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SAMARIS
Work Package 5 - Performance based specifications
Selection and evaluation of models for prediction of permanent deformations of unbound granular materials
in road pavements
Draft report
Pierre Hornych Absamad El Abd LCPC – Pavement Materials Divison February 2004
1
1. INTRODUCTION............................................................................................................. 3
2. MECHANICAL BEHAVIOUR OF UNBOUND GRANULAR MATERIALS.......... 3 2.1 PERFORMANCE OF UNBOUND GRANULAR MATERIALS IN PAVEMENTS ......................................................... 3 2.2 CYCLIC BEHAVIOUR OF UNBOUND GRANULAR MATERIALS......................................................................... 4 2.3 RESILIENT BEHAVIOUR ............................................................................................................................... 5
2.3.1. The non-linear elastic Boyce model .................................................................................................. 6 2.3.2. Simplified description using the K-Theta model ............................................................................... 7
2.4 PERMANENT DEFORMATION BEHAVIOUR .................................................................................................... 8 2.4.1. Experimental behaviour .................................................................................................................... 8 2.4.2. Influence of stress level ..................................................................................................................... 8 2.4.3. Influence of material characteristics............................................................................................... 10
3. MODELLING OF PERMANENT DEFORMATIONS OF UNBOUND GRANULAR MATERIALS.................................................................................................. 11
3.1 EMPIRICAL RELATIONSHIPS ...................................................................................................................... 12 3.1.1. Relationships describing the influence of the number of load cycles .............................................. 12 3.1.2. Relationships describing the influence of the level of stress ........................................................... 13
3.2 ELASTO-PLASTIC MODELS......................................................................................................................... 15 3.2.1. The elasto-plastic model of Chazallon ............................................................................................ 15
4. EVALUATION OF SELECTED PERMANENT DEFORMATION MODELS...... 18 4.1 EMPIRICAL APPROACHES .......................................................................................................................... 20
4.1.1. Variation with stress level ............................................................................................................... 20 4.1.2. Variation with the number of load cycles........................................................................................ 23
4.2 ELASTO-PLASTIC MODEL .......................................................................................................................... 26 4.2.1. Prediction of monotonic triaxial tests ............................................................................................. 26 4.2.2. Prediction of cyclic triaxial tests..................................................................................................... 27
5. METHODS OF PREDICTION OF PERMANENT DEFORMATIONS IN PAVEMENT STRUCTURES ............................................................................................... 28
5.1 GENERAL PRINCIPLES AND ASSUMPTIONS................................................................................................. 28 5.2 MODELLING OF RESILIENT BEHAVIOUR AND DETERMINATION OF STRESS PATHS IN GRANULAR PAVEMENT LAYERS.............................................................................................................................................................. 29
5.2.1. Finite element modelling of the resilient behaviour of pavement structures................................... 29 5.2.2. Typical stress paths under moving wheel loads .............................................................................. 30
5.3 PREDICTION OF PERMANENT DEFORMATIONS ........................................................................................... 31 5.3.1. Evaluation of the risk of rutting using a stress criterion................................................................. 32 5.3.2. Simplified layer strain procedures .................................................................................................. 32 5.3.3. Rational rut depth prediction methods ............................................................................................ 33
6. EVALUATION OF A SIMPLIFIED RUT DEPTH PREDICTION METHOD ...... 33
7. 34
7. BIBLIOGRAPHIC REFERENCES.............................................................................. 34
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Selection and evaluation of models for prediction permanent deformations of unbound granular
materials in road pavements
1. INTRODUCTION
The objective of this report is to make a review of existing models for the prediction of permanent deformations of unbound granular materials, in order to select models for the SAMARIS project, which will be used for the prediction of permanent deformations in different experimental pavement structures.
The report presents first some general aspects of the behaviour of unbound granular materials, and makes a review of available models to describe permanent-deformation behaviour. Both simple, empirical relationships and more elaborate incremental, elasto-plastic models are reviewed. Then, several models are selected and evaluated, by comparing their predictions with results of existing repeated load triaxial tests. Based on this evaluation, two models are selected for the modelling work in SAMARIS. Finally, methods for applying these models for calculation of rut depths in pavement structures are discussed.
2. MECHANICAL BEHAVIOUR OF UNBOUND GRANULAR MATERIALS
2.1 performance of unbound granular materials in pavements
Unbound granular materials (UGMs) are continuously graded granular materials, consisting in general of crushed rock particles. They usually contain a certain amount of fines (typically 4% to 10 %) and water (they are generally partially saturated). Such materials are used for mainly for base and subbase layers of low traffic pavements, and also for capping layers.
In Pavements, the performance of unbound granular materials is characterised by the accumulation of permanent deformations, leading to rutting of the pavement. Despite this importance of rutting of unbound granular materials, especially in low traffic pavements, there is presently no well established method to study the permanent deformations of unbound granular materials in the laboratory, and to predict there rutting in the pavement.
In the absence of satisfactory method to predict the rut depth, the design of pavements with unbound granular layers remains, in most design methods, very empirical. They are considered as linear elastic materials, the values of their elastic moduli are often determined on the basis of empirical rules, and the design criterion used for these materials is generally a criterion limiting the maximum vertical elastic strains at the top of the unbound layers and/or at the top of the subgrade, of the form :
bez NA −⋅≤ε (1)
This criterion assumes that rutting is governed by the vertical elastic strains at the top of the unbound material (granular base or subgrade), and is independent of the nature of the material (parameters A and b have a unique value, independent of the material).
3
Thus, today, due to these over-simplified design methods, it is not possible to take fully advantage of the real performance of unbound granular materials in pavements. There is a strong need to improve this situation, and to develop and introduce in current practice : • appropriate mechanical performance tests to determine the resistance to permanent
deformations of unbound granular materials ; • more appropriate models to predict their permanent deformations in pavements.
These new approaches are particularly needed with the increasing use, in pavements, of other sources of materials than crushed rock unbound granular materials, like recycled materials (demolition waste) and various by-products. For these novel materials, the empirical knowledge available for traditional materials, based on simple empirical identification tests (aggregate resistance tests, cleanliness of fines, etc..), and conventional design procedures cannot be employed.
2.2 Cyclic behaviour of unbound granular materials
UGMs are materials consisting of individual grains, with little or no cohesion. Therefore, these materials have very little or no tensile strength and their behaviour depends strongly on the stress state, and also on the grading of the material and the shape of the grains.
The most widely used test to study the mechanical behaviour of unbound granular materials is the cyclic triaxial test. The advantage of this test is the possibility to study the behaviour of the material under cyclic loadings, simulating accurately the in-situ conditions. Cyclic triaxial apparatuses for UGMs are now widely available in Europe, but they differ largely by their characteristics (specimen size, loading capabilities).
The principle of the cyclic triaxial test is recalled on figure 1. In the tests performed at LCPC, and which will be used for this project, the material is submitted to a cyclic vertical stress σ1 and a cyclic horizontal stress σ3 which vary in phase (this type of loading is called variable confining pressure loading (VCP)). But other, simpler test equipment, allow to apply only a cyclic vertical stress, the horizontal stress being held constant (constant confining pressure loadings (CCP). VCP tests present the advantage of simulating more closely the in situ loading conditions.
Figure 2 presents a typical example of response of a granular material in a cyclic triaxial test. The response of the material is essentially elasto-plastic, and it can be observed that : • The permanent strains accumulate rapidly during the first load cycles, but then they tend
to stabilise (at least for stress levels under the critical state), and the response of the material becomes essentially elastic.
• The elastic part of the response of the material is strongly non-linear (stress-dependent).
