Application of a High-cycle Accumulation Model to the Analysis of Soil Liquefaction

7/30/2019 Application of a High-cycle Accumulation Model to the Analysis of Soil Liquefaction

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R E S E A R C H P A P E R

Application of a high-cycle accumulation model to the analysisof soil liquefaction around a vibrating pile toe

V. A. Osinov

Received: 28 September 2012 / Accepted: 31 January 2013

Springer-Verlag Berlin Heidelberg 2013

Abstract High residual pore pressure observed in the

vicinity of piles driven in saturated soil indicates that the soilaround the pile may be liquefied. In the present paper, the

problem of deformation of saturated sand around a vibrating

pile is formulated with the use of a high-cycle accumulation

model capable of describing a large number of cycles. The

problem is solved numerically for locally undrained condi-

tions in spherically symmetric formulation suitable for the

lower part of a cylindrical closed-ended pile near the toe. The

aim of the study is to calculate the evolution of the lique-

faction zone around the pile for a large number of cycles. A

parametric study is carriedout to show how the growth of the

liquefaction zone depends on the pile displacement ampli-

tude, the relative soil density, the effective stress in the far

field and the pore fluid compressibility.

Keywords Cyclic model Liquefaction Saturated soil Vibratory pile driving

1 Introduction

It is known from numerous field measurements that the

installation of piles in saturated soils may lead to a significant

increase in the pore water pressure in the vicinity of a driven

pile [1, 8]. The residual pore pressure developed around a

pile can exceed the initial overburden pressure in the soil.

High pore pressure indicates that the effective stresses in the

soil are likely to be reduced to zero resulting in soil lique-

faction. The effective stress reduction, especially in the case

of soil liquefaction, may affect the adjacent piles and struc-

tures, the bearing capacity of the installed pile and the pileinstallation process itself.

Numerical modelling of the effective stress evolution

around a pile is determined by the pile installation method.

This paper is concerned with the deformation of saturated

soil during vibratory pile driving. Except for a few

numerical studies where a decrease in the effective stresses

is obtained for impact-driven [24] and vibrating [9] piles,

there is generally a lack of detailed theoretical investiga-

tions into the behaviour of saturated soil around dynami-

cally driven piles.

An insight into the problem was recently given by a

finite-element study of the dynamic deformation of satu-

rated sand around a vibrating pile [7]. The soil behaviour

was modelled by an extended version of the hypoplasticity

theory with intergranular strain [5] capable of describing

the cyclic deformation of granular soils. The numerical

calculations with locally undrained conditions revealed a

permanent liquefaction zone formed at a certain distance

from the pile after several cycles of vibration. Figure 1

shows the calculated distribution of the mean effective

stress in saturated dense sand around a cylindrical pile after

30 cycles (compressive stresses are negative). The darkest

area in the figure can be considered as a liquefaction zone.

Although the mean effective stress in the liquefaction zone

slightly changes during a cycle, it does not exceed 2% of

the initial effective stress. The effective stress in the

immediate vicinity of the pile does not vanish because of

the large strain amplitudes. The inner boundary of the

liquefaction zone (closer to the pile) remains stationary

with time, while the outer boundary spreads farther from

the pile making the liquefaction zone wider.

The finite-element calculations performed in [7] cover few

tens of cycles. The modelling of a real vibro-driving process

V. A. Osinov (&)

Institute of Soil Mechanics and Rock Mechanics, Karlsruhe

Institute of Technology, 76128 Karlsruhe, Germany

e-mail: [email protected]

123

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DOI 10.1007/s11440-013-0215-x


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requires at least several thousands of cycles in order to esti-

mate the size of the liquefaction zone produced by a driven

pile. The use of incremental constitutive models such as el-

asto-plasticity or hypoplasticity for calculations with large

numbers of cycles entails high computational costs and may

be impracticable for applications. Another drawback of

incremental models concerns weak accumulation effects at

small strain amplitudes of the order of 10-4 or less. Cyclic

deformation with small amplitudes is accompanied by the

gradual compaction of dry granular soil or the effective stress

reduction in saturated soilunder undrained conditions. Even if

an incremental model may correctly reproduce the plastic soilbehaviour under multi-cycle loading in general, it may be

difficult or impossible to calibrate an incremental model with

respect to the accumulation effects for small strain amplitudes

and large numbers of cycles. This especially concerns the

strong dependence of the accumulation effects on the soil

density. The growth of the liquefaction zone around a pile

after a large number of cycles is determined by the rate of the

effective stress reduction behind the outer boundary of the

liquefaction zone where strain amplitudes are small. There-

fore, the weak accumulation effects are responsible for the

final size of the liquefaction zone developed around a pile.

