soilflex: a model for prediction of soil stresses and soil compaction due to agricultural field...

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Soil & Tillage Research 93 (2007) 391–411

SoilFlex: A model for prediction of soil stresses and soil

compaction due to agricultural field traffic including

a synthesis of analytical approaches

Thomas Keller a,*, Pauline Defossez b, Peter Weisskopf c,Johan Arvidsson a, Guy Richard b,d

a Department of Soil Sciences, Swedish University of Agricultural Sciences (SLU),

P.O. Box 7014, 75007 Uppsala, Swedenb Institut National de la Recherche Agronomique (INRA), Unite d’Agronomie de Laon, Rue Fernand Christ,

02007 Laon Cedex, Francec Swiss Federal Research Station for Agroecology and Agriculture (FAL), Reckenholzstrasse 191, 8046 Zurich, Switzerland

d Institut National de la Recherche Agronomique (INRA), Soil Science Department of Orleans,

Avenue de la Pomme de Pin, BP 20619, Ardon, 45166 Olivet Cedex, France

Received 27 October 2005; received in revised form 19 May 2006; accepted 27 May 2006

Abstract

Soil compaction is one of the most important factors responsible for soil physical degradation. Soil compaction models are

important tools for controlling traffic-induced soil compaction in agriculture. A two-dimensional model for calculation of soil

stresses and soil compaction due to agricultural field traffic is presented. It is written as a spreadsheet that is easy to use and therefore

intended for use not only by experts in soil mechanics, but also by e.g. agricultural advisers. The model allows for a realistic

prediction of the contact area and the stress distribution in the contact area from readily available tyre parameters. It is possible to

simulate the passage of several machines, including e.g. tractors with dual wheels and trailers with tandem wheels. The model is

based on analytical equations for stress propagation in soil. The load is applied incrementally, thus keeping the strains small for each

increment. Several stress–strain relationships describing the compressive behaviour of agricultural soils are incorporated.

Mechanical properties of soil can be estimated by means of pedo-transfer functions. The model includes two options for

calculation of vertical displacement and rut depth, either from volumetric strains only or from both volumetric and shear strains. We

show in examples that the model provides satisfactory predictions of stress propagation and changes in bulk density. However,

computation results of soil deformation strongly depend on soil mechanical properties that are labour-intensive to measure and

difficult to estimate and thus not readily available. Therefore, prediction of deformation might not be easily handled in practice. The

model presented is called SoilFlex, because it is a soil compaction model that is flexible in terms of the model inputs, the constitutive

equations describing the stress–strain relationships and the model outputs.

# 2006 Elsevier B.V. All rights reserved.

Keywords: Model; Soil compaction; Soil stress; Soil displacement; Traffic; Contact area; Bulk density

* Corresponding author. Tel.: +46 18 67 12 10; fax: +46 18 67 27 95.

E-mail address: [email protected] (T. Keller).

0167-1987/$ – see front matter # 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.still.2006.05.012

1. Introduction

Soil degradation is a major environmental problem

worldwide. Soil compaction, i.e. decrease in pore space,

is one of the most important factors responsible for soil

mailto:[email protected]

http://dx.doi.org/10.1016/j.still.2006.05.012

T. Keller et al. / Soil & Tillage Research 93 (2007) 391–411392

Nomenclature

a half axis of the super ellipse (m)

A compactibility coefficient

Acontact contact area (m2)

Ai area carrying a point load (m2)

b half axis of the super ellipse (m)

B compactibility coefficient (kPa�1)

c cohesion (kPa)

C compactibility coefficient (kPa�1)

CA amplitude of the decay function (kN m�3)

Cc compression index (Mg m�3 kPa�1)

d rut depth (m)

D compactibility coefficient

E Young’s modulus of elasticity (kPa)

G shear modulus (kPa)

Hi horizontal point load (kN)

ji shear displacement (cm)

J1 first invariant of the strain tensor

k shear displacement coefficient (cm)

K modulus of the shear stress–strain curve

l(y) length of contact at certain position of y

(m)

m distance between the yield line and the

virgin compression line on the v� ln pplane

n parameter governing the shape of the

super ellipse

N specific volume at a mean normal stress

of 1 kPa

p mean normal stress (kPa)

Pi vertical point load (kN)

q deviator stress (kPa)

qa asymptotic value of the shear strength

(kPa)

qf failure deviator stress (kPa)

r distance between the point load and the

desired point (m)

Rf failure ratio

S1 desired degree of saturation (%)

Sk desired saturation at a reference stress (%)

v specific volume

vinit initial specific volume

vYL specific volume at the intersection of the

yield line and the recompression line

wðxÞ width of contact at certain position of x (m)

x horizontal distance along the driving

direction (m)

y lateral distance at right angles to the

driving direction (m)

z soil depth (m)

Greek letters

a order of the power-law function

g parameter governing the shape of the

decay function

d angle between the shear load vector and

the vertical plane that contains the posi-

tion vector from the shear load to the

desired point (8)DT slope of the bulk density versus degree of

water saturation curve (Mg m�3)

e strain

en natural volumetric strain

es shear strain

ev volumetric strain

ez vertical strain

u angle between the normal load vector and

the position vector from the point load to

the desired point (8)k recompression or swelling index (kPa�1)

k0 slope of the steeper recompression line

(kPa�1)

ln compression index (kPa�1)

m Poisson’s ratio

j concentration factor

r bulk density (Mg m�3)

rk reference bulk density (Mg m�3)

rs density of solids (Mg m�3)

r0 initial bulk density (Mg m�3)

s normal stress (kPa)

sk reference stress (kPa)

soct octahedral normal stress (kPa)

sr radial normal stress (kPa)

s1 major principal stress (kPa)

s3 minor principal stress (kPa)

sa applied stress (kPa)

t shear stress (kPa)

toct octahedral shear stress (kPa)

w angle of internal friction (8)

physical degradation (Pagliai et al., 2003). Therefore, it

is important to control the mechanical impacts of

agricultural machinery on soil structure in order to

reduce the risk of soil compaction.

Soil compaction models are important tools for

controlling soil compaction due to agricultural field

traffic. By means of such models, strategies and

recommendations for prevention of soil compaction

may be developed and specific advice may be given to

farmers and advisers.

Soil compaction models can be divided into two

categories, analytical and numerical (finite element)

T. Keller et al. / Soil & Tillage Research 93 (2007) 391–411 393

models (Defossez and Richard, 2002; Abu-Hamdeh and

Reeder, 2003). Analytical models are based on the work

of Boussinesq (1885), who developed equations for the

propagation of stress through elastic, homogeneous,

semi-infinite media. Frohlich (1934) introduced the so-

called concentration factor to account for the fact that

stresses in soils are more concentrated below the centre

of the applied load than in a fully elastic medium. From

the calculated stress, strain is then obtained by applying

a stress–strain relationship. Finite element models apply

continuum mechanics and may have the potential to

describe the mechanical behaviour of soil more

accurately. Stresses and strains are calculated simulta-

neously. However, compared with analytical models,

numerical models usually require more input para-

meters, which may be difficult to measure. A review of

existing soil compaction models can be found in

Defossez and Richard (2002).

Despite the limitations of the model assumptions,

analytical models have yielded satisfactory predictions of

stress propagation through soil and changes in soil

volume (e.g. Gupta et al., 1985; Johnson and Burt, 1990;

Koolen et al., 1992; O’Sullivan et al., 1999; Arvidsson

et al., 2001; Defossez et al., 2003; Keller and Arvidsson,

2004). Therefore, the analytical model approach is

justified. In addition, analytical models are usually easier

to use than finite element models, and can therefore be

used not only by experts in soil mechanics, but also by

e.g. agricultural advisors. This is a very important aspect

for the control of soil compaction in practice.

1.1. Characteristics of some existing analytical

models

In this section, we discuss the analytical soil

compaction models of Gupta and Larson (1982), van

den Akker (2004), Johnson and Burt (1990) and

O’Sullivan et al. (1999). The model of O’Sullivan

et al. (1999) is a further development of the model of

Table 1

Description of surface stress in some analytical soil compaction models

Reference Contact area Shape of stress d

Vertical stress

Gupta and Larson (1982) Elliptical Power-law functi

Johnson and Burt (1990) Rectangular Uniform or powe

(maximum stress

centre or under th

O’Sullivan et al. (1999) Circular Power-law functi

van den Akker (2004) Rectangular, elliptical,

or user-defined

Uniform, parabol

or user-defined

Smith (1985); therefore, we only discuss the former

here. The model of van den Akker (2004) is known as

‘SOCOMO’, while the model of O’Sullivan et al.