4
σ3
σ3
q stress
timecycle 1 cycle 2
σ3
q
strain
permanentstrain
resilientstrain
Figure 1. Principle of the repeated load triaxial test
g q p
0
50
100
150
200
250
0 10 20 30 40 50 60 70Axial strain ε1 ( 10- 4 )
Axi
al s
tres
s σ
1 (kP
a)
1 - 5 5000200 20000cycles :
Figure 2. Example of stress-strain cycles obtained in a repeated load triaxial test
on a granular material
The approach generally used to study this complex behaviour consists in studying and modelling separately : • The stable resilient behaviour obtained after a large number of load cycles, which can
be described by non-linear elastic models. Such models begin to be implemented in pavement analysis programs and used for pavement modelling and design
• The accumulation of permanent strains, which is more complex to describe. Development of realistic permanent deformation models (elasto-plastic models) remains a subject for research.
2.3 Resilient behaviour
This review concerns mainly the prediction of permanent deformations. However, as the objective of this work is to develop methods for analysis and design of pavements, it is also necessary to take into account, in our modelling, the resilient (or elastic) part of the behaviour of UGMs.
Modelling of the resilient behaviour of unbound granular materials was largely studied in the European IVth framework project COURAGE [1999]. In this project, a review of available resilient behaviour models was performed, and several, widely used models were evaluated, by comparison with triaxial test results on different unbound granular materials.
5
It was concluded that the non-linear elastic model proposed by Boyce [1980], and modified by Hornych and al. [1998], to take into account anisotropy, describes well the resilient behaviour of unbound granular materials. Therefore, it is proposed to use this model to describe the resilient behaviour of unbound granular materials in the work of SAMARIS. At LCPC, this model has been implemented in the finite element program CESAR-LCPC, and thus can be used for pavement structure calculations. Applications of these finite element calculations are presented in Hornych and al. [2002].
2.3.1. The non-linear elastic Boyce model
This Boyce model is expressed in terms of Bulk Modulus, K, and Shear Modulus, G, with :
v
pKε
= and q3
qGε
= (2) and (3)
with : p : mean normal stress, q : deviator stress;
εv : volumetric strain, ε ε εv = +1 2 3 , εq : shear strain, ( )ε ε εq = −23 1 3
The values of K and G are stress dependent according to the following relationships :
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛β−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
− 2n1
aa p
q1ppKK
and n1
aa p
p.GG−
⎟⎟⎠
⎞⎜⎜⎝
⎛= (4) and (5)
with : ( )a
aG6Kn1−=β (6)
Ka, Ga, n : parameters of model; pa : constant equal to 100 kPa;
To introduce anisotropy in the model, Hornych and al. [1998] proposed to multiply the principal stress, σ1 by a coefficient of anisotropy γ. This leads to the following stress-strain relationships :
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+=ε
−
2
a
a1n
a
n
av *p
*qG6
K1n1p
*pK1* and
*p*q
p*p
G31* 1n
a
n
aq −
=ε (7) and (8)
with : ( ) 32*p 31 σ+γσ= and 31*q σ−γσ=
31v 2* ε+γε=ε and ( )31q 32* ε−γε=ε
In this model, the vertical elastic modulus Ev and the horizontal elastic modulus Eh are linked by the relationship :
2
v
hEE
γ= (9)
In triaxial tests, values of γ are generally lower than 1, which means that the material is stiffer in the vertical direction (Ev > Eh ).
An example of adjustment of the anisotropic Boyce model on results of a repeated load triaxial test on an unbound granular material is shown on figure 3. This figure shows the adjustment of volumetric resilient strains and shear resilient strains (strains during unloading) obtained for different stress paths q/p.
6
Volumetric strains
-15-10-505
10152025303540
0 20 40 60 80 100 120P (kPa)
v (10
-4 )
q/p =1
q/p =2
q/p = 3
q/p = 0
anisotropicmodel
Shear strains
-10
-5
0
5
10
15
20
0 20 40 60 80 100 120
P (kPa)
q (1
0-4 )
q/p = 1
q/p = 2
q/p = 3
q/p = 0
Anisotropicmodel
Figure 3 : example of adjustment of experimental results with the anisotropic Boyce model
2.3.2. Simplified description using the K-Theta model
A proper determination of the parameters of the Boyce model requires repeated load triaxial tests with variable confining pressure, and with measurement of radial strains. However, some triaxial apparatuses do not have these capabilities, and allow to do only constant confining pressure tests, and measure only axial strains. In this case, the most suitable model to describe the test results is the K-Theta model, although this model is less accurate than the Boyce model. The expression of the K-Theta model is :
n
app3KE ⎟⎟
⎠
⎞⎜⎜⎝
⎛= (10)
with E : elastic modulus (also called resilient modulus) K, n model parameters, pa = 100 kPa
7
It should be noted that the K-theta model represents in fact a particular case of the Boyce model, when the parameter β is equal to zero. Therefore, modelling approaches based on the Boyce model, which will be used in SAMARIS, can also use the simpler k-Theta model.
2.4 Permanent deformation behaviour
2.4.1. Experimental behaviour
The most common procedure to study permanent deformations of UGMs using the repeated load triaxial apparatus consists in applying a large number of load cycles (105 and more), with a single level of stress.
Figure 4 shows examples of curves of variation of permanent axial strains with the number of load cycles obtained in such permanent deformation tests. Usually, for stress levels found in granular pavement layers, the permanent deformations increase rapidly during the first few thousand load cycles, and then tend to stabilise (shakedown). For higher stress levels, this stabilisation is not observed, and permanent deformations continue to increase, eventually leading to failure.
0
250
500
750
0 25 50 75 100 125
N CYCLES (103)
DEFORMATION PERMANENTEAXIALE (10-4)
q = 120 kPa
q = 80 kPa
q = 40 kPa
Figure 4 : Evolution of permanent axial strains with the number of load cycles, for different levels of deviatoric stress (after Martinez, 1980).
2.4.2. Influence of stress level
Generally, in repeated load triaxial testing, the applied stresses state is described in terms of mean stress p and deviatoric stress q. Early research on permanent deformations has shown that permanent axial strains increase with increasing deviator stress q, and decrease with increasing mean stress p. This has led to propose relationships, relating the permanent axial strains or shear strains to p and q, such as :
( )2,8
nps p
q.L.Nf ⎟⎟⎠
⎞⎜⎜⎝
⎛=ε (Pappin, 1979) (11)
where : is the permanent shear strain, psε
8
fn(N) is a shape function, depending on the number of cycles N,
L = 2q2p +
Other researchers have tried to relate the accumulation of permanent strains with the shear strength of the material, assuming that the level of permanent strains is a function of the proximity of the applied stresses to the failure line of the material. Barksdale [1972], for example, expressed the permanent axial strain ε1
p as a function of the stress ratio Rf = q/qf, where q is the applied shear stress and qf the shear stress at failure for the same confining stress σ3. This type of approach has been questioned by Lekarp and Isacsson [1998], who argued that failure in granular materials under repeated loading is a gradual process, and not a sudden collapse as in monotonic failure tests, and that there is no direct relationship between the monotonic shear strength, and the accumulated permanent strains.
More recently, Arnold an al. [2002] and Werkmeister and al. [2002], have performed permanent deformation tests with different stress level, and have identified three different types of evolution of permanent strains with the number of load cycles, depending on the level of stress, as shown on figure 5.
• Range A, called the plastic shakedown range, where a complete stabilisation of permanent strains is observed after a finite number of load cycles, and the behaviour becomes entirely resilient (plastic shakedown).
• Range B, called the intermediate range. In this range, the permanent strain rate per cycle either continuously decreases, or becomes constant, but the permanent strains continue to increase, at a very slow rate, without complete stabilisation.
• Range C, called the incremental collapse range. In this range, after a first decrease, the permanent strain rate per cycle starts to increase, leading progressively to failure (very high strains).
Figure 5 : different types of permanent deformation behaviour, depending on stress level. (after Arnold and al., 2002).