Besides high computational costs and the calibration

problems, calculations with an incremental model and

small strain amplitudes may produce an accumulation of

numerical errors after a large number of cycles.

Problems of cyclic soil deformation can also be solved

with the use of so-called explicit cyclic models in which

accumulation rates are defined with respect to the number of

cycles. A model of this kind, called high-cycle accumula-

tion model, is elaborated in [6, 10]. Explicit cyclic models

make it possible to calculate tens of thousands of cycles or

more in a reasonable computing time and thus to cover the

whole pile installation process. Since the constitutive

parameters control accumulation effects rather than incre-

mental stiffness, explicit cyclic models are easier to cali-

brate with respect to accumulation effects when compared

to incremental plasticity models. A drawback of explicit

cyclic models is that they are valid only for small strain

amplitudes below 10-3 and, for this reason, cannot beapplied to the immediate vicinity of a pile where defor-

mations are large. A way to circumvent this difficulty is

proposed in [7]. The approach consists in introducing an

auxiliary boundary surface around the pile in order to

exclude the region with large amplitudes from the compu-

tational domain. The strain amplitudes in the outer domain

must be small enough for a boundary value problem with an

explicit cyclic model to be posed in that domain. The

required boundary conditions on the auxiliary surface can

be obtained from the solution of a boundary value problem

for the whole domain with an incremental plasticity model

for a limited number of cycles. It is proposed in [7] tointroduce the auxiliary boundary inside the incipient liq-

uefaction zone as shown, for instance, in Fig. 1 by the white

dashed line. As follows from the solutions obtained in [7],

the varying part of the total stresses in the liquefaction zone

is nearly hydrostatic. This allows us to prescribe a simple

boundary condition for the outer domain.

The objective of the present paper is to apply the high-

cycle accumulation model [6, 10] to the calculation of the

evolution of the liquefaction zone around a vibrating pile

for a large number of cycles. The general formulation of

the problem is described in Sect. 2. The problem is for-

mulated with locally undrained conditions, assuming that

the soil permeability is low enough. Solutions with locally

undrained conditions are expected to give the highest rate

of the effective stress reduction and therefore the largest

liquefaction zone because of no pore pressure dissipation

due to seepage. The problem is solved numerically in Sect.

3 in spherically symmetric formulation. This simplification

restricts us to the consideration of the lower part of the

liquefaction zone where spherically symmetric solutions

may give a reasonable approximation, see Fig. 1. A para-

metric study is carried out to show how the growth of the

liquefaction zone depends on the pile displacement

amplitude, the relative soil density, the effective stress in

the far field and the pore fluid compressibility.

2 Formulation of the problem for saturated soil

2.1 First boundary value problem

As outlined above, the application of the cyclic model to

the pile vibration problem can be made possible by

Fig. 1 Mean effective stress in saturated sand around a pile after 30

cycles of vibration calculated for a cylindrical pile with a diameter of

30 cm, a pile displacement amplitude of 2 mm, a hydrostatic initial

effective stress of -50 kPa and a frequency of 34 Hz [7]

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introducing an auxiliary boundary surface which envelopesthe region of large strain amplitudes around the pile where

the cyclic model is inapplicable. The computational domain

is thus bounded by the auxiliary surface and a remote

boundary as shown in Fig. 2.

The calculation of stresses and deformations in saturated

soil with the use of the high-cycle accumulation model [6,

10] consists in the concurrent solution of two boundary

value problems and integration over time. The calculation

cycle between times tand t Dt is shown schematically inFig. 3 and is described below in detail.

The total stress tensor in saturated soil is the sum of the

effective stress tensor r (compressive stresses are negative)and an isotropic tensor -pI, where p is the pore pressure

(p[ 0 for compression), and I is the unit tensor. Let the

effective stress tensor r(x) and the pore pressure p(x),

where x denotes the position vector, be known at time t.