(1999) is known as ‘Compsoil’. The four models

discussed are widely used and referred to in the

literature (see Defossez and Richard, 2002).

The characteristics of the upper boundary condition

of the models discussed are presented in Table 1. In the

model of O’Sullivan et al. (1999), the contact area is

calculated from tyre and loading characteristics and

then converted to an equivalent circular area, while the

stress distribution is dependent upon soil water content

and bulk density as suggested by Sohne (1953).

The model of O’Sullivan et al. (1999) is one-

dimensional, i.e. stress is calculated under the tyre

centre only, whereas the models of Gupta and Larson

(1982), Johnson and Burt (1990) and van den Akker

(2004) are two-dimensional and stress is calculated in a

vertical plane parallel to the transverse or longitudinal

axis of the tyre.

O’Sullivan et al. (1999) use the analytical equations

proposed by Sohne (1953) to calculate the vertical stress

below a circular contact area. Gupta and Larson (1982),

Johnson and Burt (1990) and van den Akker (2004) use

the summation procedure as proposed by Sohne (1953).

Whereas Johnson and Burt (1990) and van den Akker

(2004) calculate the complete stress state including the

principal and octahedral stresses, Gupta and Larson

(1982) and O’Sullivan et al. (1999) only calculate the

major principal stress. O’Sullivan et al. (1999) use then

an empirical relationship to determine the intermediate

and minor principal stress in order to calculate the mean

normal stress.

Soil deformation is not calculated in the models of

Johnson and Burt (1990) and van den Akker (2004).

However, van den Akker (2004) estimates soil failure:

soil failure occurs when the precompression stress and/

or the shear strength are exceeded by the soil stresses.

Gupta and Larson (1982) describe volume change with

istribution No. of wheels

Horizontal stress

on (Sohne, 1953) Not considered 1

r-law function

either in the

e edge of the tyre)

Uniform or as a function

of soil shear strength

1

on (Sohne, 1953) Not considered 4 � 1

ic, trapezoidal, As for vertical stress 1


a relationship between the bulk density and the major

principal stress according to Larsson et al. (1980),

whereas O’Sullivan et al. (1999) use the relationship of

O’Sullivan and Robertson (1996) in terms of the

specific volume and the mean normal stress. O’Sullivan

et al. (1999) calculate vertical displacement from

volumetric strain, assuming the vertical strain to be

equal to the volumetric strain.

1.2. Objectives of the new model presented

in this paper

The analysis of the soil compaction models of

Gupta and Larson (1982), Johnson and Burt (1990),

O’Sullivan et al. (1999) and van den Akker (2004)

reveals that each of these models has its limitations.

Some models include stress–strain relationships, while

others do not. The same applies to the shear stress on

the soil surface. The contact area and the distribution

of stress over the contact area are described in many

different ways.

Therefore, the objectives of this work were to

include and further develop all useful aspects from these

existing models in a new, two-dimensional model for

prediction of stresses and compaction due to agricul-

tural field traffic. A further objective was to provide

model functions for estimation of the upper boundary

condition and soil mechanical parameters. The main

goal of the new model is to provide accurate predictions

of stress propagation and changes in bulk density. The

model, which we call ‘SoilFlex’, is presented in this

paper. We included different approaches for character-

isation of the stress–strain relationship that were

developed from compression tests on different agri-

cultural soils and comprise slightly different soil

mechanical parameters. The model allows for a direct

comparison between these stress–strain relationships.

We developed a model that allows for the simulation of

the passage of machine combinations (several wheels),

as well as different wheel configurations (dual wheels,

tandem wheels). Furthermore, we included functions to

estimate the contact area and the contact stresses from

tyre and loading characteristics, and pedo-transfer

functions to estimate soil mechanical properties that

are required for the stress–strain relationships. Our new

model is intended to be easy to use and can therefore be

used in practice, e.g. by agricultural advisers.

2. Structure of SoilFlex

SoilFlex is a two-dimensional model that calculates

the stress state, changes in bulk density and vertical

displacements in soil due to agricultural field traffic.

The model contains three main components. Firstly,

stress on the surface is described; both normal and shear

stresses are considered. Secondly, stress propagation

through soil is calculated analytically. Thirdly, soil

deformation is calculated as a function of stress. This

approach was also used by O’Sullivan et al. (1999). The

flow chart of SoilFlex is illustrated in Fig. 1.

2.1. Upper boundary condition: contact area and

contact stresses

The stresses are represented by an array of i point

loads, each having normal (Pi) and shear (Hi)

components, and acting at the centre point of their

respective areas, Ai. In SoilFlex, the vertical and

horizontal point loads are represented as matrices on

separate Excel sheets.

In addition to the possibility of entering any user-

defined contact area and stress distribution, the model

also offers several options to generate stress distribu-

tions (Fig. 1, Table 2). The shape of the contact area can

generally be described by a so-called super ellipse

(Hallonborg, 1996), which in an orthogonal co-ordinate

system is given as

xn

anþ yn

bn ¼ 1 (1)

where a and b are the half axes of the super ellipse and n

is a positive real number that governs the shape. For

a = b and n = 2, Eq. (1) defines a circle. In the general

case of a 6¼ b, the curve is a pure ellipse for n = 2,

whereas it grows towards a rectangle as n!1. In the

model, the contact width is assumed to be equal to the

tyre width (unless a circular contact area is chosen). The

contact area is calculated either (i) by dividing wheel

load by tyre inflation pressure, or from tyre and loading

characteristics according to (ii) O’Sullivan et al. (1999)

or (iii) Keller (2005). The parameter n of Eq. (1) has a

value of 2 in the cases (i) and (ii), and is a function of

tyre dimensions in case (iii). The distribution of vertical

stress can either be uniform, linear (the stress declines

linearly from a maximum at the contact area centre to

zero at the contact area edge), described by a power-law

function (the stress declines continuously from a max-

imum at the contact area centre to zero at the contact

area edge) according to Sohne (1953), or generated

from tyre properties according to Keller (2005). In all

cases, the wheel load is equal toP

Pi. If the stress

distribution is chosen according to Sohne (1953) in

SoilFlex, the order of the power-law function can be

calculated from soil conditions according to O’Sullivan


Fig. 1. Flow chart of SoilFlex. For explanations see Table 2 (input parameters) and Table 3 (pedo-transfer functions (ptfs)).


Table 2

Options, characteristics and input parameters SoilFlex

Description Equations Input parameters Reference

Upper boundary condition

Contact area Distribution of vertical stress

Circular Uniform or linear Fwheel: wheel load (kg);

ptyre: tyre inflation pressure (kPa)

Power-law function Fwheel: wheel load (kg);

ptyre: tyre inflation pressure (kPa)

Sohne (1953)

Power-law function,

calculated from soil

conditions

Fwheel: wheel load (kg); ptyre: tyre inflation

pressure (kPa); ttyre: tyre type (X-ply or radial);

rsurf: bulk density surface layer (Mg m�3)

O’Sullivan et al.

(1999) and Sohne

(1953)

Elliptical Uniform Fwheel: wheel load (kg); ptyre: tyre inflation

pressure (kPa); wtyre: tyre width (m)

Power-law function Fwheel: wheel load (kg); ptyre: tyre inflation

pressure (kPa); wtyre: tyre width (m)

Sohne (1953)

Superelliptical Calculated from

tyre parameters

Eqs. (1)–(3) Fwheel: wheel load (kg); ptyre: tyre inflation

pressure (kPa); prec: recommended ptyre (kPa);

wtyre: tyre width (m); dtyre: tyre diameter (m)

Keller (2005)

User-defined User-defined

Contact area Distribution of

horizontal stress

Any shape Uniform or linear Ftraction: net traction (kN)

Calculated from

soil strength

Eq. (4) c: cohesion (kPa); w: angle of internal

friction (8); k: shear displacement coefficient

(cm); s: wheel slip (%)

Janosi (1962)

User-defined

Stress

propagation

Eqs. (5),

(A.1)–(A.16)

j: concentration factor Boussinesq

(1885), Cerruti

(1888), Frohlich

(1934) and

Sohne (1953)

Stress–strain

relationships

General input

parameters

rinit: initial dry bulk density (Mg m�3);

rs: density of solids (Mg m�3)

Larsson et al. (1980) Eq. (6) C: slope of VCL (Mg m�3 kPa�1);

S1: desired degree of saturation (%),

Sk: degree of saturation at sk and rk (%);

DT: slope of r vs. S curve at sk (kPa);

rk: reference bulk density (Mg m�3);

sk: reference stress (kPa)

Larsson et al.

