between the driving machine, the pile andthe soil, and (3) the propagation of the waves through the soil.Three main driving techniques to install foundation piles and sheet piles can be distinguished:impact pile driving, vibratory pile driving, and jacking. Impact pile driving is known as theoldest technique of driving and produces a transient vibration in the ground. The requireddriving energy is provided using a gravity principle, in which a ram mass drops from a specificheight and strikes the pile head with a downward impact velocity. As impact driving generates

∗Corresponding author. E-mail: hamidreza.masoumi@bwk.kuleuven.be

1

high noise and vibration levels, it is not preferred in urban area and its use is more and morelimited.During the last decades, the vibratory pile driving technique has been preferred to other drivingtechniques due to economical, practical and environmental concerns. Vibratory pile drivingproduces a harmonic vibration in the ground and is used to install piles in granular soils andwith low operating frequency, also in cohesive soils. The vibratory motion is generated bycounter-rotation of eccentric masses actuated within an exciter block.Jacking does not induce environmental disturbance but is very expensive. This technique is onlyused where sensitive environmental conditions are encountered.The vibration transmission through the soil is complex, resulting from the difficulty of modelingprecisely the soil behavior in the near and far field. Research on pile driving problems has con-centrated on two main directions. Setting up numerous field measurements, several authors havebeen focused on environmental effects (far field or external effects) [1, 2, 3]. Some efforts havebeen also made to develop models to assess the driving efficiency, investigating the driveabilityand the bearing capacity of driven piles (near field or internal effects) [4, 5].During pile driving, the transmitted energy through the soil is very high and cause plasticdeformations in the near field. In the far field, however, reported data show that the inducedvibrations cause deformations in the elastic range [6]. As the focus is on the response in the farfield, where deformations are relatively small, a linear elastic constitutive behavior is assumedfor the soil.In this study, a model for both vibratory and impact driving is proposed to predict free fieldvibrations. Using a subdomain formulation for dynamic soil-structure interaction, developed byAubry et al. [7, 8], the dynamic soil-pile interaction problem is investigated. A coupled FE-BE model is presented, in which the pile (bounded domain) and the soil (unbounded domain)are modeled using the finite element (FE) method and the boundary element (BE) technique,respectively. The subdomain formulation has been implemented in a computer program MISS(Modelisation d’Interaction Sol-Structure) [9]. The program MISS is used to determine theimpedance functions as well as modal responses of the soil in the frequency domain. Thedynamic impedance of the soil is calculated by means of a boundary element formulation thatuses the Green’s function of a horizontally stratified soil.Free field vibrations in the soil are evaluated due to impact and vibratory driving of a concretepile in a homogeneous half space. Results are discussed for different penetration depths of thepile.

2 Numerical modeling

2.1 Problem outline

The proposed model is based on the following hypotheses: (1) the soil medium is elastic withfrequency independent material damping (hysteresis damping), (2) no separation is allowedbetween the pile and soil medium, (3) all displacements and strains remain sufficiently small,and (4) the pile is embedded in a horizontally layered soil.The domain is decomposed into two subdomains: the unbounded semi-infinite layered soil de-noted by Ωext

s and the bounded structure (pile) denoted by Ωp. The interface between the soiland the pile is denoted by Σ (figure 1).

2.2 Governing equations

First, the pile Ωp is considered. The boundary Γp = Γpσ ∪ Σ of the pile Ωp is decomposed intoa boundary Γpσ where tractions tp are imposed and the soil-pile interface Σ. The displacement

S

Gps

Gs

Gss

Wp

Ws

ext

Figure 1: Geometry of the subdomains.

vector up of the pile satisfies the following Navier equation and boundary conditions:

divσp(up) = −ρpω2up in Ωp (2.1)

tp(up) = tp on Γpσ (2.2)

where t(u) = σ(u) · n is the traction vector on a boundary with a unit outward normal vectorn.Second, the exterior soil domain Ωext

s is taken into consideration. The boundary Γexts = Γsσ ∪

Γs∞ ∪ Σ of the soil domain Ωexts is decomposed into the boundary Γsσ where tractions are

imposed, the outer boundary Γs∞ where radiation conditions are imposed and the soil-structureinterface Σ. A free boundary or zero tractions are assumed on Γsσ. The displacement vector us

of the soil satisfies the Navier equation and the following boundary conditions:

divσs(us) = −ρsω2us in Ωext

s (2.3)

ts(us) = 0 on Γsσ (2.4)