Having defined these 3 ranges of behaviour, Arnold and al. have tried to determine the boundaries between these different ranges, in the (p,q) stress space. An example of results, obtained for a granodiorite crushed rock granular material is shown on figure 6. The boundaries are defined as straight lines in the (p,q) space.
9
Figure 6 : Range A and range B behaviour boundaries in the (p,q) stress space for a granodiorite unbound granular material (after Arnold and al., 2002).
Another important factor, for the accumulation of permanent deformations, is the rotation of principal stresses. In the pavement, the material is subject to moving wheel loads, which produce a change in the direction of the principal stresses. This rotation is not simulated in cyclic triaxial tests. However, research using a hollow cylinder triaxial apparatus (Chan 1990), has shown that permanent strains increase significantly when a rotation of the principal stresses is applied (for the same principal stress amplitude). Similar observations have been made by Hornych and al. [2000], in an accelerated loading experiment on a low traffic pavement. They compared the effect of a moving wheel load, and of a cyclic load applied on a fixed circular plate (with the same vertical stress level and load duration), and found that the moving wheel load produced about two times higher permanent deformations than the cyclic plate load.
2.4.3. Influence of material characteristics
Research performed in France, on a large variety of unbound granular materials, has shown that the following parameters have a significant influence on the resistance to permanent strains :
• The water content : Permanent deformations increase significantly when the water content increases, particularly at high water contents (around or above optimum water content). This moisture sensitivity is strongly related to the fines content of the material. Figure 7 shows an example of the influence of water content on the characteristic permanent axial strain A1c (obtained after a standard loading procedure) , for three crushed rock granular material of different origin (hard limestone, soft limestone and micro-granite). For all three materials, there seems to be a critical water content, above which the permanent strains start to increase rapidly.
10
Permanentstrain
A1c(10-4)
0
100
200
300
400
500
0-1-2-3-4w - wOPM (%)
Measured water content in granular base layer
Hard limestone
Soft limestone
Microgranite
Figure 7 : Sensitivity of permanent axial strains to water content, for 3 different unbound granular materials (after Gidel and al., 2001)
• The density : Clearly, an increase of the degree of compaction of the material reduces the void ratio, and increases the number of contacts between the grains, thus increasing the resistance to permanent deformations. More generally, the resistance to permanent deformations strongly depends on the void content. For the same degree of compaction, a modification of the grading, reducing the void content, increases the resistance to permanent deformations.
• The mineralogical nature of the material : The mineralogy influences the shape of the particles, their surface roughness, and also the quality of the fine fraction and its sensitivity to water, and this can lead to significant differences in behaviour, for the same grading.
3. MODELLING OF PERMANENT DEFORMATIONS OF UNBOUND GRANULAR MATERIALS
The resilient response of unbound granular materials has been widely studied, and is well described by the non-linear elastic models presented previously. Modelling of permanent deformations is less advanced, for several reasons : • The experimental study of permanent deformations Is much more time-consuming than
the study of the resilient behaviour. Whereas the resilient behaviour of a material can be determined using a single triaxial test (with different stress paths), the study of permanent deformations requires an important number of tests (in each test, only one stress path is generally applied, to avoid influence of previous stress-history), with large numbers of load cycles (105 or more). In addition, results of permanent deformation tests present a much larger experimental scatter than results of resilient behaviour tests.
11
• Permanent deformation characteristics are often considered to be less important than resilient behaviour : resilient strains are directly related to fatigue cracking of the bituminous layers, which is almost impossible to remedy once it has occurred, whereas permanent deformations only create progressive deformations of the surface, and can be remedied by overlaying the pavement
• Finally, permanent deformation models are more difficult to apply to pavement structure calculations than resilient behaviour. Modelling of rutting requires the simulation of large numbers of load cycles, with varying environmental conditions (temperature, moisture content , different loads).
Existing permanent deformation models are mostly based on repeated load triaxial testing, and are of two main types : • Empirical laws, describing the variation of permanent strains with the number of load
cycles and the maximum applied stresses. • Incremental models, generally based on the theory of elasto-plasticity .
3.1 Empirical relationships
3.1.1. Relationships describing the influence of the number of load cycles
One of the first interpretation applied to results of permanent deformation tests in the triaxial consisted in describing the variation of permanent deformations (generally only axial deformations) with the number of load cycles N. A summary of various relationships of this type, proposed by different authors, is presented in table 1. Table 1 : Relationships describing the variation of permanent axial deformations with the number of load cycles
Author relationship Parameters
Barksdale [1972] ( )Nlog b a p1 +=ε
a, b
Khedr [1985] b-p1 N . A N
=ε
A,b
Paute and al [1988] DN
N A *p1 +
=ε
*p1ε permanent deformation after
the first 100 cycles A parameter function of stress level, D
Sweere [1990] bp1 N a =ε
a, b
Hornych and al. [1993] ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛=
−B*p
1 100N-1A ε
*p1ε permanent axial strain after
the first 100 cycles A, B
Vuong [1994] cr1
p1 N
ba ⎟
⎠⎞
⎜⎝⎛ε=ε
r1ε resilient axial strain
a, b, c
12
Wolff and Visser [1994]
( )( )N -bp1 e-1a N c +=ε
a, b, c
Huurman [1997] ( ) )1e.(C)1000
N.(AN 1000NDBP
1 −+=ε
A, B, C, D parameters function of the level of stress
The first well known relationship describing the variation of permanent axial deformations ε1p with the number of load cycles is that proposed by Barksdale [1972], who found a linear increase of ε1p with the logarithm of the number of load cycles N :
( )Nlog b a p1 +=ε (11)
Later, other researchers (Khedr [1985], Sweere [1990], Vuong [1994]) found similar results, indicating a linear relationship between log(ε1p) and log(N), leading to relationships like that proposed by Sweere [1990] :
(12) bp
1 N a =ε
Hornych and al. [1993], who tested three French unbound granular materials, found that for typical stress levels found in pavement foundations, permanent axial strains stabilise after a large number of load cycles (about 105). They obtained good predictions with the following relationship, which assumes that ε1p has a finite limit for an infinite number of cycles :
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛=ε
−B*p
1 100N-1A (13)
3.1.2. Relationships describing the influence of the level of stress
The relationships describing the variation of permanent deformations with the number of load cycles presented above cannot be applied to the prediction of permanent deformations in pavement structures, because they do not take into account the applied stresses.
Other researchers have followed another approach, trying to relate the permanent deformation after a given number of cycles to the applied stresses (generally the maximum stresses ). Relationships of this type are listed in table 2. Some of these relationships also try to couple the effects of both stresses and number of load cycles, but they are only very few of them (see table 2).