(1980)

Bailey and

Johnson (1989)

Eqs. (7) and (8) A: compactibility coefficient; B: compactibility

coefficient (kPa�1); C: compactibility

coefficient (kPa�1); D: compactibility coefficient

Bailey and

Johnson (1989)

O’Sullivan and

Robertson (1996)

Eqs. (9)–(11) m: separation distance YL-VCLa (kPa);

N: specific volume at p = 1 kPa; k: slope

of RCLa (kPa�1); k0: slope of the ‘‘steeper’’

RCL (kPa�1); ln: slope of VCLa (kPa�1)

O’Sullivan and

Robertson (1996)

Shear failure Mohr-Coulomb Eq. (14) c: cohesion (kPa); w: angle of internal

friction (8)cited in Koolen

and Kuipers (1983)

Shear deformation Eqs. (20)–(24) c: cohesion (kPa); w: angle of internal friction

(degrees); G: shear modulus (MPa)

a YL: yield line (O’Sullivan and Robertson, 1996); VCL: virgin compression line; RCL: recompression line.


Fig. 2. Geometrical relationship for the calculation of stress at a

desired point in the soil. Pi: vertical point load; Hi: horizontal point

load; sr: radial normal stress.

et al. (1999). Keller (2005) describes the transverse

distribution of vertical stress as

sðyÞ ¼ CA

�wðxÞ

2� y

�e�gððwðxÞ=2Þ�yÞ; 0 � y � wðxÞ

2

(2)

where CA and g are parameters and wðxÞ the width of

contact area at position x (for x = 0, wðxÞ is equal to the

tyre width). The longitudinal axis of the tyre along the

driving direction is assumed to be an axis of symmetry.

The longitudinal distribution of vertical stress is

described by a power-law function:

sðxÞ ¼ sx¼0;y

�1�

�x

lðyÞ2

�a�; 0 � x � lðyÞ

2(3)

where sx = 0,y is the stress under the tyre center, l(y) the

length of contact area at position y (for y = 0, l(y) is

equal to the tyre length), and a is a parameter. The

transverse axis of the tyre (at right angles to the driving

direction) is assumed to be an axis of symmetry. The

parameters of Eqs. (2) and (3) are calculated from

readily available tyre parameters (Keller, 2005).

Horizontal stress (shear stress) on the soil surface can

be either calculated from a given traction or from soil

strength, as defined by the cohesion, c, and the angle of

internal friction, w. Distribution of horizontal stress is

either uniform, linear from zero at the front edge of the

contact area to a maximum at the rear edge, or

calculated from the distribution of vertical surface stress

and shear strength. In the latter case, Hi, is calculated

according to Janosi (1962):

Hi ¼ Ai

�cþ Pi

Aitan ’

�ð1� e� ji=kÞ (4)

where ji is the shear displacement at Ai and k is the shear

displacement coefficient. The shear displacement is

calculated as ji = sxi, where s is the slip and xi is the

distance from the front edge of the contact area to the

position of Ai (Koolen and Kuipers, 1983). The slip, s, is

a model input parameter. For all distributions, net

traction is equal toP

Hi.

In contrast to many soil compaction models, the

input of e.g. dual and triple wheels, as well as double

and triple tandem wheels is possible and automatically

generated in SoilFlex. The additional parameters

needed for that are the gap between the dual/triple

wheels and the distance between the axles of double/

triple tandem wheels.

It is possible to input up to four different axles (one

axle can contain one single wheel or dual/triple wheels,

or tandem wheels) in SoilFlex to simulate multi-passage

of wheels.

2.2. Stress propagation

Calculation of stress is based on the work of

Boussinesq (1885), Cerruti (1888), Frohlich (1934) and

Sohne (1953). The contact area, Acontact, is divided into i

small elements with an area Ai each and a normal stress,

si, carrying the load Pi = siAi, which is treated as a point

load, following the approach of Sohne (1953). We

include shear stress on the soil surface in the same way,

i.e. each area Ai can also contain a shear point load, Hi.

The radial normal stress, sr,i, at depth z is then

sr;i ¼jPi

2pr2i

cosj�2 ui þjHi

2pr2i

sinj�2 ui cos di (5)

where j is the concentration factor (Frohlich, 1934), r

the distance from the point load to the desired point, u

the angle between the normal load vector and the

position vector from the point load to the desired point

and d is the angle between the shear load vector and the

vertical plane that contains the position vector from the

shear load to the desired point (Fig. 2). The propagation

of the stress components in the x-, y- and z-directions,

the principal stresses, the mean normal stress, p, the

deviator stress, q, and the octahedral shear stress, toct,

are calculated from Eq. (5), as given in Appendix A

(Eqs. (A.1)–(A.16)).


Fig. 3. Model of rebound and recompression according to O’Sullivan

and Robertson (1996) in terms of specific volume, v, and mean normal

stress, p (adapted from O’Sullivan et al. (1999)). VCL: virgin com-

pression line; RCL: recompression line; RCL0: steeper recompression

line; ln: compression index; k: recompression index; k0: slope of the

steeper recompression line.

2.3. Soil compaction: stress–strain relationships

We incorporate three different approaches (which can

be chosen optionally in SoilFlex, cf. Fig. 1 and Table 2)

for description of the sub-routine to calculate volumetric

strain and change in bulk density, namely the stress–

strain relationships by Larsson et al. (1980), Bailey and

Johnson (1989) and O’Sullivan and Robertson (1996),

which all describe the compressive behaviour of

agricultural soils. These three approaches may be

classified as ‘strain hardening models’, i.e. soil becomes

stronger due to straining. To our knowledge, the stress–

strain relationship of Bailey and Johnson (1989) has not

previously been included in any analytical soil compac-

tion model, but it has been incorporated in finite element

models (Bailey et al., 1995).

Larsson et al. (1980) describe volume change with a

relationship between the bulk density and the base 10

logarithm of the major principal stress:

r ¼ ½rk þDTðS1 � SkÞ� þ Cc log

�sa

sk

�(6)

where r is the compacted (final) bulk density corre-

sponding to an applied stress sa, rk the reference bulk

density corresponding to a reference stress sk on the

virgin compression line (VCL), DT the slope of the bulk

density versus degree of water saturation curve at sk, S1

the desired degree of saturation at sk, Sk the degree of

saturation corresponding to rk and sk and Cc is the

compression index, i.e. the slope of the VCL. Eq. (6)

describes the soil behaviour on the VCL only. We

assume the recompression index to equal zero in the

model of Larsson et al. (1980).

The model of Bailey and Johnson (1989) was

developed for cylindrical stress states and is given by

en ¼ ðAþ BsoctÞð1� e�CsoctÞ þ D

�toct

soct

�(7)

where en is the natural volumetric strain, soct the

octahedral (or mean) normal stress, toct the octahedral

shear stress, and A–D are the compactibility coeffi-

cients. For D = 0, the model of Bailey and Johnson

(1989) reduces to the model of Bailey et al. (1986),

which was developed for hydrostatic stress states. The

coefficients A–C have the same values in both models

(Bailey et al., 1995). In terms of bulk density, r, Eq. (7)

becomes (Bailey and Johnson, 1989):

ln r ¼ ln r0 ��ðAþ BsoctÞð1� e�CsoctÞ þ D

�toct

soct

��

(8)

where r0 is the initial bulk density.