The displacement compatibility and stress equilibrium condition are imposed on the interfaceΣ:

us = up on Σ (2.5)

tp(up) + ts(us) = 0 on Σ (2.6)

According to the compatibility condition (2.5), the displacement vector us is termed as the wavefields usc(up) that are radiated in the soil due to the pile motion up, imposed on the interfaceΣ (figure 2):

us =usc(up) in Ωexts (2.7)

The wave field usc(up), radiated in the soil, satisfies the elastodynamic equation in Ωexts and the

following boundary conditions (figure 2):

divσs(usc(up)) = −ρsω2usc(up) in Ωext

s (2.8)

ts(usc(up)) = 0 on Γsσ (2.9)

usc(up) = up on Σ (2.10)

S

Gs

Gss

Ws

ext

Figure 2: The displacement field us in term of the radiated wave fields usc(up).

2.3 Variational formulation

Accounting for the equilibrium of the tractions on the soil-pile interface Σ, the equation ofmotion of the dynamic soil-pile interaction problem can be formulated in a variational form forany virtual displacement field v imposed on the pile:

∫

Ωp

ǫ(v) : σp(up)dΩ − ω2

∫

Ωp

v · ρpupdΩ

+

∫

Σ

v · ts(usc(up))dΣ =

∫

Γpσ

v · tpdΓ (2.11)

In a FE formulation, the displacement field up is approximated as up = Npup, where Np arethe globally defined shape functions. The stress vector is defined as σp = Dǫp, with D theconstitutive matrix for an isotropic linear elastic material. Substituting the strain-displacementrelations and the linear constitutive equation, the virtual work equation (2.11) must hold forany virtual displacement field v and can be elaborated as follows:

[

Kp − ω2Mp + Ks

]

up = fp (2.12)

where the stiffness matrix Kp is defined as follows:

Kp =

∫

Ωp

BTp DBp dΩ (2.13)

with the matrix Bp = LNp. The mass matrix Mp of the pile is:

Mp =

∫

Ωp

NTp ρpNp dΩ (2.14)

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

fp =

∫

Γpσ

NTp tpdΓ (2.15)

The dynamic stiffness matrix Ks of the semi-infinite layered half space is equal to:

Ks =

∫

Σ

NTp ts(usc(Np))dΣ (2.16)

2.4 Modal decomposition

A modal decomposition is introduced, based on the modes ψm(m = 1, . . . , q) of the pile with freeboundary conditions. The modal decomposition for the displacement vector up can be writtenas follows:

up =

q∑

m=1

ψmαm in Ωp (2.17)

Applying a finite element approximation of the modes ψm ≃ Npψm, the structural displacement

vector can be written as follows:

up ≃

q∑

m=1

Npψmαm = NpΨα (2.18)

where the modes ψm

(m = 1, . . . , q) are collected in a matrix Ψ and the modal coordinates

αm(m = 1, . . . , q) are collected in a vector α.Substituting the displacement vectors up, us and the virtual vector v = NpΨαv in the varia-tional equation (2.11) results into:

[

ΨTKpΨ + iωΨTCpΨ− ω2ΨTMpΨ + ΨTKsΨ]

α = ΨTfp (2.19)

Using proportional damping, the equilibrium equation is also uncoupled, which is an advantage.The corresponding damping matrix Cp is computed as follows:

Cp = MpΨdiag(2ξmωm)ΨTMp (2.20)

and the equilibrium equation (2.19) can be elaborated as follows:

(

Λqq + iωΥqq − ω2Iqq + ΨTKsΨ)

α = ΨTfp (2.21)

where the q-dimensional diagonal matrix Λqq = diag(ω2m) contains the squares of the eigenfre-

quencies ωm(m = 1, . . . , q). Proportional damping is represented by the q-dimensional matrixΥqq = diag(2ξmωm), where ξm are the modal damping ratios.