They represent average values over a cycle as defined in

[6]. The total stress must satisfy static equilibrium.

The first boundary value problem is solved in order to

find a scalar strain amplitude field eampx in the soil at time

t caused by given periodic boundary conditions on the

auxiliary surface. The boundary conditions must yield

sufficiently small strain amplitudes eamp (\10-3) in the

computational domain for the cyclic model to be applica-

ble. This boundary value problem is independent of the

cyclic model and may be solved in dynamic or quasi-static

formulation depending on the actual rate of loading in the

physical problem under study. In the dynamic case, non-reflecting boundary conditions should be prescribed at the

remote boundary to avoid the influence of reflected waves

on the strain amplitudes near the pile.

The first boundary value problem can be solved with the

use of any appropriate constitutive model. However, using

an incremental model to find amplitudes during cyclic

deformation would require high computational costs. We

assume that the response of the soil in the first boundary

value problem is linearly elastic and isotropic. This allows

us to solve the problem in dynamic steady-state formula-

tion with time-harmonic boundary conditions. The current

effective stress r and the bulk modulus of the pore fluid, Kf,determine the soil stiffness. The small-strain stiffness of a

soil skeleton as a function of the effective pressure is

known to follow a power law. For locally undrained con-

ditions, the Lame constants of the soil for small strain

amplitudes may be written in the form

k k0 rr0

m1 e

eKf; l l0

r

r0

m; 1

where r is the mean effective stress, e is the void ratio of

the skeleton, and k0, l0, r0, m are parameters. The term

with Kf is responsible for the contribution of the pore fluidcompressibility to the change in the total stresses. The soil

stiffness is spatially inhomogeneous because of the inho-

mogeneity ofr. The latter varies from a nearly zero value

in the liquefaction zone to a prescribed value in the far

field. Numerical calculations in Sect. 4 are performed with

k0 = 120 MPa, l0 = 80 MPa, r0 = -100 kPa, m = 0.6.

For sinusoidallyvarying strain components, the scalar strain

amplitude eamp required for the cyclic model is calculated as

Fig. 2 Computational domain for the problem with the high-cycle

accumulation model

Fig. 3 Solution scheme for saturated soil with the high-cycle accumulation model

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eamp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie

ampij e

ampij

q; 2

where eampij are the amplitudes of the strain components in a

rectangular coordinate system [6]. Relation (2) is valid

independently of the phase shifts between the components.

2.2 Strain accumulation rate

The strain amplitude eampx calculated in the firstboundary value problem determines a tensorial strain

accumulation rate _eaccx in the high-cycle accumulationmodel, see Fig. 3. The meaning of the tensorial quantity

_eacc is that it gives the rate of accumulated deformation in a

dry soil subjected to cyclic loading with the strain ampli-

tude eamp under the condition that the average values of the

stress components do not change. The rate of eacc in the

accumulation model is defined with respect to the number

of cycles, N, treated as a real variable. The connection

between the rate deacc=dN and the time derivative _eacc isgiven by

_eacc x

2p

deacc

dN; 3

where x is angular frequency.

The relation between eamp and deacc=dN constitutes themain part of the high-cycle accumulation model. This

relation also involves the stress tensor and the void ratio

and is written as [6, 10]

deacc

dN fampf0NfefpfYfpm: 4

The tensor m in (4) is a homogeneous function of degree

zero in r. It has unit Euclidean norm and determines the

direction of strain accumulation in the strain space. In the

problem considered in Sect. 3, the stress tensor is always

isotropic. For isotropic stress states, m I= ffiffiffi3p .The norm of the accumulation rate in (4) is determined

by the scalar factors famp;f0N;fe;fp;fY;fp. The factor famp

depends on the current strain amplitude eamp :

famp eamp

eamp

ref

Campif

eamp

eamp

ref

Camp\100;

100 otherwise:

8>:

5

The factor f0N depends on the number of cycles and alsotakes into account the changes in eamp during previous

deformation. It is given by

f0N CN1CN2 exp gA

CN1famp

CN1CN3; 6

where gA is a function of the number of cycles N. This

function is found from the solution of the differential

equation

dgA

dN fampCN1CN2 exp gA

CN1famp 7

with the initial condition gA(0) = 0.