The model of O’Sullivan and Robertson (1996) is

illustrated in Fig. 3. The VCL, the recompression line

(RCL) and the steeper recompression line (RCL0) are

given by

VCL : v ¼ N � ln ln p (9)

RCL : v ¼ vinit � k ln p (10)

RCL0 : v ¼ vYL � k0 ln p (11)

where v is the specific volume, p the mean normal

stress, N the specific volume at p = 1 kPa, ln the

compression index, vinit the initial specific volume,

k the recompression index, vYL the specific volume at

the intersection of the yield line and the recompression

line and k0 is the slope of the steeper recompression

line, which is given by (O’Sullivan and Robertson,

1996):

k0 ¼ffiffiffiffiffiffiffilnkp

(12)

The parameter m in Fig. 3 was found to be 1.3 (O’Sul-

livan and Robertson, 1996). Bulk density, r, is calcu-

lated from specific volume, v, according to

r ¼ rs

v(13)

where rs is the density of solids. Note that the VCL in

the model of O’Sullivan and Robertson (1996) is

defined in v� ln p space, whereas Larsson et al.

(1980) define the VCL in r � log s space, and hence

Cc in Eq. (6) is different from ln in Eq. (9).


2.3.1. Pedo-transfer functions to estimate soil

mechanical parameters

In SoilFlex, the parameters of the stress–strain

relationships can be estimated by means of pedo-transfer

functions (ptfs). Table 3 gives an overview of ptfs for soil

mechanical parameters found in the literature. These

functions correlate soil mechanical properties with soil

physical properties, soil texture, etc. They may be used to

estimate soil mechanical properties or to analyse the

effect of e.g. soil texture on soil mechanical properties in

an average case. Not all ptfs of Table 3 were included in

SoilFlex, because some require many soil parameters

and/or complicated measurements.

The parameters of the model of Larsson et al. (1980;

Eq. (6)) can be estimated by the ptfs described in Gupta

and Larson (1982). Alternatively, the compression

index, Cc (Eq. (6)), can be estimated from the ptfs

developed by Lebert and Horn (1991) and Imhoff et al.

(2004). Parameters A–C of the model of Bailey and

Johnson (1989; Eq. (8)) can be estimated by the ptfs by

McBride (1989). We could not find any ptfs for

prediction of D (Eq. (8)). The parameters of the model

of O’Sullivan and Robertson (1996; Eqs. (9)–(11)) can

be estimated by the ptfs developed by O’Sullivan et al.

(1999) and Defossez et al. (2003).

2.4. Shear failure

In SoilFlex, shear failure can be calculated option-

ally (Table 2). Shear failure is calculated according to

the Mohr–Coulomb failure criterion. No failure of the

soil occurs if the following criterion is satisfied (Koolen

and Kuipers, 1983):

s3� s1 tan2

�45� � ’

2

�� 2c tan

�45� � ’

2

�(14)

where s1 and s3 are the major and minor principal

stress, respectively, w the angle of internal friction, and c

is the cohesion (kPa).

Note that the shear parameters cohesion and angle of

internal friction are no required input parameters for

model calculations of compaction, since shear para-

meters are not included in the calculations of changes in

bulk density (Section 2.3). We could not find any ptfs

for estimation of the shear parameters. Typical values

for w and c may be found in textbooks.

2.5. Vertical soil displacement and rut depth

Two options to calculate vertical soil displacement

are incorporated in SoilFlex. In the first option, uniaxial

strain state is assumed. According to Koolen and Kuipers

(1983), lateral strains are negligible in the subsoil, i.e.

the strain state in the subsoil is with good approximation

uniaxial. Assuming uniaxial strain state is a simple

approximation that has been used in analytical soil

compaction models (e.g. O’Sullivan et al., 1999).

However, different authors including O’Sullivan et al.

(1999), Defossez and Richard (2002) and Defossez et al.

(2003) have reported that this may result in under-

estimation of rut depth and that considering lateral

strains in the calculation would probably increase the

accuracy of the predictions. While lateral strains may

be small in the subsoil, they may be significant in the

topsoil. Therefore, a second option for calculation of

vertical displacement is included in the model. This

second option includes shear strains due to shear stresses

andallows for lateral deformation. To ourknowledge, this

aspect has not been considered in analytical soil com-

paction models, but is common in finite element models.

The first invariant of the strain tensor, J1, equals the

volumetric strain, ev, and yields (see e.g. Koolen and

Kuipers, 1983):

J1 ¼ ev ¼ e1 þ e2 þ e3 ¼ ex þ ey þ ez (15)

where e1, e2 and e3 are the normal strains in the principal

directions and ex, ey and ez are the normal strains in x-,

y- and z-direction. Note that ev is calculated as described

in Section 2.3.

In the first option, the strain state in soil is assumed to

be uniaxial, i.e. ex = ey = 0. Hence, Eq. (15) reduces to

ez ¼ ev (16)

where ez is the vertical strain.

In the second option, volumetric and shear strains are

calculated (following Atkinson and Bransby (1978), we

associate like invariants of stress and strain, i.e. shear

strain, es, with deviator stress, q, and volumetric strain,

ev, with mean normal stress, p). The strain state is

assumed to be biaxial (plane strain), i.e. the horizontal

strain in the plane of calculation is assumed to be zero.

This restriction has to be made due to numerical

reasons. Therefore, the model does not predict

horizontal displacements. However, this is an accep-

table simplification if the main goal is to predict vertical

displacements. Furthermore, the x-, y-, and z-directions

are assumed to be principal directions. Shear strain, es,

is then given as

es ¼ ez � ez (17)

where ez is the horizontal strain perpendicular to the

vertical plane of calculation. The vertical strain, ez, is

T.

Keller

eta

l./So

il&

Tilla

ge

Resea

rch9

3(2

00

7)

39

1–

41

14

00

Table 3

Pedo-transfer functions for estimation of soil mechanical properties

Reference Property No. of

samples

Variables Soil group

description

Regressions equations (pedo-transfer functions) r2

Gupta and

Larson (1982)

Slope of r vs. degree of

saturation curve at sk, DT

54 Silt content, %silt

(g 100 g�1); clay content,

%clay (g 100 g�1)

Fine-textured soils DT = 3.461 � 10�3 + 1.742 � 10�4 (%silt) � 2.980 � 10�6 (%silt)2 0.63

Coarse-textured soils DT = 3.217 � 10�3 + 3.251 � 10�4 (%clay) � 5.385 � 10�6 (%clay)2 0.70

Compression index, Cc

(uniaxial tests)

54 Clay content, %clay

(g 100 g�1)

Expanding clay Cc = 2.033 � 10�1 + 1.423 � 10�2 (%clay) � 1.447 � 10�4 (%clay)2 0.79

Non-expanding clay Cc = 1.845 � 10�1 + 1.205 � 10�2 (%clay) � 1.108 � 10�4 (%clay)2 0.89

McBride (1989) The three parameters of

the Bailey et al. (1986)

relationship: A–C

(uniaxial tests)

34 Sand content, SC (g kg�1);

clay content, CC (g kg�1);

organic matter, OM (g kg�1);

particle density, rP (Mg m�3);

Atterberg lower plastic limit,

LPL (g 100 g�1); plasticity

index, PI; initial bulk density,

rb (Mg m�3); initial void ratio,

e; gravimetric water content,

w (g g�1); degree of saturation, S

Non-plastic (low w) A = 0.184 � 0.0529 log CC � 0.203e 0.973

B ¼ �0:887þ 1:946 log rP þ 0:557 log rb þ 0:0895 log w 0.909

C ¼ 19:87� 10:11 log SCþ 1:355 log OMþ 14:55 log w 0.807

Plastic ðw>PLÞ A ¼ 12:88� 0:130wþ 0:003w2 0.927

B = �0.766 + 1.649 log rP 0.667

C = 16.35 � 1.34 log(OM � CC) � 7.574 log PI 0.987

Plastic

(w< PL, PI < 15)

A = 1.036 � 0.062 log(OM � CC) � 1.086 logL PL 0.968

B = �2.269 � 0.049 log(OM � CC) + 5.075 log rP + 0.190 log LPL 0.995

C = �175.8 + 348.7[log(CC)]�1 + 1.746 log(OM � CC)

+ 14.05 log PI + 11.11S

0.998

Plastic

(w< PL, PI � 15)A ¼ 6:14� 13:74½logðCCÞ��1 � 0:950 log LPL � 0:345 log w 0.967

B = 0.551 � 0.477 log LPL 0.614

C = 25.30 � 5.896 log SC � 8.622 log PI 0.974

Lebert and

Horn (1991)a

Precompression stress,

pc (uniaxial tests)