The term ΨTKsΨ =∫

Σ(NpΨ)Tts(usc(NpΨ))dΣ denotes the impedance matrix of the un-

bounded soil domain Ωexts due to an imposed displacement field ψm. This matrix is calculated

using a boundary element technique. The solution of the dynamic soil-pile interaction problemin terms of the modal coordinates allows to compute the soil tractions at the interface and,subsequently, in the free field.A computation procedure is developed to solve the governing system of equations. This com-putation procedure can be partitioned into two parts. First, using the Structural DynamicsToolbox (SDT) in MATLAB, the finite element model of the pile is made. In the second part,using the program MISS 6.3, the soil impedance ΨTKsΨ as well as the modal responses of thesoil usc(Ψ) are computed.

3 Numerical example

3.1 Problem outline

The free field vibrations due to driving of a concrete pile in a semi-infinite medium are in-vestigated. The pile has a length Lp = 10, m, a diameter dp = 0.50m, a Young’s modulus

Ep = 40000MPa, a Poisson’s ratio νp = 0.25, and a density ρp = 2500 kg/m3. The longitudi-nal wave velocity of the pile Cp = 4000m/s. The contributions of different types of waves areinvestigated for several penetration depths ep = 2m, 5m, and 10m (figure 3).

Fp

Lpep

dp

Figure 3: Geometry of the problem.

The pile is modeled using 8-node isoparametric brick elements. Only the axial modes of thepile are excited due to pile driving and considered in the modal superposition. The number ofrequired modes depends upon the frequency content of the dynamic force. It is suggested toemploy all modes with eigenfrequencies lower than 1.5 times the highest excitation frequency.Figure 4 shows the axial rigid body and flexible modes of the pile with free boundary conditions.The first axial mode is a rigid body mode and the second mode at 200.0Hz corresponds to thefirst compression-tension mode.

Mode 1:

0.0 Hz.

Mode 2:

200.0 Hz.

Mode 3:

400.0 Hz.

Figure 4: The axial rigid body and flexible modes of the pile with free boundary conditions.

The soil consists of a homogeneous half space with a Young’s modulus Es = 80MPa, a Poisson’sratio νs = 0.4, and a density ρs = 2000 kg/m3. The shear wave velocity of the half space is

Cs = 120m/s. In order to account for the dissipation of the internal energy in the soil (materialdamping), complex Lame coefficients parameters are used:

µ∗

s = µs(1 + 2iβs) (3.22)

λ∗

s = λs(1 + 2iβs) (3.23)

where βs is the material damping ratio. In this example, a material damping ratio βs = 0.025is used.The size of each boundary element should be sufficiently small with respect to the minimumwavelength λmin = Cs/fmax, where Cs is the shear wave velocity in the soil and fmax is thehighest frequency in the dynamic excitation. It is proposed that the element size should besmaller than λmin/8.

3.2 Ground vibrations due to vibratory pile driving

First, ground vibrations due to vibratory driving are considered. A standard hydraulic vibratorydriver ICE 44-30V is selected. It operates at the frequency f = 20Hz with an eccentric momentme = 50.7 kgm, resulting in a centrifugal force Fp = 800 kN. The results are presented inthe (r, z) plane, where the vertical coordinate varies from z = 0.0m at the ground surface toz = 30m. The horizontal coordinate varies from the pile shaft at r = 0.50m up to a distancer = 30.5m from the pile center.

0 10 20 30

0

10

20

30

r [m]

z [m

]

0 10 20 30

0

10

20

30

r [m]

z [m

]

0 10 20 30

0

10

20

30

r [m]

z [m

]

a. ep = 2m. b. ep = 5m. c. ep = 10m.

Figure 5: The norm of the particle velocity in a homogeneous half space due to vibratory piledriving at 20Hz for different penetration depths (a) ep = 2m, (b) ep = 5m, and (c) ep = 10m.

Figures 5 shows the norm of the particle velocity due to vibratory pile driving at 20Hz fordifferent penetration depths ep = 2m, 5m, and 10m. The characteristics of the propagatingwaves induced by vibratory driving can be classified as follows: (1) because of the soil-shaftcontact, vertically polarized shear waves are generated which propagate radially from the shafton a cylindrical surface; (2) at the pile toe, shear and compression waves propagate in alldirections from the toe on a spherical wave front; (3) Rayleigh waves propagate radially on acylindrical wave front along the surface. When the pile toe is near the surface, ground motions areinfluenced only by toe resistance, and the response of the soil around the pile shaft is dominatedby Rayleigh waves. As the pile is driven, the contact area along the pile shaft increases, andvertically shear waves dominate the response around the pile shaft. In an elastic half space,both body waves and Rayleigh waves decrease in amplitude with increasing distance from the