The factor fe in (4) is responsible for the dependence of

the accumulation rate on the current void ratio e:

fe Ce e21 eref

1 eCe eref2: 8

The factor fp depends on the mean effective stress r:

fp exp Cp rpref

1 !

: 9

The factor fY is a function of the invariants of r. For

isotropic stress states, fY = 1. The factor fp depends on the

evolution of the cyclic strain path in the strain space. We

put fp = 1, which means that the strain loops change

sufficiently slow with the number of cycles. For detailed

discussion of (4) and the quantities involved, see [6, 10].

The parameters of (4-9) used for the numerical

calculations are given in Table 1.

2.3 Second boundary value problem

The application of the high-cycle accumulation model is

based on the premise that the evolution of the average

effective stress tensor r during cyclic deformation is

determined by the constitutive equation

_r Er : _e _eacc ; 10where _e is the rate of accumulated deformation, E is a

stress-dependent stiffness tensor, and _eacc is found from

eamp as described above.

Equation (10) shows the meaning of _eacc already men-

tioned earlier: _eacc gives the rate of accumulated deforma-

tion if the stress tensor is maintained constant, i.e., if _r 0.On the other hand, if cyclic deformation is applied without

accumulation of deformation, then _e 0, and _eacc deter-mines directly the stress rate according to the stiffness E.

Based on these relations, Eq. (10) can be calibrated for E

by comparing results of drained and undrained test. It is

Table 1 Constitutive parameters of the cyclic model (sand L12 from [ 11])

Camp eampref

CN1 CN2 CN3 Ce eref Cp pref (kPa)

1.6 10-4 3.6 9 10-3 0.016 1.05 9 10-4 0.48 0.829 0.44 100

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proposed in [10] to take the tensor E as in an isotropic

elastic solid. For the spherically symmetric problem con-

sidered in Sect. 3, only the bulk modulus Kcorresponding

to E is needed. This modulus is taken as a function of the

mean effective stress r in the form

Kr Ap1natm rn; 11

with A = 467, n = 0.46 and patm = 100 kPa [10].The second boundary value problem in Fig. 3 is quasi-

static and consists in the determination of the rates of the

effective stress, _r (x), and the pore pressure, _px, for agiven field _eaccx. Under the assumption of locallyundrained conditions, the governing equations are the static

equilibrium equation

div _r grad _p 0; 12the constitutive equation (10) for the effective stress tensor

and the evolution equation for the pore pressure

_p 1 ee

Kf tr _e: 13

Boundary conditions on the auxiliary surface can

be specified in terms of displacement or traction. It is

reasonable to prescribe zero displacement or constant

traction at that boundary, although further numerical

studies with incremental models such as in [7] may show

the appropriateness of other (e.g. time-dependent) boundary

conditions. The remote boundary is intended to imitate

an infinite domain and may be supplemented with zero

displacements, constant tractions or more sophisticated

boundary conditions such as infinite elements used in finite-

element analyses.The last step in the calculation cycle shown in Fig. 3 is

the integration of the fields _rx; _px and _gAx (see 7)over a time increment Dt. When rx;px and gA(x) attime t Dt have been found, a new calculation cyclebegins with the solution of the first boundary value prob-

lem. An optimum time increment may vary substantially

during calculation and, for this reason, should be steadily

estimated and updated. The time increment may be

increased if the change in the effective stresses in one

increment is too small. At the same time, care should be

taken in increasing the time increment because the func-

tions f0N and gA are nonlinear in Nand one time incrementmay contain a large number of deformation cycles.

3 Spherical problem

The problem of the evolution of the liquefaction zone for

large numbers of cycles as described in Sect. 2 is solved in

this paper under the assumption of spherical symmetry

as an approximation suitable for the lower part of the

liquefaction zone, see Fig. 1. The computational domain is

shown in Fig. 4. The domain is bounded by two spheres of

radii RA and RB which represent, respectively, an auxiliary

and a remote boundaries as discussed earlier. The mean

effective stress r is understood as an average value over a

cycle as defined in [6]. An inhomogeneous initial distri-

bution ofr assumed for time t= 0 reflects the fact that an

incipient liquefaction zone has already been formed. The

effective stress at each point will decrease with time due to

the cyclic loading resulting in the widening of the lique-

faction zone as shown in Fig. 4 for t[ 0. The meaning of a

stress rliq which defines the liquefaction zone will be dis-

cussed below. The boundary of the liquefaction zone is

denoted by Rliq.