307 Internal friction, f (8); cohesion,

c (kPa); bulk density, rb (Mg m�3);

air capacity, Lk (%, v/v); available

water capacity, nFk (%, v/v);

non-available water capacity,

TW (%, v/v); saturated hydraulic

conductivity, kf (�103 cm s�1);

organic matter, OM (g 100 g�1)

Sand pc(pF = 1.8) = 438.10rb � 0.0008(f1.8)3 � 3.14TW

� 0.11(nFk1.8)2 � 465.60

0.778

pc(pF = 2.5) = 410.75rb � 0.0007(f1.8)3 � 3.41TW

� 0.35(nFk2.5)2 � 384.71

0.710

Sandy loam pc(pF = 1.8) = 169.30rb � 29.03(OM)0.5 + 6.45kf + 32.18 log(C1.8)

� 9.44f1.8 + 27.25sin(TW) + 119.74 log(nFk1.8) + 19.51

0.828

pc(pF = 2.5) = 89.50rb � 23.99(OM)0.5 � 2.89kf + 125.76 log(C2.5)

� 1.14f1.8 + 26.90 sin(TW) + 51.46 log(nFk2.5) + 77.25

0.874

Silt pc(pF = 1.8) =374.15rb � 4.10OM + 3.38Lk1.8 � 1.58(kf)�0.5

+ 1.79C1.8 + 1.09(TW) � 6.37(f1.8)0.67 + 0.088(nFk1.8)2 � 472.77

0.765

pc(pF = 2.5) = 460.71rb � 20.33OM + 9.08Lk2.5 � 2.38(kf)�0.5

+ 2.86C2.5 + 4.50(TW) � 20.96(f2.5)0.67 + 0.304(nFk2.5)2 � 610.62

0.847

Clay and clay

loam (<35%)

pc(pF = 1.8) = 0.843rb � 0.544(kf)0.33 + 0.022TW + 7.03 (C1.8)�1

+ 0.024f1.8 � 0.015nFk1.8 + 0.725

0.808

pc(pF = 2.5) = 0.844rb � 0.456(kf)0.33 � 0.026TW + 12.88 (C2.5)�1

+ 0.003f2.5 � 0.016nFk2.5 + 1.419

Clay and clay

loam (�35%)

pc(pF = 1.8) = 4.59rb � 1.02OM � 16.43(kf)0.33 + 0.31TW

� 1.57nFk1.8 + 3.55C1.8 + 1.18f1.8 � 18.03

0.774

pc(pF = 2.5) = 70.65rb � 0.55OM � 7.01(kf)0.33 + 1.32TW

� 1.08nFk2.5 + 1.72C2.5 + 1.05 f2.5 � 100.94

0.763

T.

Keller

eta

l./So

il&

Tilla

ge

Resea

rch9

3(2

00

7)

39

1–

41

14

01


(uniaxial tests)

307 Clay and clay

loam (�35%)

Cc = 0.302 0.696

Clay and clay

loam (<35%)

Cc = 0.260 0.446

Sandy loam Cc = 0.235 0.523

Sand (�1%) Cc = 0.138 0.224

Sand (<1%) Cc = 0.145 0.146

Silt Cc = 0.168 0.510

Clay, loam and silt

with platy structure

Cc = 0.129 0.419

Kirby (1991)b Precompression stress, pc

(uniaxial tests)

170 Liquid index IL, void ratio at

pc, epc; saturation at epc, Sepc;

All soils pc = exp(5.856 � 4.352IL � 1.074epc) 0.682


(uniaxial tests)

170 Void ratio at pc, epc;

saturation at epc, Sepc

All soils Cc = 0.229 + 0.1736epc � 0.400Sepc 0.514

Rebound index, k

(uniaxial tests)

170 Void ratio at pc, epc All soils k = �0.00905 + 0.0153epc 0.205

O’Sullivan

et al. (1999)

Specific volume

at p = 1 kPa, N

Gravimetric water content,

w (g 100 g�1)

Sandy loam N ¼ 2:430� 0:0055ðw� 11:2Þ2

Clay loam N ¼ 2:813� 0:0128ðw� 17:4Þ2

Slope of the VCL, l Specific volume at p = 1 kPa, N Sandy loam l = (N � 1.572)/17

Clay loam l = (N � 1.557)/26

Slope of the RCL, k Water content, w (g 100 g�1);

slope of the VCL, l

Sandy loam k ¼ l½0:119� ð0:082w=17ÞClay loam k ¼ l½0:119� ð0:082w=26Þ�

Defossez

et al. (2003)

Specific volume at

p = 1 kPa, N

Gravimetric water content,

w (g 100 g�1)

Loss N ¼ 1:9997þ 0:2629w� 0:02753w2 þ 0:00111w3

�1:6� 10�5w4

Calcareous N ¼ 2:4196þ 0:13767w� 0:02035w2 þ 0:00121w3

�2:4� 10�5w4

Slope of the VCL, l Gravimetric water content,

w (g 100 g�1)

Loss l ¼ 0:1402þ 0:00192wþ 0:00021w2 � 1:3� 10�5w3

Calcareous l ¼ 0:08051þ 0:03131w� 0:00502w2 þ 0:00031w3

�6� 10�6w4

Imhoff

et al. (2004)a

Precompression stress,

pc (uniaxial tests)

50 Clay content, CC (g 100 g�1);

initial bulk density, rb (Mg m�3);

water content, w (g g�1)

All soils pc ¼ �566:764þ 442:891rb þ 4:338CC� 773:057w 0.70

Compression index,

Cc (uniaxial tests)

Clay content, CC (g 100 g�1);

initial bulk density,rb (Mg m�3)

CC < 29.42 Cc = 0.248 + 0.006CC � 0.121rb 0.77

CC > 29.42 Cc = 0.416 � 0.121rb 0.77

r: bulk density; sk: reference stress on the virgin compression line (Larsson et al. (1980), Gupta and Larson (1982)); pF: logarithm of soil water potential in kPa; p: mean normal stress; VCL: virgin

compression line; RCL: recompression line.a The equations for the precompression stress were not included in SoilFlex.b Not included in SoilFlex.


Fig. 4. Deviator strain, es, as a function of deviator stress, q; measure-

ments (grey curve) of O’Sullivan and Robertson (1996) and approx-

imation (black curve) using Eq. (19). qa: asymptotic value of the shear

strength; qf: ultimate (failure) deviator stress.

obtained from Eqs. (15) and (17):

ez ¼ 12ðev þ esÞ (18)

Shear stress–strain relationships were measured by

O’Sullivan and Robertson (1996) in tri-axial pressure

cells. An example of a q � es curve is given in Fig. 4. We

propose to describe this relationship by the following

equation:

es ¼ �K ln

�1� q

qa

�(19)

where K is the modulus of the shear stress–strain curve

and qa is the asymptotic value of the shear strength. The

initial shear modulus, G, is the slope of the initial

section of the shear stress–strain curve. For tri-axial

strain conditions with ex = ey, the initial part yields

(Atkinson and Bransby, 1978):

G ¼ dq

3des

(20)

Hence, with respect to Fig. 4:

K ¼ qa

3G(21)

The shear modulus, G, may be calculated as (see e.g.

Atkinson and Bransby, 1978):

G ¼ E

2ð1þ mÞ (22)

where E is the Young’s modulus of elasticity and m is

the Poisson’s ratio. The asymptotic stress, qa, is defined

as

qa ¼qf

Rf

(23)

where Rf is the failure ratio, which is chosen to be

Rf = 0.9 (Brinkgreve, 2002) and qf is the ultimate (fail-

ure) deviator stress (kPa), which is obtained from

Eq. (14) as

qf ¼ s3½tan2ð45� þ 12’Þ � 1� þ 2c tanð45� þ 1

2’Þ (24)

where c is the cohesion and w is the angle of internal

friction. During elastic unloading a different path is

followed (Fig. 4), which can be formulated as

es � e0s ¼

K

qa

ðq� q0Þ (25)

where the superscript 0 indicates the state at the end of

unloading. The relationship for shear deformation pro-

posed here is similar to the hyperbolic relationship

proposed by Duncan and Chang (1970) for the hyper-

bolic model used in several finite element soil compac-

tion models (e.g. Chi et al., 1993).

From the vertical strain calculated either by

assuming uniaxial strain state (Eq. (16)) or by assuming

plane strain (Eq. (18)), vertical displacement and rut

depth are calculated by summation of vertical strains.