1 10 1001

10

100

1000

PP

V [m

m/s

]r [m]

Figure 6: Peak particle velocity (PPV) on the surface versus the distance from the pile dueto vibratory pile driving at 20Hz. The comparison is made for different penetration depthsep = 2m (− ⋄ −), ep = 5m (− −), and ep = 10m (− ∗ −) with the results (dash-dotted line)reported by Wiss [2].

pile due to the geometrical damping. Figure 5 shows how Rayleigh waves attenuate slower thanbody waves and propagate in a restricted zone close to the surface of the half space.Figure 6 illustrates the attenuation of the peak particle velocity (PPV) on the surface with thedistance r from the pile. The maximum vibration amplitude in the near field occurs when thepile toe is near the ground surface (ep = 2m). However, reported data on pile driving show thatground vibrations are mostly independent of the pile penetration depth and depend on the soilproperties [3, 10]. When the soil around the pile is assumed to deform in the elastic range, groundmotions are affected by the impedance of the pile and the soil condition. The pile impedanceaffects ground vibrations in two opposite ways. Increasing the pile impedance, which occurs forincreasing penetration depth, increases the force transmitted to the pile, but decreases the peakparticle velocity [11]. In figure 6, the dash-dotted line represents approximate values obtainedby field measurements during vibratory pile driving [2] and illustrates the relative vibrationintensities. It is observed that the present model predicts more conservative ground vibrationsin the far field, especially when the penetration depth is small. This reflects the fact that duringactual driving, higher (plastic) strains are induced in the soil, leading to more material damping[3].

0 100 200 300 400 500

0

5

10

15

Particle velocity [mm/s]

Dep

th [m

]

0 10 20 30

0

5

10

15

Particle velocity [mm/s]

Dep

th [m

]

a. r = 0.1. b. r = 5.

Figure 7: Peak particle velocity (PPV) versus the depth due to vibratory pile driving at 20Hz,at (a) r = 0.1 and (b) r = 5 from the pile for different penetration depths ep = 2m (− ⋄ −),ep = 5m (− −), and ep = 10m (− ∗ −).

Figure 7 displays the variation of the peak particle velocity versus the depth at different dimen-sionless distances. As the strain level is proportional to the particle velocity [1, 2], the profile ofthe particle velocity around the pile can be interpreted as the variation of the shear deforma-tion. Figure 7a shows how this profile is uniformly distributed along the shaft, except when thepenetration depth is small.

3.3 Ground vibration induced by impact driving

Ground vibrations due to impact driving are subsequently investigated. The hammer impactforce is evaluated using a 2DOF model developed by Deeks and Randolph [12]. A BSP-357hammer is used to drive a pile with an impedance Zp = 1960 kNs/m. The hammer cushion issteel with a stiffness kc = 1.6 × 106 kN/m. The specific ram and anvil masses for this hammerare mr = 6860 kg and ma = 850 kg, respectively. The hammer stroke is h = 0.50m.

(a)

0 10 20 300

2

4

6

For

ce [M

N]

Time [ms]

(b)

0 100 200 300 400 5000

10

20

30

For

ce [k

N/H

z]

Frequency [Hz]

Figure 8: (a) Time history and (b) frequency content of the impact force .

Figure 8 shows the time history and frequency content of the impact force. As the frequencycontent of the force is mainly dominated by frequencies below 300Hz, the soil stiffness andmodal responses of the soil are calculated in the frequency domain up to 300Hz. Consideringthis excitation frequency, axial modes with eigenfrequencies up to 450Hz are included in themodel implementation. In order to obtain an accurate transform Fourier from the frequencydomain to the time domain, the frequency step f must be sufficiently small with respect toCR/rmax, where rmax represents the largest distance from the pile center.

0 100 200 3000

2

4

6

8x 10

−4

Par

ticle

vel

ocity

[mm

/s/H

z]

Frequency [Hz]

Figure 9: Frequency content of the particle velocity in a homogeneous half space at differentdistances r = 5.0m (solid line), r = 15.0m (dash-dotted line) and r = 30m (dashed line) fromthe pile for the penetration depth ep = 5m.