Let the mean effective stress r(r) as a function of radius r

at a current time be known. The soil response in the first

boundary value problem is assumed to be linearly elastic

and isotropic with the Lame constants given by (1).

Assuming harmonic excitation, we are looking for solutions

in the form urreixt;rrreixt;rureixt, where ur;rr;ruare the complex amplitudes of the radial displacement and

radial and circumferential stress components, respectively, t

is time variable, and i is the imaginary unit. Given r(r), the

first boundary value problem consists in finding

urr;rrr;rur

which satisfy the equation of motion

drr

dr 2

rrr ru x2.ur; 14

the constitutive equations

rr k 2l durdr

2k urr

; 15

ru k durdr

2k l urr

; 16

and the boundary conditions

Fig. 4 Spherically symmetric problem

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rrRA ramp; 17rrRB SurRB; 18where . is the soil density, and ramp is a given amplitude

at the auxiliary boundary. The dynamic stiffness coefficient

S in (18) is

S 4lRB

x2

.cpRB cp ixRB c2p x2R2B

; 19

where cp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2l=.

pis the longitudinal wave speed

[12]. Relation (18) with (19) is a nonreflecting boundary

condition for outgoing spherical waves. It exactly imitates

an infinite domain provided k and l are homogeneous for

rC RB. Equations (1418) are solved by a finite-difference

technique.

The strain amplitude (2) in the spherical problem is

given by

eamp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidurdr 2

2urj j

2

r2s

: 20

The solution of the second boundary value problem in

the spherically symmetric case can be simplified by taking

_e in (10) to be equal to zero. This approximation is justified

if the bulk modulus of the pore fluid is much higher than

the stiffness of the skeleton, and the soil permeability is

low enough. With _e 0 in (10), the second boundary valueproblem degenerates and reduces to relation (10) at each

point to find the radial and circumferential stress

components. The equilibrium equation can be satisfied

through the proper distribution of the pore pressure.

Equation (13) is not used in this case as it becomesindeterminate with _e 0 and Kf ! 1:

The initial effective stress is taken to be isotropic. As

follows from (4) and (10) with _e 0, the effective stresswill then always remain isotropic. The rate of the mean

effective stress is determined by the relation

_r x2p

ffiffiffi3

pKfampf

0Nfefp; 21

where Kis given by (11). Using (21), the stress field can be

integrated over a time increment Dt to find a new distri-

bution r(r) at time t Dt:In specifying the inner radius RA and the loading

amplitude ramp for the spherical problem, we make use of

the numerical study of soil liquefaction around the toe of a

vibrating cylindrical pile with a diameter of 30 cm per-

formed with a hypoplastic constitutive model [7]. The

frequency of vibration in the present study is taken to be

34 Hz and is the same as in [7]. The auxiliary boundary

shown in Fig.1 by the white dashed line is assumed to lie

inside the incipient liquefaction zone formed around the

pile toe after several cycles of vibration starting from a

homogeneous stress state. The lower part of the auxiliary

boundary is approximated by a spherical surface with a

radius RA. This radius is determined mainly by the pile

displacement amplitude and the soil density and depends

only slightly on other parameters. In the numerical exam-

ples presented in [7] for dense sand, RA increases from

3540 to 6570 cm as the pile displacement amplitude

changes from 1 to 4 mm. For the present calculations, wetake RA = 50 and 70 cm, which corresponds, respectively,

to a pile displacement amplitude of 2 and 4 mm. As found

in [7], the varying part of the total stress in the liquefaction

zone is nearly hydrostatic. The total stress amplitude at the

auxiliary boundary, ramp, depends on the pile displacement

amplitude and on the position at the auxiliary boundary.

The latter dependence violates the spherical symmetry in

the lower part of the liquefaction zone and is the only

adverse factor for the spherical approximation.