The bottom layer is fully fixed, i.e. it cannot move in any

direction. Therefore, it is important to choose the lower

boundary at a depth great enough where displacements

are zero. The rut depth, d, is given as

d ¼Xi¼n

i¼1

ez;i (26)

where ez,i is the vertical strain of layer i, i = 0 the bottom

layer and i = n is the surface layer.

2.6. Numerical aspects

The model is implemented in an Excel file. It

contains several macros that are written in Visual

Basic. As an option, the user may calculate the vertical

stress only, the stress state (without consideration of

soil deformation) or the complete stress state

including calculation of soil deformations and

displacements (Fig. 1). This has the advantage that

input and output can be chosen according to the

requirements needed. For example, for an a priori

comparative assessment of the impact of different

machines, the calculation of the vertical stress may be

sufficient. The computational time of such a calcula-

tion is short, and no soil mechanical parameters are

required. In a second step, site-specific calculations

(with the most suitable machine) may be made by

considering stress–strain relationships and calculation

of soil deformations.


For the calculation of soil stress, the soil profile is

assumed to be homogeneous, i.e. the concentration factor

is the same at any depth. This might be a poor

approximation of a real soil. However, the introduction

of layers is in conflict with the analytical equations that

were developed by Boussinesq (1885) and Cerruti (1888)

for a semi-infinite, elastic, homogeneous, isotropic

medium. Frohlich (1934) introduced the concentration

factor to account for inelasticity of soil but the equations

are still only strictly valid for semi-infinite conditions.

Therefore, only one layer can be considered.

For the calculation of soil compaction and soil

deformation, the soil profile can be divided into a

number (maximum 10) of separate soil layers for

description of soil mechanical properties, i.e. input

parameters for the three different models to describe the

compressive behaviour of soil and the parameters

needed to describe the shear behaviour of soil. The

thickness of each layer can be chosen individually.

The load (surface stresses) is applied in a number of

increments as suggested by O’Sullivan et al. (1999). In

doing so, the strains are kept small for each increment,

thus overcoming conflicts with the basic assumptions of

the stress calculus. After each increment, displacement

is calculated and rut depth is updated before the next

increment. The contact area is divided into i small

squares, Ai, the size of which can be chosen (by default,

all Ai are squares of 0.05 m � 0.05 m).

The model is principally three-dimensional. By

default, stresses and strains are calculated in two

dimensions, namely in a plane perpendicular to the

driving direction (i.e. under the transverse axis of a tyre)

and/or in the driving direction (i.e. under the long-

itudinal axis of a tyre).

As mentioned elsewhere, the simulation of the

passage of several axles (each of which may contain one

single wheel or dual/triple wheels) is possible in

Fig. 5. Wheeling experiment on a moist Eutric Cambisol with a tyre of size 1

(a) measured vertical stress (symbols) and calculated stress (curve); (b) mea

uniaxial strain state (black curve), and assuming plane strain state (grey cu

SoilFlex. In this case, the soil undergoes elastic

unloading prior to the passage of the subsequent wheel;

the irrecoverable strains and displacements due to the

previous wheel form the initial soil conditions for the

subsequent wheel.

3. Model evaluation

3.1. Comparison of simulated and measured stress

and displacement

We compared measured with calculated vertical

stress (Fig. 5a) and vertical displacement (Fig. 5b)

due to a single passage of a wheel of a sugar beet

harvester (wheel load 86 kN, tyre inflation pressure

100 kPa) on a moist Eutric Cambisol. Soil parameters

are given in Table 4. A value for j of 5 was used. The

experiment is described in detail in Keller and

Arvidsson (2004). The stress distribution at the tyre–

soil interface was calculated according to Eqs. (2) and

(3). Bulk density was calculated applying the stress–

strain relationship of Robertson (1996; Eqs. (9)–(13)).

Vertical displacement was calculated using the two

options described in Section 2.5: either the vertical

strain was equal to the volumetric strain (Eq. (16)), or

Eq. (18) was used where strains due to shear stresses

are included in the calculation of vertical displace-

ment. The parameters used for the simulations are

given in Table 4.

The simulated stress agreed very well with the

measured stress (Fig. 5a). With both options, vertical

displacement in the subsoil was overestimated, whereas

it was underestimated in the topsoil, which resulted in

an underestimation of the rut depth (i.e. the vertical

displacement of the soil surface) (Fig. 5b). Similar

results were reported by Defossez et al. (2003) who

compared measured with simulated rut depth using

050/50 R32 with a load of 86 kN and an inflation pressure of 100 kPa;

sured vertical displacement (symbols) and calculated values assuming

rve). For soil parameters see Table 4. The error bars indicate S.E.M.


Table 4

Soil parameters used for simulations shown in Figs. 5 and 10

Parameter Symbol

(units)

Topsoil

(0–0.3 m depth)

Subsoil

(>0.3 m depth)

Texture (FAO) Loam Silty clay loam

Initial bulk density r0 (Mg m�3) 1.66 1.67

Model of O’Sullivan and Robertson (1996)

Specific volumea at p = 1 kPa N 1.811 1.777

Compression indexa ln (kPa�1) 0.068 0.060

Swelling indexa k (kPa�1) 0.0194 0.0137

Slope of the ‘steeper recompression line’a k0 (kPa�1) 0.0363 0.0286

Parametera m 0.8 0.9

Shear parameters

Cohesionb c (kPa) 60 100

Angle of internal frictionc w (8) 30 30

Elastic shear modulusd G (kPa) 500 800

a Calculated from oedometer tests by assuming s2 = s3 = 0.5s1 (Koolen and Kuipers, 1983).b Measured in situ by means of a shear vane apparatus.c Estimated.d Calculated from the Young’s modulus derived in oedometer tests by assuming a Poisson’s ratio of 0.3.

Fig. 6. Distribution of vertical stress on the soil surface (upper

boundary condition) below a tyre of size 710/70 R38 with a load

of 107 kN and an inflation pressure of 220 kPa in Gysi (2001) and with

SoilFlex.

‘Compsoil’ (O’Sullivan et al., 1999), which applies

Eq. (16). The underestimation of rut depth may be due

to the fact that ‘Compsoil’ neglects lateral displacement

(Defossez et al., 2003). However, rut depth was also

underestimated by considering lateral displacement

(Fig. 5b). This may be due to uncertain values of some

parameters, especially of the shear parameters. There-

fore, we performed a sensitivity analysis of the critical

mechanical parameters, which is described later in

Section 3.3.3.

3.2. Comparison of predicted with measured values

and with values predicted by other models

We compared simulations with SoilFlex with

measurements and simulations by Gysi (2001) who

measured and modelled stress, rut depth and change in

bulk density due to the passage of a single wheel of a

sugar beet harvester. Gysi (2001) used a critical state

soil mechanics model which was based on the finite

element method. Soil parameters for the simulations in

SoilFlex were taken from Gysi (2001). Avalue for j of 3

was used for the simulations in SoilFlex. The

distribution of the vertical stress on the soil surface

as used in Gysi (2001) and the distribution generated

from tyre parameters (Keller, 2005) used in SoilFlex are

shown in Fig. 6. The comparison of measured and

predicted mean normal stress and bulk density are

shown in Fig. 7. The values predicted by means of

SoilFlex agreed well with measurements by Gysi

(2001), as shown in Fig. 7a. It is interesting to note that

the mean normal stress calculated with SoilFlex using

the upper boundary condition (surface stresses) of Gysi

(2001) does not differ from the mean normal stress

calculated by Gysi (2001), who used a finite element

model. This shows that (i) the analytical approach for

calculation of stress distribution is justified and (ii) the

distribution of stresses on the soil surface strongly

affects the stress propagation. The latter is in agreement

with earlier studies (e.g. Keller and Arvidsson, 2004).

3.3. Sensitivity analyses

3.3.1. Concentration factor

The concentration factor, j (Eq. (5)), may often

be used as a fitting parameter, since it is a parameter


Fig. 7. Wheeling experiment on a Eutric Cambisol with a tyre of size 710/70 R38 with a load of 107 kN and an inflation pressure of 220 kPa (Gysi, 2001);

(a) measured (white circles) and simulated (grey rhombi) mean normal stress, p, of Gysi (2001), simulated p with SoilFlex (black curve), and (black

broken curve) simulated p with SoilFlex using the upper boundary condition (surface stresses) of Gysi (2001); (b) measured initial bulk density (circles)

and final bulk density (triangles) of Gysi (2001), and simulated final bulk density with SoilFlex (black curve). The error bars indicate S.D.

that is not directly measurable. Horn (1990) showed

that j is greater the smaller the precompression stress

(i.e. the weaker the soil) and the greater the applied

load. This implies that j is not only dependent on

Fig. 8. Effect of concentration factor, j, on propagation of (a) the major pr

density with depth with the stress–strain relationship of (c) Larsson et al

characteristics as in Fig. 5.

soil properties, but also on the loading intensity.