Table 2 : Relationships describing the variation of permanent axial deformations with applied stresses
Author relationship Parameters
Lashine et al. [1971] 3
p1
q a σ
=ε a, σ3 confining stress, q deviator stress
13
Barksdale [1972] ( )( )⎥⎦
⎤⎢⎣
⎡φσ+φ
φ−−
σ=ε
sinC.cos2sin1 . .qR1
aq
3
f
b3p
1
a, b , Rf = ratio of applied deviator stress q to deviator stress at failure qf , φ friction angle, C cohesion
Shenton [1974] a
3
maxp1
qK ⎟⎟⎠
⎞⎜⎜⎝
⎛σ
=ε
K ,a , qmax maximum applied deviatoric stress
Pappin [1979] ( )2,8
nps p
q.L.Nf ⎟⎟⎠
⎞⎜⎜⎝
⎛=ε
psε permanent shear strain,
fn(N) shape function, depending on the number of cycles N, p mean normal stress,
L = 2q2p +
Lentz et Baladi [1981]
( ) ( )( ) ( )Nln
Sqm-1Sqn Sq-1ln 15,0
0,95Sp1 ⎥
⎦
⎤⎢⎣
⎡+ε=ε −
S 95,0ε axial strain at 95% of the deviatoric stress at failure,m slope of the failure line,S deviatoric stress at failure
Paute et al. [1994] ( ) ( )
( )⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
+=ε
*ppqmb
*ppq
Nf *p1
*p1ε permanent axial strain after
the first 100 cycles b,p*, m slope of the failure line in p,q space f(N) function of the number of cycles N
Nishi [1994] b
ap
ult,1 pq k =ε
a, b ultimate permanent axial strain
pult,1ε
Lekarp et al. [1998]
( )( )
b
0
refp1
pq a
pLN
⎟⎟⎠
⎞⎜⎜⎝
⎛=
ε
a,b
( ) Nrefp1ε : permanent strain after
a reference number of cycles Nref
L = 2q2p +
p0 reference mean stress
Several relationships are very simple, and are expressed as a power relationship between
ε1p and the stress ratio q/p or q/σ3 (for example b
p1 p
qK ⎟⎟⎠
⎞⎜⎜⎝
⎛=ε or b
ap1 p
qK =ε )
Recently, Gidel and al. [2001], performed tests on two granular materials, and tried to model variation of permanent axial strains ε1
p with stress level and number of load cycles. By describing separately the variation of ε1
p with the number of cycles N and the maximum cyclic stresses pmax and qmax, he arrived to the following expression :
14
)pq
psm(
1p
LNN1)N(
maxmax
max
n
amax
B
0
p01
p1
−+⋅⎥
⎦
⎤⎢⎣
⎡⋅
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=
−εε
with : 2maxq2
maxpmaxL += , pa = 100 kPa,
ε1p0, B, n, m, s parameters of the model.
3.2 Elasto-plastic models
In the field of soil mechanics, various elasto-plastic models simulating accurately the monotonic and cyclic behaviour of soils and granular materials have been developed. The plasticity models separate the strains into elastic and plastic parts : pe ε+ε=ε where ε is the
total strain, eε is the elastic strain and pε is the plastic strain. The models link the stress tensor to the strain tensor with the incremental equation: ε=σ dHd , where H is a fourth order tensor. This equations system is solved by incremental calculations. The major advantage of this incremental approach is that it is possible to describe the response to any type of loading history (whereas the empirical models describe only the response to a cyclic loading of constant stress amplitude).
The modelling of cyclic behaviour requires elaborate elasto-plastic models, which can generate plastic strains during loading and also during unloading, like models with kinematic hardening. Various models of this type exist in soil mechanics, but generally, the available models are used to describe relatively low numbers of load cycles (less than 100 cycles), and would need to be adapted to pavement loading conditions, where the number of loads is much larger (typically 105 to 107). Another characteristic of these models is that they generally have much more parameters than the empirical relationships presented in §3.1, which need to be determined using several different types of tests (monotonic and cyclic tests, with different stress paths).
Few authors have developed specific elasto-plastic models for pavement applications. Recent developments in this field have been proposed by Bonaquist and Witczak (1997), Hicher and al. (1999) and Chazallon (2000).
3.2.1. The elasto-plastic model of Chazallon
The model developed by Chazallon (2000, 2002) is based on the work of Hujeux (1985). It is an elasto-plastic cyclic model with kinematic hardening, applicable to large numbers of load cycles.
The model uses the yield function and plastic potential of the non-associated model of Hujeux (Hujeux 1985) in its simplest formulation. The formulation used by the author and proposed by Hujeux is a one-mechanism model used for monotonic loading of sands based on the critical state concept (Schoffield et al 1968). The model has been first presented in (Chazallon 2000), and some modifications have been added. To take into account the unsaturated state of the material, leading to a macroscopic cohesion, a constitutive parameter C0 has been added. It appears in the expression of the yield surface, plastic potential, the kinetic laws and the elasticity law, by adding C0 to the mean stress p .
The elastic part of the behaviour is considered non-linear and is described by the anisotropic Boyce model [Hornych and al., 1998] described previously. In the expression of this model, the mean stress p is replaced by :
15
031 C
32
p ++
=σγσ(
where : C0 is the cohesion due to the unsaturated state of the material
The yield function f is written:
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−+−⎟
⎠⎞
⎜⎝⎛=
*c
11cII
2/1
p3
)X~~(Iln3
)X~~(IMbr)X~~(S
23f σσ
σ (14)
where I1 is the Trace operator, SII is the deviatoric stress operator, b is a parameter which controls the shape of the yield surface in the (p,q) plane. M is the slope of the critical state line in the (p,q) plane. pc
* is the critical pressure corresponding to the actual void ratio.
The hardening is given by:
)exp(pp pv0cc εβ= (15)
where εvp is the volumetric plastic strain. pc0 and β are constant which determine the position
of the critical state line in the (e,ln(p)) plane. pc0 is the initial critical pressure corresponding to the initial critical void ratio ec0. rc is a hardening variable associated to the deviatoric plastic strain.
elpd
pd
c ra
r ++
=ε
ε (16)
The kinetic of the hardening variable rc is:
)X~~(aMI/)r1(3drd 12
cc −−= σλ (17)
Initially, under monotonic loading rc = rmel , where rm
el represents the initial elastic domain, (rc ≤ 1) and a = am governs the evolution of the hardening variable rc. When unloading occurs, rc = rc
el represents the initial elastic domain and a = ac governs the evolution of the hardening variable rc. dλ is the plastic multiplicator, and it can be determined by the consistency condition df = 0 and dλ ≥ 0. The former equations are completed by the definition of a non-associated plastic potential g and the following kinetic:
σσλε ~
)X~~(gd~d p∂
−∂= (18)
where : ⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+−−⎟
⎠⎞
⎜⎝⎛=
*c
11II
2/1
p3
)X~~(Iln)X~~(MI/)X~~(S2
27g σσσ (19)
X~ is the tensorial kinematic hardening variable. Its kinetic is given by
and (20) σµ ~dX~d = 10 ≤µ≤
If one follows a stress path AB from A to B, and one unloads from B to C, X~ is the stress of the origin of the new stress plane (p,q) during unloading.
16
and I~pP~X~ *clc/ucB += σ 1P0 lc/uc ≤≤ (21)
Where I~ is the second order identity tensor, Puc and Plc are two parameters which take into account the position of the yield surface and the plastic potential during unloading (subscript uc) or reloading (subscript lc).
Figure 8 shows the yield surface of the model, and its evolution during a cyclic loading between points A and B in the (p,q) stress space. During the first loading from A to B, the origin of the surface is at point O, and the size of the surface grows during loading (isotropic hardening). When unloading starts, from point B, the origin of the surface moves to point O2, and then the position of the surface continues to change, during unloading and subsequent reloading (isotropic and kinematic hardening).
p (unloading)
M
O Pc
O2
A
B
p (loading)
q (loading)
q (unloading)
Yield surface at point AYield surface at point B, when unloading starts
New origin ofyield surface
Figure 8 : yield surface of the model of Chazallon , and hardening mechanism.
The parameters of the model are summarised in table 3. They can be divided into 4 non-linear elastic parameters, 7 plastic parameters describing monotonic loading and 5 plastic parameters describing cyclic loading; these three groups of parameters are uncoupled, and can be determined separately. Table 3 : parameters of the elasto-plastic model of Chazallon
Non-linear elastic parameters
Monotonic loading parameters Cyclic loading parameters
n Ka MPa
Ga MPa
γ C0
kPa M
0cp
MPa
β ma
b elmr ucP
lcP
elcr ac µ
The elastic parameters (Ka ,Ga , n) and the anisotropic parameter γ are determined using the results of cyclic triaxial tests used to determine the resilient behaviour. These tests include a
17
cyclic conditioning, which aim is to stabilise the permanent deformations, and then cyclic loadings with various (q/p) ratios.