4 Numerical results

As follows from (21), the average effective stress falls to

zero in a finite time and remains zero thereafter. This

implies rliq = 0 in Fig. 4. In a real situation, however, the

effective stress in the liquefied soil around a vibrating pile

would oscillate with a small amplitude about a small

nonzero average value. Such oscillations cannot be taken

directly into account within the framework of the cyclic

model if the strain amplitude is determined from steady-

state solutions. An approximate way is to take a small

nonzero rliq as a limit for the effective stress reduction and

not to reduce r below rliq regardless of (21). Note that a

small nonzero effective stress assumed for liquefied soil

can also be found in other models dealing with soil liq-

uefaction (cf. [13]). The question is how a small nonzero

stress rliq in the liquefaction zone influences the solution

when compared to rliq = 0.

To reveal the influence of rliq, consider the first

boundary value problem with a given distribution of the

effective stress r(r) as shown in Fig. 5. Besides rliq, the

Fig. 5 Prescribed distribution of the effective stress r for the

solutions in Figs. 68

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parameters of the distribution are RA;Rliq;DR and rini. The

change in r between Rliq and Rliq DR is taken to follow asinusoidal curve from rliq to rini. For fixed loading

amplitude ramp and DR, let us increase Rliq starting from RAto imitate the widening of the liquefaction zone. Figure 6

shows the strain amplitude (20) at r= RA as a function of

Rliq for two values ofrliq. The curves in the figure exhibit a

resonance-like phenomenon with eamp reaching a maximum

at a certain Rliq. The maximum value of eamp is 50 times

higher than that at the beginning when Rliq = RA. Similarcurves are obtained for points with r[RA. For brevity, the

strong increase in eamp at a certain Rliq will subsequently be

referred to as resonance. Another important feature of the

solutions is that the size of the liquefaction zone, Rliq,

which corresponds to the resonance turns out to depend

strongly on rliq, so that rliq becomes an additional

parameter in the modelling. Figure 7 shows eamp as a

function of both Rliq and rliq. The function for larger RAand DR is shown in Fig. 8.

In order to trace the evolution of the effective stress, we

now solve both the first and the second boundary value

problems in parallel as described in Sects. 2 and 3. Considerthe case with RA = 50 cm. The initial distribution of the

effective stress is taken with Rliq DR 50 cm. Theamplitude of the boundary loading, ramp, is kept constant

and equal to 3 kPa. Figure 9 shows the boundary of the

liquefaction zone, Rliq, as a function of time for different

values of rliq. Three stages can be distinguished in the

motion ofRliq. After an initial increase, Rliq undergoes an

abrupt jump within few seconds followed by a very slow

increase. The jump is a consequence of the resonance that

occurs at a certain Rliq. The speed of propagation ofRliq is

determined by the strain amplitude eamp in front ofRliq. For

a fixed time, eamp decreases with the distance from the

boundary. This explains why the growth ofRliq in the post-

resonance stage is very slow. The time at which resonance

occurs (subsequently referred to as resonance time) depends

on rliq. For instance, as seen from Fig. 9, the resonance time

for rliq = -0.6 kPa is larger than 3 min, that is why the

resulting Rliq in 3 min is much smaller than in the other

cases in the figure. Thus, the boundary of the liquefaction

zone after a given time of vibration essentially depends on

whether the resonance occurs within this time or later.

10-4

10-3

10-2

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

amp

Rliq [m]

liq = 0liq = -0.6 kPa

Fig. 6 Strain amplitude eamp at r= RA as a function ofRliq for given

r(r) as shown in Fig. 5, for a frequency of 34 Hz and RA = 50 cm,

DR 50 cm, rini = -50 kPa, ramp = 3 kPa, Kf = 2.2 GPa

Fig. 7 Strain amplitude eamp at r= RA as a function ofrliq and Rliqfor given r(r) as shown in Fig. 5. RA = 50 cm, DR 50 cm,rini = -50 kPa, ramp = 3 kPa, Kf = 2.2 GPa

Fig. 8 The same as in Fig. 7 for RA = 70 cm, DR 70 cm, ramp = 6kPa

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m]

time [min]

liq = 0liq = -0.2 kPaliq = -0.4 kPaliq = -0.6 kPa

Fig. 9 Boundary of the liquefaction zone as a function of time.