Sohne (1953) and O’Sullivan et al. (1999) related j to

the bulk density and the water content of the soil in such

a way that j is greater the softer the soil. However,

incipal stress, s1, and (b) the mean normal stress, p; changes in bulk

. (1980) and (d) O’Sullivan and Robertson (1996); soil and loading


Trautner (2003) found an opposite behaviour, i.e. that j

is greater the harder the soil. Therefore, it is important

to analyse the sensitivity of the predicted stress

propagation on j.

It is well known that j changes the shape of ‘pressure

bulbs’ in such a way that the stress is more concentrated

around the load axis and therefore reaches greater

depths at greater values of j. It is worthwhile

mentioning that this holds true for the major principal

stress (Fig. 8a), but not necessarily for other stress

components. For example, the opposite is true for

horizontal stresses. Therefore, p may become smaller or

larger with increasing j, depending on the soil depth

(Fig. 8b) and as a consequence, the effect of j on soil

compaction is dependent on whether soil compaction is

described as a function of s1 (e.g. Larsson et al., 1980)

or p (e.g. O’Sullivan and Robertson, 1996) as shown in

Fig. 8c and d. This has to be considered when drawing

conclusions on the effect of the concentration factor on

soil compaction.

3.3.2. Compressive stress–strain relationship

We studied the effect of the stress–strain relation-

ship (as described in Section 2.3) on the change in

bulk density. Uniaxial strain state was assumed, i.e.

vertical displacement was calculated according to

Eq. (16). The soil mechanical parameters required for

the different models (i.e. the models of Larsson et al.

(1980), Bailey and Johnson (1989) and O’Sullivan

and Robertson (1996)) were measured for a chalky

soil with an initial bulk density of 1.1 Mg m�3 and an

Table 5

Measured soil mechanical parameters of a chalky soil used for simulations

Parameter

Model of Larsson et al. (1980)

Reference stress

Reference bulk density at sk

Degree of saturation corresponding to rk and sk

Desired degree of saturation

Slope of the bulk density vs. Sk curve at sk

Compression index

Model of O’Sullivan and Robertson (1996)

Specific volume at p = 1 kPa

Compression index

Swelling index

Slope of the ‘steeper recompression line’

Parameter

Model of Bailey et al. (1986)

Compactibility coefficient



initial water content of 0.20 g g�1 (corresponding to a

water potential of �10 kPa). The soil is described in

Defossez et al. (2003). The parameters of the model

of Larsson et al. (1980; Eq. (6)) were measured by

means of oedometer tests, while those of the models

of Bailey and Johnson (1989; Eq. (8)) and O’Sullivan

and Robertson (1996; Eqs. (9)–(13)) were measured

using tri-axial cells. Details of the soil mechanical

tests are given in Defossez et al. (2003). The

measured parameters are summarised in Table 5.

We used a value for j of 5. We did not measure the

compactibility coefficient D (Eq. (8)); therefore, we

applied the model of Bailey et al. (1986) rather than

the model of Bailey and Johnson (1989). For the

simulations we applied a total load of 10 kN with

a uniform surface stress of 100 kPa over a circular

area.

The results of the simulations are shown in Fig. 9.

The differences in simulated changes in bulk density

(Fig. 9a) are small between the simulations using the

models of Bailey et al. (1986) and O’Sullivan and

Robertson (1996) at depths shallower than 0.3 m. The

predicted change in bulk density was lowest when using

the model of Larsson et al. (1980). At depths greater

than 0.3 m, the model of Larsson et al. (1980) predicts

no volume change, while the model of O’Sullivan and

Robertson (1996) predicts volume changes down to

0.85 m depth. The model of Bailey et al. (1986) predicts

volume changes at any depth. This is obvious from

Eq. (8): as soon as soct > 0, r > r0 with the parameters

used (Table 5). An additional simulation using the

shown in Fig. 9

Symbol (units) Value

sk (kPa) 100

rk (Mg m�3) 1.25

Sk (%) 51.4

S1 (%) 44.2

DT (Mg�1 m3) 0.054

Cc (Mg m�3 kPa�1) 0.393

N 2.813

ln (kPa�1) 0.176

k (kPa�1) 0.0058

k0 (kPa�1) 0.0319

m 1.3

A �0.244

B (kPa�1) �1.62 � 10�4

C (kPa�1) 1.175 � 10�2


Fig. 9. Changes in (a) bulk density and (b) vertical displacement of a chalky soil below a circular area that is uniformly loaded (load = 10 kN, contact

stress = 100 kPa) calculated with the stress–strain relationship of (light grey curve) Larsson et al. (1980); (dark grey curve) Bailey and Johnson

(1989); (black curve) O’Sullivan and Robertson (1996); (black broken curve) O’Sullivan and Robertson (1996) assuming k = k0 = 0.

model of O’Sullivan and Robertson (1996) was

performed with k = k0 = 0, as assumed for the other

models. This resulted in compaction to a smaller depth

compared with the ‘original’ simulation (k0 > k > 0,

Table 5), which demonstrates that in the model of

O’Sullivan and Robertson (1996) it is assumed that

compaction occurs even if the applied stress is

lower than the precompression stress (Fig. 3). The

differences in changes in bulk density are reflected

in the simulated vertical displacement and hence rut

depth (Fig. 9b).

3.3.3. Soil mechanical parameters

Based on the results of the simulation experiment

described in Section 3.1 and shown in Fig. 5, we

performed a sensitivity analysis to evaluate the two

Fig. 10. Measured (symbols) and calculated (lines) vertical displacement be

pressure of 100 kPa on a moist Eutric Cambisol (as in Fig. 5); (a) effect o

assuming uniaxial strain state: reference simulation with parameter values giv

(black curve) and ln = 0.07 (grey curve); (b) effect of the shear parameter

reference simulation with parameter values given in Table 4 (black broken c

(dark grey curve) and G = 100 kPa (black curve). The error bars indicate s

options for calculation of vertical displacement. For the

sensitivity analysis, we studied the following soil

mechanical parameters: the compression index, ln

(Eq. (9)); the cohesion, c (Eq. (24)); the angle of internal

friction, w (Eq. (24)); the elastic shear modulus, G

(Eq. (20)). Each of these parameters was analysed

individually, while the values of the other parameters

were as given in Table 4. The simulations were performed

with the stress–strain relationships of O’Sullivan and

Robertson (1996; Eqs. (9)–(13)). Loading characteristics

were as described in Section 3.1.

Fig. 10a shows the effect of the compression index,

ln, of the subsoil on the vertical displacement assuming

uniaxial strain state. We made simulations with

ln = 0.05 and 0.07, which had a strong effect on the

vertical displacement. Furthermore, the reference

low a tyre of size 1050/50 R32 with a load of 86 kN and an inflation

f the compression index, ln, of the subsoil on vertical displacement

en in Table 4 (black broken curve) and simulations made with ln = 0.05

s of the topsoil on vertical displacement assuming plane strain state:

urve), and simulations made with c = 10 kPa (light grey curve), w = 0

tandard error of the mean.


simulation assuming uniaxial strain state, with a value

for ln of 0.06 (Table 4) is shown in Fig. 10a.

Fig. 10b shows the effect of the cohesion, c, angle of

internal friction, w, and elastic shear modulus, G of the

topsoil on the vertical displacement assuming biaxial

strain state. Parameters c, w and G are the additional

input parameters needed when assuming biaxial strain

state compared with when assuming uniaxial strain

state. In addition to the reference simulation, we made

three different simulations where either c only, w only,

or G only was changed compared with the reference

simulation, and all other parameters had the values

given in Table 4. The values of the changed parameters

used in these three simulations were c = 10 kPa, w = 0

and G = 100 kPa, respectively. A decrease in soil

strength, i.e. a smaller c, w or G, significantly increases

shear deformation and thus the vertical displacement

compared with the simulated displacement obtained

from the measured soil parameters (reference simula-

tion assuming biaxial strain state) (Fig. 10b). Simula-

tions with higher c, w and G resulted in insignificantly

smaller displacements compared with the reference

simulation and are therefore not shown. This indicates

that the shear strength was probably overestimated.