The monotonic loading parameters are determined using monotonic triaxial shear tests till failure at different initial confining pressures (for pavement applications, low confining pressures of 0 kPa, 10 kPa and 20 kPa are used). C0 is the cohesion. M is the slope of the critical state line in p,q space, separating contractant ad dilatant behaviour. In the Hujeux model, the initial critical pressure pc0 and the plastic compressibility modulus β require, for their determination, isotropic compression tests performed till at least 20 MPa, which can not be performed using triaxial cells for unbound granular materials for roads. Alternatively, their values can be estimated using correlations (Rahma 1992). The hardening parameters am and b are determined by fitting the σ1 - ε1 stress strain curves of the monotonic triaxial tests.
The cyclic parameters (Puc , Plc , rcel , ac, µ) are determined using cyclic triaxial tests with
large numbers of load cycles. The model assumes that the plastic strains tend to stabilise (elastic shakedown or plastic shakedown). The cyclic plastic parameter rc
el is determined from the level of axial plastic strain obtained at stabilisation in the (N, ε1
p) plane. The parameters Puc and Plc determine how rapidly the shakedown state will be attained.
4. EVALUATION OF SELECTED PERMANENT DEFORMATION MODELS
In SAMARIS, the objective is to propose two approaches for prediction of permanent deformations . One “routine level” approach, based on empirical relationships, relating permanent deformations to the applied stresses and number of load cycles, and one more advanced approach, based on elasto-plastic modelling. As a first step, several different models have been selected from the literature study and evaluated, by comparison with results of existing repeated load triaxial tests.
For the evaluation, it was decided to use two series of cyclic triaxial tests, performed on a 0/20 mm granular material (crushed microgranite), and a 0/4 mm sand, containing approximately 8 % of fines.
The microgranite specimens were compacted at a density equal to 97 % of the Modified Proctor Optimum density, and a water content 2 % below optimum. The test program is summarised in table 4. Five tests were performed, each following a different stress path q/p, and each test included several loading stages, with increasing stress levels. The stress levels are representative of those found in the granular base layer of a low traffic pavement.
Table 4 : program of permanent deformation tests on the microgranite
Test Loading sequence
Number of cycles N
pmax (kPa)
qmax (kPa)
q/p ε1p
(10-4) 02-256-1 1 10000 250 250 1 26,6 2 10000 400 400 1 47,8 3 10000 500 500 1 67,2 02-263-1 1 10000 166 250 1.5 43,3 2 10000 266 400 1.5 71,4 3 10000 333 500 1.5 85,6
18
4 10000 400 600 1.5 107,5 02-269-1 1 10000 125 250 2 51,4 2 10000 200 400 2 94,2 3 10000 250 500 2 127,5 4 10000 300 600 2 164,6 02-284-1 1 10000 60 150 2,5 53,5 2 10000 100 250 2,5 170,6 3 10000 166 400 2,5 417,8 4 10000 200 500 2,5 506,0 02-277-1 1 10000 33 100 3 40,1 2 10000 66 200 3 82,7 3 10000 100 300 3 185,9 4 10000 133 400 3 366,5
The tests on the sand were performed on specimens compacted at the normal Proctor optimum density, and at the optimum water content (w = 11 %). Four tests were performed, using a test procedure similar to that used for the microgranite. The applied stress levels and the permanent strains obtained for each load sequence are presented in table 5. Here, the stress levels are representative of values found in a capping layer, or a subgrade.
Table 5 : program of permanent deformation tests on the sand
Test Loading sequence
Number of cycles N
pmax (kPa)
qmax (kPa)
q/p ε1p
(10-4) GSM13 1 10000 20 20 1 2 10000 40 0 1 3 10000 60 60 1 4 10000 80 80 1 5 10000 120 120 1 6 10000 180 180 1 GSM14 1 10000 13.3 20 1.5 2 10000 26.6 40 1.5 3 10000 40 60 1.5 4 10000 53.3 80 1.5 5 10000 66.6 100 1.5 6 10000 80 120 1.5 7 10000 93.3 140 1.5 GSM11 1 10000 10 20 2 2 10000 20 40 2 3 10000 30 60 2 4 10000 40 80 2 5 10000 50 100 2 6 10000 60 120 2 7 10000 70 140 2 GSM12 1 10000 6.6 20 3
19
2 10000 13.3 40 3 3 10000 20 60 3 4 10000 23.3 70 3
4.1 Empirical approaches
The idea is to propose, on the basis of the literature review, a relationship describing the variation of permanent axial strains with both stress level and number of load cycles, using an expression of the form :
( ) ( ) ( )maxmaxp1 q,pgNfN ⋅=ε
where : f(N) is a function of the number of load cycles, N; g(pmax,qmax) is a function of the maximum mean stress pmax and maximum deviatoric stress qmax.
The two functions f and g will be selected from those presented above.
4.1.1. Variation with stress level
To describe the variation of permanent deformations with stress level, the three relationships proposed by Nishi [1994], Lekarp and al. [1998], and Gidel and al. [2001] have been evaluated. The relationships, and the parameter values obtained for each tested material, are shown in table 6.
Table 6 : empirical permanent deformation relationships selected for evaluation, and parameters obtained for the two materials.
Relationship UGM Sand
ε1p = ε1p0. nmax )
paL
( . 1
(max
max
maxm
sp
qp
+ - ) (22)
Gidel (2001)
ε1p0 = 29,32
n = 0,62 m = 2,41 s = 46,82 R = 0,83
ε1p0 = 31,45
n = 0,36 m = 1,60 s = 26,21 R = 0,80
ε1p = ε1p0.
bmax
amax
p
q (23)
Nishi (1994)
ε1p0 = 0,0012
a = 5,55 b = 4,14 R = 0,69
ε1p0 = 0,18
a = 3,30 b = 2,17 R = 0,76
ε1p = ε1p0. b
maxmaxmax )
pq
.(pa
L (24)
Lekarp (1998)
ε1p0 = 1,83
b = 3,76 R = 0,64
ε1p0 = 23,78
b = 2,42 R = 0,76
R : correlation index
20
The relationships were adjusted on the final strain values obtained at the end of each loading stage, using a least squares method. The coefficient R is a “correlation index”, indicating the quality of the adjustment. It is defined by :
( )∑
∑
−
−
−=
i
2i
i
2ii
yy
))x(fy(1R 1≤R
where : yi are the experimental values obtained for each stress level, f(xi) are the values predicted by the model, and y is the average of the experimental values yi .
As can be seen in table 6, the relationship proposed by Gidel gives the best results, for both materials. The predictions obtained with this relationship are shown on figures 9 and 10. These figures compare the experimental strains obtained at the end of each loading stage (for each of the tests, with different values of q/p), with the predicted values.
This relationship will be retained for the rest of the work.
0
50
100
150
200
250
300
350
400
450
0 100 200 300 400 500 600
p (kPa)
e1p
(10-
4)
mesuresmodèle
q/p=3
q/p=2,5
q/p=2
q/p=1,5
q/p=1
Figure 9 : Predictions obtained with the model proposed by Gidel and al., for the 0/20 mm UGM
21
0
50
100
150
200
250
300
0 20 40 60 80 100 120 140
p (kPa)
ε1p
(10-4
)
mesuresmodèle
q/p=1
q/p=1,5
q/p=2
q/p=3
Figure 10 : Predictions obtained with the model proposed by Gidel and al., for the sand.
Similarly, figures 11 and 12 show the results obtained with the model proposed by Nishi (the second, in terms of values of correlation index). The predictions remain acceptable for the sand, but not for the UGM.
0
50
100
150
200
250
300
350
400
450
0 100 200 300 400 500 600
p (kPa)
e1p
(10-
4)
mesuresmodèle
q/p=1
q/p=1,5
q/p=2
q/p=2,5
q/p=3
Figure 11: Predictions obtained with the model proposed by Nishi for the 0/20 mm UGM.