Figure 9 display the frequency content of the particle velocity at different distances from the pile.It can be observed that the frequency content of ground vibrations reduces as the distance fromthe pile is increased. Such behavior is as expected because of material damping which resultsin a reduction of the frequency content in the far field. The ratio of the dominant frequencyof ground vibrations to the natural frequency of the adjacent structures is one of the mostimportant parameters to analyse. For most sandy and clayey soils, the principal frequency dueto pile driving is generally reported between 15 and 35Hz [2, 6].

ep = 2.0m

0 10 20

0

5

10

15

20

r [m]

z [m

]

0 10 20

0

5

10

15

20

r [m]

z [m

]

0 10 20

0

5

10

15

20

r [m]

z [m

]ep = 5.0m

0 10 20

0

5

10

15

20

r [m]

z [m

]

0 10 20

0

5

10

15

20

r [m]

z [m

]

0 10 20

0

5

10

15

20

r [m]

z [m

]

ep = 10.m

0 10 20

0

5

10

15

20

r [m]

z [m

]

0 10 20

0

5

10

15

20

r [m]

z [m

]

0 10 20

0

5

10

15

20

r [m]

z [m

]

a. t = 30 ms. b. t = 80 ms. c. t = 160 ms.

Figure 10: The norm of the particle velocity in a homogeneous half space due to impact piledriving at time steps (a) t = 30ms, (b) t = 80ms, and (c) t = 160ms for different penetrationdepths ep = 2.0m, 5.0m, and 10.0m.

Figure 10 shows the norm of the particle velocity in a homogeneous half space due to impactdriving at different time steps t = 30, 80, and 160ms and for different penetration depths. Thefollowing observations can be made: (1) body waves dominate around the toe and propagateon a spherical wave front; (2) vertically shear waves dominate around the pile and propagateradially on a cylindrical wave front; and (3) Rayleigh waves propagate near the surface with avelocity slightly less than shear waves.Figure 11 displays the attenuation of the peak particle velocity (PPV) with the distance r

1 10 1001

10

100

1000

PP

V [m

m/s

]

r [m]

Figure 11: Peak particle velocity (PPV) versus the distance from the pile due to impact piledriving.The comparison is made for different penetration depths ep = 1.0m (−⋄−), ep = 5.0m(− −), and ep = 10.0m (− ∗ −) with the field measurements (dash-dotted line) reported byWiss [2] .

from the pile due to impact pile driving. The maximum vibration amplitude occurs when thepenetration depth is small (ep = 2m). When no separation along the soil-pile interface isassumed (linear model), the peak particle velocity reduces for increasing penetration depth. Inpractice, during pile driving, a large portion of the energy is dissipated to overcome the soilfriction, moving the pile downwards (non-linear behavior). Therefore, less energy is transmittedin comparison with the case where the pile has arrived at a deeper layer with a higher resistance.In figure 11, the results have also been compared with field measurements (dash-dotted line)reported by Wiss [2], in which the vibrations have been measured due to an actual impact piledriving with a diesel hammer with an energy of 48 kJ. It can be observed that the predictedvibration levels attenuate slower than the field measurements. This reflects the fact that higherstrain levels during actual driving result in a higher attenuation.

4 Conclusions

A coupled FE-BE model has been developed to predict ground vibrations due to vibratoryand impact pile driving. Using a subdomain formulation, a linear model for dynamic soil-pileinteraction has been implemented. The pile is modeled using the finite element method. Thesoil is modeled as a horizontally layered elastic half space using a boundary element method.The characteristics of the propagating waves around the pile have been investigated. In the caseof vibratory driving, the induced waves can be classified into: (1) the vertically polarized shearwaves around the shaft; (2) the body waves around the pile toe, and (3) the Rayleigh waves onthe surface. Propagating waves due to impact driving are more complex. It is observed thatdeformations around the shaft are dominated by the vertically shear deformation where thenormal and the radial deformations are small compared to shear deformations. In the far field,however, body waves are attenuated and Rayleigh waves dominate the zone near the surface,independent of the penetration depth. Furthermore, it is observed that the frequency contentof ground vibrations reduces and peak values occur at lower frequency as the distance from thepile is increased.

Acknowledgements

The results presented in this paper have been obtained within the frame of the SBO projectIWT 03175 ” Structural damage due to dynamic excitation: a multi-disciplinary approach ”.This project is funded by IWT Vlaanderen, the Institute of the Promotion of Innovation byScience and Technology in Flanders. Their financial support is gratefully acknowledged.

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