Cohesion was obtained from measurements with a vane

shear apparatus, which may overestimate cohesion

(e.g. Schjønning and Rasmussen, 2000). Furthermore,

we did not measure w, and for the calculation of G we

had to assume a value for the Poisson’s ratio. Therefore,

there is a large uncertainty in the parameter values,

which may explain discrepancies between measured

and simulated vertical displacements.

Note that the reference simulations in Figs. 10a and

b slightly differ from each other, as also shown in

Fig. 5b. This is because the reference simulation shown

in Fig. 10a represents the reference simulation for

uniaxial strain state (no shear parameters needed),

while the one shown in Fig. 10b represents the

reference simulation for biaxial strain state (shear

parameters are required).

It may be concluded that accurate prediction of

displacements is strongly dependent on accurate values

of the soil mechanical parameters. However, some

parameters are difficult and/or work-intensive to

measure. Including shear deformations in the calcula-

tion of vertical displacement (Eqs. (17)–(25)) may have

the potential to provide more accurate predictions, but

requires additional soil parameters (c, w and G), which

in many cases are not readily available. Therefore, two

options are available in SoilFlex, which can be utilised

according to the available soil parameters and the

required accuracy.

4. Summary and conclusions

A two-dimensional soil compaction model, ‘Soil-

Flex’, is presented. We use the name SoilFlex, because

it is a soil compaction model that is flexible in terms of

the description of the stresses on the soil surface,

modelling of the stress–strain relationship, estimation

of mechanical soil properties using pedo-transfer

functions, and model output; and because the user

can e.g. easily add pedo-transfer functions to the model.

With the model, the mechanical flexibility of soil may

be studied. The flow chart of SoilFlex is illustrated in

Fig. 1. The model options and their respective input

parameters are summarised in Table 2. A copy of the

Excel spreadsheet is available by e-mail from the

authors. SoilFlex is not a commercial product. No

liability is accepted by the authors.

The model is easy to use, which makes it applicable

not only in education and research, but also for

agricultural advisers. This is of great importance for the

control of soil compaction in practice. With SoilFlex, it

is possible to simulate the passage of machine

combinations that are used in practice, including e.g.

tractors with dual wheels, which is a significant aspect

and an omission in previous models. The model allows

for a realistic estimation of the stresses at the tyre–soil

interface that can be estimated from tyre parameters.

This is not accounted for in either existing finite

element or analytical soil compaction models. How-

ever, it has been shown that this is an important aspect

for accurate predictions of stress propagation (e.g.

Keller and Arvidsson, 2004). Soil mechanical para-

meters required for characterisation of the stress–strain

relationship can be estimated by means of pedo-

transfer functions, which is a great help in many

situations where data are missing. However, there are

only a few pedo-transfer functions for soil mechanical

parameters available; the performances of which are

not well evaluated and therefore their reliabilities are

not well known.

With regard to the accuracy of predicted stresses and

volume changes, the analytical approach as used in

SoilFlex was shown not to be inferior to finite element

models. A weak point of the analytical solution may be

the concentration factor, as it is not a directly

measurable soil parameter. However, finite element

models include e.g. the Poisson’s ratio, which may be

very difficult to measure with standard testing equip-

ment. With the analytical approach, we can only

consider one homogeneous layer for stress propagation,

which may be a serious limitation. However, the error

may not be large in many cases. Nevertheless, for


strongly heterogeneous soil conditions the model may

yield less accurate predictions, and it could be the

subject of future research to define the limits of

application of analytical models. In SoilFlex, we

consider strain due to shear stresses and lateral

displacement, thus increasing the accuracy of the

prediction of the rut depth, which is not done in previous

analytical models. However, computation results of soil

deformation strongly depend on soil mechanical

properties. The lack of easily accessible and represen-

tative soil mechanical properties is a major obstacle to

the prediction of soil deformation in practice. As stated

by van den Akker (2004), there is a need for

development of pedo-transfer functions for estimation

of soil mechanical properties. Some generally valid

aspects on the reliability and use of pedo-transfer

functions are discussed for soil hydraulic properties by

e.g. Wosten et al. (1999).

Future refinements of the model could include a

function for estimation of the traction (shear stresses on

the soil surface) from draught force requirement of tillage

implements, the incorporation of more (better) pedo-

transfer functions for estimation of the soil mechanical

parameters and the consideration of the factor of time

(deformation is a time-dependent process).

Agricultural field traffic is an important aspect of soil

management. Future applications may therefore include

coupling SoilFlex with SISOL (Roger-Estrade et al.,

2000), a model that simulates the structural changes of

soil over time as a function of management practices.

Acknowledgments

Mr. M.F. O’Sullivan of the Scottish Agricultural

College (SAC), Midlothian, U.K., is thanked for kindly

providing the code of the Compsoil model (O’Sullivan

et al., 1999). Dr. A.J. Koolen of the Wageningen

University, Wageningen, The Netherlands, and Mr.

Hubert Boizard of the Institut National de la Recherche

Agronomique (INRA), Estrees-Mons, France, are

thanked for helpful discussions. Financial support

was received from the French National Institute for

Agricultural Research (INRA), the Swedish Research

Council for Environment, Agricultural Sciences and

Spatial Planning (Formas) and the Swedish Farmers’

Foundation for Agricultural Research (SLF), which is

gratefully acknowledged.

Appendix A

The stress components in the x-, y- and z-directions at

depth z are calculated from Eq. (5), following the

approach of Sohne (1953):

sz ¼Xi¼n

i¼0

ðszÞi ¼Xi¼n

i¼0

sr;i cos2 ui (A.1)

sx ¼Xi¼n

i¼0

sh;i cos2 di ¼Xi¼n

i¼0

sr;i sin2 ui cos2 di (A.2)

sy ¼Xi¼n

i¼0

sh;i sin2 di ¼Xi¼n

i¼0

sr;i sin2 ui sin 2di (A.3)

txy ¼Xi¼n

i¼0

sh;i cos di sin di ¼Xi¼n

i¼0

sr;isin2 ui cos di sin di

(A.4)

txz ¼Xi¼n

i¼0

ti cos2 di ¼Xi¼n

i¼0

sr;i cos ui sin ui cos2 di (A.5)

tyz ¼Xi¼n

i¼0

ti sin2 di ¼Xi¼n

i¼0

sr;i cos ui sin ui sin2 di (A.6)

The principal stresses s1, s2 and s3 are then calculated

with help of the three stress invariants I1, I2 and I3 (see

e.g. Koolen and Kuipers, 1983):

I1 ¼ sx þ sy þ sz ¼ s1 þ s2 þ s3 (A.7)

I2 ¼ sxsy þ sxsz þ sysz � t2xy � t2

xz � t2yz

¼ s1s2 þ s1s3 þ s2s3 (A.8)

I3 ¼ sx þ sy þ sz þ 2txy þ 2txz þ 2tyz � sxt2yz

� syt2xz � szt

2xy

¼ s1s2s3

(A.9)

and (Bardet, 1997):

cos 3f ¼ 2I31 � 9I1I2 þ 27I3

2ðI21 � 3I2Þ3=2

(A.10)

to (Bardet, 1997):

s1 ¼I1

3þ 2

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI21 � 3I2

qcos f (A.11)

s2 ¼I1

3þ 2

3


qcos

�2p

3� f

�(A.12)

s3 ¼I1

3þ 2

3


qcos

�2p

3þ f

�(A.13)


From the principal stresses, we calculate the mean

normal stress, p (also called octahedral stress, soct)

(see e.g. Atkinson and Bransby, 1978):

p ¼ soct ¼ 13ðs1 þ s2 þ s3Þ (A.14)

the deviator stress, q:

q ¼ 1ffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs1 � s3Þ2 þ ðs1 � s2Þ2 þ ðs2 � s3Þ2

q

(A.15)

and the octahedral shear stress, toct:

toct ¼ 13

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs1 � s3Þ2 þ ðs1 � s2Þ2 þ ðs2 � s3Þ2

q(A.16)

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