22
0
50
100
150
200
250
300
0 20 40 60 80 100 120 140
p (kPa)
ε1p
(10-4
)
mesuresmodèle
q/p=1
q/p=1,5
q/p=2
q/p=3
Figure 12 : Predictions obtained with the model proposed by Nishi for the sand.
4.1.2. Variation with the number of load cycles
For describing the variation of axial permanent strains with the number of load cycles, two relationships have been tested: the one proposed by Sweere [1990], which is very widely used, and the one proposed by Hornych and al. [1993] (see § 3.1.1). The main difference is that the relationship of Hornych predicts a stabilisation of permanent strains (finite limit for N infinite), contrary to that of Sweere.
The two relationships, and the parameter values and correlation indexes obtained with them, for the two materials, are presented in table 7. As previously, the adjustments were performed using a least squares method. The two functions give relatively similar correlations indexes R.
Table 7 : Parameters obtained for the two selected functions of the number of load cycles, and correlation indexes
Relationship UGM Sand f(N) = A.[1-(N/N0)-B] (N >N0=1) Hornych (1993) (25)
A = 2,67 B = 0,04 R = 0,66
A = 1,004 B = 0,60 R = 0,63
f(N) = A.NB
Sweere (1990) (26)
A = 0,198 B = 0,175 R = 0,62
A = 0,773 B = 0,028 R = 0,66
R : correlation index
23
Finally, figures 13 and 14 show the results obtained for prediction of the experimental results on the microgranite, using the complete relationship, , with the function g proposed by Gidel (selected in 4.1.1), and the two functions f(N) selected here (Hornych, and Sweere). For the microgranite, the evolution with the number of load cycles seems better predicted by the function f(N) of Sweere.
( ) ( ) ( )maxmax1 q,pgNfN ⋅=ε p
0
50
100
150
200
250
300
350
400
450
0 5000 10000 15000 20000 25000 30000 35000 40000 45000
Nombre de cycles (-)
e1p(
10-4
)
mesuremodèle
q/p=1
q/p=1,5
q/p=2
q/p=2,5
q/p=3
Figure 13 : Microgranite - Predictions obtained with the function of variation with the number of load cycles f(N) proposed by Hornych
0
100
200
300
400
500
600
0 5000 10000 15000 20000 25000 30000 35000 40000 45000
Nombre de cycles (-)
e1p
(10-
4)
mesuremodèle
q/p=3
q/p=2,5
q/p=2
q/p=1,5
q/p=1
Figure 14 : Microgranite - Predictions obtained with the function of variation with the number of load cycles f(N) proposed by Sweere.
24
Similarly, figures 15 and 16 show the results obtained for prediction of the experimental results on the sand, with the two different f(N) functions. The two functions give similar results, and predict the experimental results relatively well, except for the highest q/p ratio (q/p = 3)/.
0
50
100
150
200
250
300
0 10000 20000 30000 40000 50000 60000 70000
Nombre de cycles (-)
e1p(
10-4
)
mesuremodèle
q/p=1
q/p=1,5
q/p=2
q/p=3
Figure 15 : Sand - Predictions obtained with the function of variation with the number of load cycles f(N) proposed by Hornych.
0
50
100
150
200
250
300
0 10000 20000 30000 40000 50000 60000 70000
Nombre de cycles (-)
e1p(
10-4
)
mesuremodèle
q/p=3
q/p=2
q/p=1,5
q/p=1
Figure 16 : Sand - Predictions obtained with the function of variation with the number of load cycles f(N) proposed by Sweere
25
4.2 Elasto-plastic model
As for the empirical relationships, it was also decided to evaluate the elasto-plastic model, by comparison with experimental results; However, for this model, the previous tests could not be used, because the determination of the model parameters requires not only cyclic load triaxial tests, but also monotonic triaxial shear tests. So, in connection with another research project, a specific test program was set up, in order to evaluate the elasto-plastic model. These tests were performed on a 0/10 mm crushed gneiss material, at a density of 97 % of the Modified Proctor Optimum density, and at a water content of 6 % (close to optimum).
4.2.1. Prediction of monotonic triaxial tests
4 monotonic triaxial tests were performed on the material, with confining pressures of 0, 10, 20 and 50 kPa, in drained conditions (the material is unsaturated). These tests allow to determine the monotonic loading parameters of the model.
The values of the monotonic parameters are given in table 8. The values of the parameters Pc
0 and β, impossible to determine with the test equipment used, have been assumed identical to those used by Chazallon and al.[2002]. The predictions obtained with the model for the axial stress – axial strain (σ1 = f(ε1) ) monotonic loading curves are shown on figure17. The model predicts quite well the experimental results, for the different confining pressures. Table 8 : values of the elasto-plastic model parameters obtained for the gneiss
Non-linear elastic parameters
Monotonic loading parameters Cyclic loading parameters
n Ka
MPa Ga
MPa γ C0
kPa M
0cpMPa
β ma
b elmr ucP
lcP
elcr ac µ
0.125 6.47 25.8 0.625 14 2.46 10 400 6.10-3 0.188 0.112 3.10-3 1,7.10-3 0.01 2.10-5 0.5
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
1000000
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
eps1
q (P
a) mes
modèle
Pcf = 50 kPa
Pcf = 20 kPa
Pcf = 10 kPa
26
Figure 17 : prediction of monotonic triaxial shear tests with different confining pressures using the elasto-plastic model.
4.2.2. Prediction of cyclic triaxial tests
Four cyclic triaxial tests were also performed on the crushed gneiss material, with a procedure similar to those used for the other materials. Three tests were performed to determine the permanent deformation behaviour; each test included several load levels, with the same stress ratio q/p, but with increasing stress amplitudes. The applied stress paths are summarised in table 9. These tests allowed to determine the cyclic loading parameters, which values are given in table 8. A last cyclic triaxial test was used to determine the non-linear elastic parameters (see table 8). This test included a cyclic conditioning, followed by short load sequences (100 cycles) following different stress paths. Table 9 : program of permanent deformation tests on the crushed gneiss
Test Loading sequence
Number of cycles N
pmax (kPa)
qmax (kPa)
q/p
1 80000 600 60 1 2 80000 100 100 1
1
3 80000 150 150 1 1 80000 15 30 2 2 80000 30 60 2 3 80000 50 100 2
2
4 80000 75 150 2 1 80000 24 60 2.5 3 2 80000 40 100 2.5
An example of prediction of axial strains ε1p, for the permanent deformation test with a stress
ratio q/p = 2, is presented on figure 18. With the selected parameter values, the model predicts better the behaviour for the lower stress levels (where the permanent strains stabilise more rapidly). However, due to the large number of parameters of the model, the adjustment procedure is relatively complex. The study will have to be continued, to see if a better adjustment of the results can be obtained for the highest stress levels.
27
0
10
20
30
40
50
60
70
80
90
100
0 50000 100000 150000 200000 250000 300000 350000
Nb cycles
ε1p
(10-4
)
mesure
modèle
Figure 18 : Prediction of a cyclic permanent deformation test with the elasto-plastic model.
5. METHODS OF PREDICTION OF PERMANENT DEFORMATIONS IN PAVEMENT STRUCTURES
5.1 General assumptions
The work performed in this task WP 5.2 of SAMARIS deals only with the permanent deformations of unbound pavement layers. Therefore, in this report, only the prediction of the permanent deformations of unbound granular pavement layers is considered, and no permanent deformations are supposed to occur in the other pavement layers (bituminous layers and subgrade. Parallel work on modelling of bituminous materials is carried out in task WP 5.3 of the project. At a later stage, the possibility to associate the modelling of rutting of both bituminous and unbound materials will be examined.
For the prediction of permanent deformations of unbound pavement layers, several general assumptions will be made concerning the cyclic behaviour of UGMs, based on the experimental observations.