RA = 50 cm, rini = -50 kPa, ID = 0.7, ramp = 3 kPa, Kf = 2.2

GPa

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Figures 10, 11, 12, 13 show the same four solutions as inFig. 9 with one parameter being changed. An increase in

the loading amplitude ramp or a decrease in the relative

density ID shorten the resonance time. For instance, an

increase in ramp from 3 to 4 kPa or a decrease in ID from

0.7 to 0.6 reduce the resonance time for rliq = -0.6 kPa

to about 1 min and thus substantially increase the resulting

Rliq if the vibration time is longer than 1 min (Fig. 11).

The compression modulus of the pore fluid, Kf, in fully

saturated soil is equal to that of pure water (2.2 GPa). In

reality, it may be difficult to determine whether the soil isfully saturated or contains a small amount (a few volume

per cent) of undissolved gas entrapped in the pore space. In

the latter case, the compressibility of the pore fluid (a

mixture of water and gas) is substantially higher than that

of pure water. Figure 12 shows the curves with the same

parameters as in Fig. 9 except for Kf = 20 MPa, which

corresponds approximately to a degree of saturation of 99

%. The decrease in Kf has practically no effect on the

solution for rliq = 0, but essentially increases the reso-

nance time for nonzero rliq.

Another factor which strongly influences the solution is

the effective stress in the far field, rini, which represents theinitial stress in the soil prior to the vibration. As seen from

Fig. 13, the change in rini from -50 to -20 kPa drastically

reduces the resonance time, so that the resonance occurs at

the very beginning of the vibration.

According to the numerical modelling performed in [7],

the solutions in Figs. 9, 10, 11, 12, 13 correspond to a

pile displacement amplitude of 2 mm. For larger ampli-

tudes, both RA and ramp become larger. Figure 14 presents

an example which corresponds to a pile displacement

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m]

time [min]

liq = 0

liq = -0.2 kPaliq = -0.4 kPaliq = -0.6 kPa

Fig. 10 The same as in Fig. 9 with ramp = 4 kPa

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m]

time [min]


Fig. 11 The same as in Fig. 9 with ID = 0.6

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m]

time [min]

liq = 0liq = -0.2 kPaliq = -0.4 kPa

liq = -0.6 kPa

Fig. 12 The same as in Fig. 9 with Kf = 20 MPa

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m]

time [min]

liq = 0


Fig. 13 The same as in Fig. 9 with rini = -20 kPa

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq[m]

time [min]


Fig. 14 Boundary of the liquefaction zone as a function of time.

RA = 70 cm, rini = -50 kPa, ID = 0.7, ramp = 6 kPa, Kf = 2.2

GPa

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9/10

amplitude of 4 mm. The curves are similar to those shown

in Fig. 13. The resonance time is very small, and the res-

onance manifests itself as a rapid increase in Rliq at the

beginning. Calculations for denser soil with ID = 0.8

instead of 0.7 give a slight increase in the resonance time,

which still remains small compared to 3 min, see Fig. 15.

A much stronger increase in the resonance time is observed

by changing the stress rini from -50 to -100 kPa as seen

from Fig. 16.

The figures presented in this section show that the radius

of the liquefaction zone, Rliq, in the post-resonance stage

grows very slowly with time and, for the parameters con-

sidered, lies in the range between 1.1 and 1.6 m for a pile

displacement amplitude of 2 mm (Figs. 9, 10, 11, 12, 13)

and between 1.3 and 1.6 m for a pile displacement

amplitude of 4 mm (Figs. 14, 15, 16). The eventual size of

the liquefaction zone after a given vibration time is influ-

enced by the residual effective stress in the liquefaction

zone, rliq. A nonzero value ofrliq increases the radius Rliqwhen compared to rliq = 0 and also shifts the resonance to

a later time. The question of what value ofrliq should be

taken in applications when solving a particular problem

requires further investigation.

5 Concluding remarks

The problem of the deformation of saturated soil around a

vibrating pile, even without considering penetration, is

rather complicated for theoretical modelling because of the

fact that it involves a large number of cycles and both large

and small strain amplitudes. Neither an incremental plas-

ticity model nor an explicit cyclic model can be used toadequately describe the deformation process in the whole

region of interest from the immediate vicinity of the pile to

the far field. In the present study, the application of an

explicit cyclic model to the pile vibration problem and thus

the calculation of a large number of cycles is made possible

by introducing an auxiliary boundary around the pile and

solving the problem for the outer domain where the strain

amplitudes are small enough. The approach is implemented

for the cyclic model developed in [6, 10]. The choice of the

auxiliary boundary and the specification of the boundary

conditions are based on the solutions obtained earlier for the

whole domain with the hypoplastic constitutive model for alimited number of cycles [7]. The aim of the study was to

trace the evolution of the liquefaction zone around the pile.