1. As in all the report, it will be assumed that the behaviour of unbound granular materials is elasto-plastic and strains will be divided into an elastic part and a plastic part :
pe ε+ε=ε
2. The second assumption is that for one load cycle, the increment of permanent strains is very small, compared with the elastic strains :
ep ε<<εδ
Experimentally, this is verified except for the very first load cycles (less than 100 cycles).
28
3. The third assumption is that the elastic properties of the granular materials are independent of the number of load cycles. In triaxial tests, this is also well verified, after the first few hundred cycles (where an increase of elastic modulus is generally observed).
With these assumptions, the prediction of permanent deformations in the pavement layer can be performed in two independent steps :
• Calculation of the stress field in the pavement, considering only the resilient behaviour.
• Use of the resilient stresses to calculate the permanent strains in the granular layers, and then the resulting displacements.
Possible approaches for the calculation of the stress field in the pavement, and of the permanent strains and displacements are presented and discussed below.
5.2 Modelling of resilient behaviour and determination of stress paths in granular pavement layers
5.2.1. Finite element modelling of the resilient behaviour of pavement structures
The determination of the stress field in the pavement, using suitable models for the behaviour of the pavement materials, and realistic material properties, is an essential step for the accurate prediction of permanent strains.
Although most current pavement design methods are based on linear elastic calculations, it is well known that these calculations are not satisfactory for low traffic pavements, where the granular layers represent the main structural element; In such pavements, linear elasticity often leads to unrealistic stress states in the granular layers, in particular tensile stresses at the bottom of the unbound granular layers, which cannot be sustained by these materials.
At LCPC, an important research program has been carried out in the past few years, to study the resilient behaviour of unbound granular materials, in connection with the European research projects COST 337 and COURAGE. This work led to the choice of the non linear elastic model of Boyce [1980], modified to take into account anisotropy, to describe the resilient behaviour of unbound granular materials (see part 2.3 of the report). It has also led to the development of a new module of the finite element program CESAR-LCPC, dedicated to the modelling of the resilient behaviour of pavements under moving wheel loads (Heck and al, [1998], Hornych and al. [1998], Heck [2001]). This module, called CVCR includes several specific models for pavement materials :
• Two non linear elastic models for unbound granular materials (and also granular soils): the modified Boyce model, and the k-theta model (also described in part 2.3).
• The viscoelastic model of Huet-Sayegh for bituminous materials (Huet [1963], Sayegh [1965]), which gives very good predictions of complex modulus measurements.
The module CVCR has been validated by comparisons with results of several experiments on low traffic pavements, with granular bases (Courage [1999], Hornych and al. [2000, 2002]). These validations indicated that CVCR leads to a much better prediction of the non linear, load-dependent response of these thinly surfaced pavements, and much more realistic stress distributions in the granular layers than classical linear elastic calculations.
Therefore, the module CVCR will be used in Samaris for the calculation of stress fields in pavement structures. In addition, because the objective is to determine only the stress fields in the granular layers, a linear elastic behaviour will be assumed in the calculations for
29
bituminous layers and for the subgrade soil, and only the UGM layers will be considered non linear elastic.
5.2.2. Typical stress paths under moving wheel loads
In our study, it is assumed that the properties of the pavement are constant along the Ox axis (direction of displacement of the vehicles), and that the speed of the vehicles is constant. With these assumptions, the permanent deformations need to be calculated only in the plane (0,y,z) perpendicular to the Ox direction, due to symmetry. At a point M, of coordinates (0, y, z) in this plane, the passing of one vehicle produces a loading history, or stress path σΜ(t). For the sake of simplicity, and because our approach does not take into account principal stress rotation, this stress path will be described only by the variations of the mean stress pM(t) and of the deviatoric stress qM(t).
In our case, where all the materials are elastic, the determination of the stress paths from the results of the CVCR finite element calculations is particularly simple. In the referential of the moving load, the stress path at M (0, y, z) corresponds to the variation of the stresses along the x axis, pM(x) and qM(x), for constant values of y and z.
In order to illustrate the stress paths obtained in granular pavement layers, the approach described above was applied to the modelling of two typical low traffic pavements, consisting of :
• a bituminous wearing course, with two different thicknesses : 4 cm and 8 cm ;
• a granular base and subbase, with a total thickness of 40 cm ;
• a subgrade soil, with a total thickness of 230 cm.
The CVCR calculations were performed in 3D, with a mesh comprising 2196 elements (describing 1/4 of the structure, due to symmetry), and a load consisting of 2 twinned wheels, with a total load of 65 kN (see figure 19).
In the calculations, the soil and the bituminous concrete were considered linear elastic, and the granular material was described using the Boyce model with parameter values corresponding to a 0/20 mm crushed rock material (crushed gneiss), of medium performance.
30
Applied twin wheel load
34 cm
23 cm
Contact pressure : 415 kPa
Figure 19 : Finite element mesh and load used for the determination of the stress paths in unbound granular pavement layers (calculations with the finite element module CVCR).
Figure 20 shows the stress paths obtained under the centre of one wheel and at different depths z, for the two pavement structures. In the (p,q) stress space, these stress paths are practically linear, with values of slopes q/p between approximately 2 and 2.5. Similar results are also obtained at other points in the granular layer. Thus, The stress paths in the granular layers are very close to those applied in the repeated load triaxial tests.
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0 0,05 0,1 0,15p (MPa)
q (M
Pa)
z=4cmz=14cmz=24cmz=34cmz=44cm
0
20
40
60
80
100
120
140
160
0 20 40p (kPa)
q (k
Pa)
60
z = -0.08mz = -0.18mz = -0.28mz = -0.38mz = -0.48m
Asphalt thickness 4 cm Asphalt thickness 8 cm
Figure 20 : stress paths q = f(p) produced by moving wheel loads in an unbound granular pavement layer, at different depths z, for two pavement structures with asphalt covers of 4 cm and 8 cm.
5.3 Prediction of permanent deformations
A literature study has been performed, to identify the possible methods for predicting rut depths in pavement structures. Three main types of methods have been found :
• The first and simplest method consists in defining a design criterion, in terms of stress level, leading to a given level of permanent deformations.
• A second, simplified approach, consist in calculating the permanent strains using a “layer strain” approach. This consists in dividing the granular layer in a number of horizontal sub-layers, and calculating the permanent strains on the basis of the maximum stresses
31
in each sub-layer. Such methods are generally based on empirical permanent deformation models, such as those described in part 3.1 of the report
• Finally, the most elaborate methods are based on a structural calculation of the permanent strains in the pavement (or granular layer), using incremental permanent strain models.
5.3.1. Evaluation of the risk of rutting using a stress criterion
One approach of this type is presented in the work of Arnold and al. [2002] and Werkmeister and al. [2002]. As discussed in 2.4.2, these researchers have define 3 ranges of permanent strain behaviour under cyclic loading, depending on whether permanent strains stabilise (range A), increase indefinitely, but at a very slow rate (Range B), or increase rapidly to failure (range C). From these results, they have concluded that in a pavement, only range A behaviour should be allowed, to avoid the risk of development of excessive permanent deformations. On the basis of triaxial tests on different materials, with different stress levels, they have then defined the stress boundaries corresponding to this stable behaviour; These boundaries are represented by straight lines, in the p,q stress space (see figure 21).
Their method of evaluation of the risk of rutting is based on this stress boundary (or stress criterion) : it consists in calculating the stress distribution in the granular pavement layer, using linear elastic or non-linear elastic models, and comparing the maximum stresses with the proposed stress criterion. If the stresses exceed the stress criterion, the design is not acceptable, and the thickness of the bituminous cover of the pavement has to be increased. The principle of the method is shown on figure 21. This figure shows the maximum stresses obtained at the top of a granular pavement layer, for different thicknesses of asphalt cover (ac between 40 mm and 110 mm). These stresses can be compared with the stress criteria obtained for different materials (represented on the same figure), in order to determine the minimum thickness of asphalt cover required to ensure stable (range A ) behaviour.