The solutions obtained in the spherically symmetric

approximation reveal a resonance-like increase in the strain

amplitude at a certain distribution of the effective stress

and the resulting rapid increase in the current size of the

liquefaction zone at the resonance. The ultimate boundary

of the liquefaction zone for a given vibration time is

essentially determined by the time at which the resonance

occurs. The liquefaction zone becomes much bigger if the

resonance occurs within the vibration time. For certain sets

of parameters, the resonance is observed at the very

beginning of the vibration. The parametric study has shown

how the evolution of the liquefaction zone around a

vibrating pile toe is influenced by the pile displacement

amplitude, the relative soil density, the effective stress in

the far field, the pore fluid compressibility and the residual

effective stress assumed for the liquefaction zone.

Acknowledgements The study has been carried out within the

framework of the Research Unit FOR 1136 Simulation of geotech-

nical construction processes with holistic consideration of the stress

strain soil behaviour, Subproject 6, financed by the Deutsche

Forschungsgemeinschaft.

References

1. Hwang J-H, Liang N, Chen C-H (2001) Ground response during

pile driving. J Geotech Geoenviron Eng 127(11):939949

2. Mabsout M, Sadek S (2003) A study of the effect of driving on

pre-bored piles. Int J Numer Anal Meth Geomech 27:133146

3. Mabsout ME, Reese LC, Tassoulas JL (1995) Study of pile driving

by finite-element method. J Geotech Eng ASCE 121(7):535543

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m]

time [min]

liq = 0


Fig. 15 The same as in Fig. 14 with ID = 0.8

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5 3

Rliq

[m

]

time [min]


Fig. 16 The same as in Fig. 14 with rini = -100 kPa

Acta Geotechnica

123


10/10

4. Mabsout ME, Sadek SM, Smayra TE (1999) Pile driving by

numerical cavity expansion. Int J Numer Anal Meth Geomech

23:11211140

5. Niemunis A, Herle I (1997) Hypoplastic model for cohesionless

soils with elastic strain range. Mech Cohesive Frict Mater

2(4):279299

6. Niemunis A, Wichtmann T, Triantafyllidis T (2005) A high-cycle

accumulation model for sand. Comput Geotech 32:245263

7. Osinov VA, Chrisopoulos S, Triantafyllidis T (2012) Numerical

study of the deformation of saturated soil in the vicinity of a

vibrating pile. Acta Geotech (Accepted)

8. Pestana JM, Hunt CE, Bray JD (2002) Soil deformation and

excess pore pressure field around a closed-ended pile. J Geotech

Geoenviron Eng 128(1):112

9. Schumann B, Grabe J (2011) FE-based modelling of pile driving

in saturated soils. In: De Roeck G, Degrande G, Lombaert G,

Muller G (eds) Proceedings of the 8th international conference on

structural dynamics, EURODYN 2011, pp 894900

10. Wichtmann T, Niemunis A, Triantafyllidis T (2010) On the

elastic stiffness in a high-cycle accumulation model for sand:

a comparison of drained and undrained cyclic triaxial tests. Can

Geotech J 47(7):791805

11. Wichtmann T, Niemunis A, Triantafyllidis T (2011) Simplified

calibration procedure for a high-cycle accumulation model based

on cyclic triaxial tests on 22 sands. In: Gourvenec S, White D

(eds) Frontiers in offshore geotechnics II.. Taylor & Francis,

London, pp 383388

12. Wolf JP (1988) Soil-structure-interaction analysis in time

domain. Prentice Hall, New Jersey

13. Zhang J-M, Wang G (2012) Large post-liquefaction deformation

of sand, part I: physical mechanism, constitutive description and

numerical algorithm. Acta Geotechnica 7:69113

Acta Geotechnica

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Application of a High-cycle Accumulation Model to the Analysis of Soil Liquefaction