This method has for advantage of being very simple, because only the stress distribution in the pavement needs to be calculated, to evaluate the risk of rutting. Its disadvantage is that the criterion does not take into account the number of applied loads, and thus leads to the same design whatever the traffic. It can be useful, however, as a first approximation, to check if there is a risk of rapid failure of the pavement, due to excessive stresses.
5.3.2. Simplified layer strain procedures
This approach, which allows to calculate only the vertical permanent strains and displacements is widely used because of it is simple and can work with both simple empirical permanent deformation laws and more elaborate models. The general principle of this procedure is the following :
• The strains are calculated in the plane (0,y,z) perpendicular to the axis of the road
• The granular layer is divided into j horizontal sub-layers of thickness hj.
• Several points Mij (yi,zj) are defined in the plane (0,y,z), at mid-height of each sub-layer, and at different lateral positions yi (only one point per layer can be used, if only the maximum rut depth under the centre of the load is calculated).
• The maximum stresses at each point Mij (yi,zj) are then determined, by a calculation of the resilient behaviour of the pavement. In this approach , the entire stress path at each point is not considered, but only the maximum stresses pmax(yi,zj) and qmax(yi,zj).
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• the permanent vertical strain at each point Mij , after N cycles, is then calculated using the permanent deformation model. With a relationship of the type
, this calculation is very simple. )q,p(g).N(f)N( maxmax1 =ε p
• finally, the rut depth d(yi) at a given lateral position yi is obtained by cumulating the vertical strains in each sub-layer along the vertical direction z :
(16) ∑=
ε=n
1jj
p1i h)y(dij
Where : ε1pij is the vertical permanent strain in each sub-layer j, for y = yi;
hj is the thickness of each sub-layer j.
5.3.3. Rational rut depth prediction methods
The above layer-strain prediction method appears as a simple, practical alternative, but has several important limitations :
• Only the vertical permanent strains and displacements are calculated.
• Only one type of load is taken into account.
• Only the maximum stresses produced by the passage of the load are taken into account, and not the entire stress path.
This approach can be improved in two main ways :
• With an incremental permanent deformation model (elastoplastic model, in particular), it is possible to take into account the exact stress paths induced by the moving loads, and to cumulate the effect of different loads.
• Instead of using the layer-strain method, it is possible to calculate the complete permanent strain field (and not only the vertical strains), and then the complete displacement field, using a structural calculation procedure (finite element method, for example).
5.3.4 Rut depth prediction methods selected in SAMARIS
Routine level approach
Advanced level approach
6. EVALUATION OF THE “ROUTINE LEVEL” RUT DEPTH PREDICTION METHOD
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7. BIBLIOGRAPHIC REFERENCES Arnold, G. K., Dawson, A. R., Hughes, D., Werkmeister, S., Robinson, D., (2002) Serviceability Design of Granular Pavement Material, Proc. 6th Int. Conf. on Bearing Capacity of Roads and Airfields, Lisbon, Portugal. Barksdale R.D. (1972) Laboratory evaluation of rutting in base course materials, Proc. 3rd Int. Conf on the Structural Design of Asphalt Pavements, London.
Bonaquist R.F., Witczak M.W. (1997), A comprehensive constitutive model for granular materials in flexible pavement structures, Proc. 8th Int. Conference on Asphalt Pavements, Seattle, August 1997, vol1, pp 783-802. Boyce JR, (1980) A non-linear model for the elastic behaviour of granular materials under repeated loading, Proc. Int. Symp. Soils under Cyclic & Transient Loading, Swansea, pp 285-294. Chan, F. W. K. (1990) Permanent deformation resistance of granular layers in pavements, PhD thesis, Dept. Of Civil Engineering, University of Nottingham, England.
Chazallon C. (2000), An elastoplastic model with kinematic hardening for unbound aggregates in roads, UNBAR 5 Conference, Nottingham, 21-23 June, pp 265 – 270.
Chazallon C., Habiballah T., Hornych P. (2002) Elastoplasticity framework for incremental or simplified methods for unbound granular materials for roads, BCRA workshop on modelling of flexible pavements, Lisbon, June 2002. COURAGE (1999) Final report of the Ivth Framework European research project,
Gidel G., Hornych P., Chauvin J.J., Breysse D., Denis A. (2001) Nouvelle approche pour l’étude des déformations permanentes des graves non traitées à l’appareil triaxial à chargements répétés, Bulletin des LPC n°233, pp 5-21 Heck, J. V., (2001) Modélisation des déformations réversibles et permanentes des enrobés bitumineux – Application à l’orniérage des chaussées, PhD Thesis, University of Nantes, France.
Hicher, P.Y., Daouadji, A., Fedghouche, D., (1999) Elastoplastic modelling of cyclic behaviour of granular materials, Unbound Granular Materials, , 21-22 January, Lisbon. Balkema, pp 161-168. Hornych P., Corté J.F., Paute J.L. (1993) Etude des déformations permanentes sous chargement répétés de trois graves non traitées, Bull liaison Labo P et Ch, 184, pp 45-55 Hornych P, Kazai A & Piau J-M , (1998) Study of the resilient behaviour of unbound granular materials, Proc. 5th Conference on Bearing Capacity of Roads and Airfields, Trondheim, Norway, July 1998, vol 3, pp 1277 – 1287.
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Hornych P., KazaÏ A., Quibel A., (2000) Modelling a full scale experiment on two flexible pavements with unbound granular bases, UNBAR5, Int. Symposium on Unbound Aggregates in Roads, Nottingham, , pp 359-367. Hornych P., Kerzrého J.P., Salasca S. (2002) Prediction of the behaviour of a flexible pavement using finite element analysis with non-linear elastic and viscoelastic models., 9th International Conference on Asphalt Pavements, Copenhagen, August 2002.
Hujeux, J.C., (1985), Une loi de comportement pour le chargement cyclique des sols, Génie parassismique, Presse des Ponts et Chaussées, Paris, pp 316-331. Lentz R.W., Baladi G.Y. (1980) Simplified procedure to characterize permanent strain in sand subjected to cyclic loading, Proceedings Int. Symposium on Soils Under Cyclic and Transient Loading, Swansea, pp 89-95. Martinez, J. (1980) Nishi M., Yoshida N., Tsujimoto T., Ohashi K. (1994) Prediction of rut depth in asphalt pavements, Proceedings 4th Int. Conf. on the Bearing Capacity of Roads and Airfields, Minneapolis, USA, pp 1007-1019. Pappin J.W. (1979) Charcteristics of a granular material for pavement analysis, Doctoral thesis, University de Nottingham. Paute J.L., Hornych P., Benaben J.P. (1994) Comportement mécanique des graves non traitées, Bulletin de Liaison des Laboratoires des Ponts et Chaussées, n°190, pp 27-38. Rahma, A., (1991) Modélisation probabiliste des paramètres des lois élasto-plastiques., Doctoral Thesis, Ecole Centrale de Paris, France.
Schoffield, A.N., and Wroth, C.P., (1968), Critical state soil mechanics, Mc Graw Hill [eds], London. Shenton M.J. (1974) Deformation of railway ballast under repeated loading (triaxial tests), Rapport RP5, British Railways Research Department. Sweere, G.T.H. (1990) Unbound granular bases for roads, Doctoral thesis, University of Delft, 431 p. Vuong, B. (1994) Evaluation of back-calculation and performance models using a full scale granular pavement tested with the accelerated loading facility (ALF), Proceedings 4th Int. Conf. on the Bearing Capacity of Roads and Airfields, Minneapolis, USA, pp 183-197. Werkmeister S., Numrich R., Wellner F. (2002) Modelling of granular layers in pavement construction, Proceedings Int. Conf
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