2003 hadi fem pavements
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Non-linear finite element analysis of flexible pavements
Muhammad N.S. Hadi*, B.C. Bodhinayake
Faculty of Engineering, University of Wollongong, Wollongong, NSW 2522, Australia
Abstract
A research study is being undertaken to incorporate the realistic material properties of the pavement layers and the moving traffic load, inthe analysis of flexible pavements, using the finite element theory. As a preliminary step taken herein in this direction, a pavement structure
where field measurements have been carried out when subjected to a cyclic loading, is selected and modelled as a finite element model.
The analysis is being carried out using the finite element computer package ABAQUS/STANDARD, when this pavement model is subjected
to static and cyclic loading while considering the linear and non-linear material properties of the pavement layers. The results indicate that
displacements under cyclic loading when non-linear materials are present, are the closest to field measured deflections.
q 2003 Elsevier Ltd. All rights reserved.
Keywords: Pavement analysis; Flexible pavement; Finite element; Cyclic loading; Non-linear; ABAQUS
1. Introduction
In mechanistic methods used in the analysis of layeredpavement systems under traffic load, the pavement layers
are considered as homogenous, linear elastic and isotropic
and the loading is considered as static[1]. These mechan-
istic methods work reasonably well, if the pavement
subgrade system behaves as a linear elastic system [2].
The use of multi-layer elastic theory together with static
loading is a rational approach compared with older
empirical pavement design methods. However, in the real
situation, these heterogeneous pavement layers behave far
from these ideal conditions and are subjected to dynamic
and cyclic loading. Researchers diverted their research to
the finite element method, which provides a better solution
in the dynamic analysis of pavements while considering theheterogeneity, non-linearity and orthotropy condition of
the pavement structure at the same time [3,4]. With the
availability of high-speed computers, finite element
methods are gaining acceptance as the finite element
analysis programs can handle complex geometry, boundary
conditions and material properties with ease [5]. But still
these research efforts are in their early stages.
Research is being undertaken to model the flexible
pavement as a finite element model, with defined boundary
conditions and to investigate the effects of static and cyclic
loading when combined with linear and non-linear
characteristics of pavement materials, in the analysis of
flexible pavements. A pavement section where Accelerated
Loading Facility (ALF) trial has been carried out atCallington, South Australia (Site No. 5 of ALF trial at
Callington), is selected for this study [6]. The reason for
selecting Australian Road Research Board (ARRBs)
accelerated loading facility is its capability of applying a
cyclic loading on pavement structure. At this site the
existing cracked asphalt surface course and granular base
course have been removed and replaced with two new
asphalt layers before the ALF trial, so that the behaviour of
the new asphalt layers can be considered as linear and the
granular layers below the new asphalt layers can be
considered as non-linear.
2. Background
Design methods for flexible pavements have evolved
since the turn of the last century. Empirical methods with or
without a strength test were the early methods employed in
the design of flexible pavements. The method without
strength test refers to the soil classification system provided
by Hogentogler and Terzaghi [7]. In the method with
strength test the thickness of pavement was related to CBR
[8]. By 1950, two methods based on limited deflections and
limited shear failure were presented. The method based on
limited deflection was presented by Kansas State Highway
Commission[9]. The method based on limited shear failurewas first presented by Barber[10]and later by McLeod[11].
0965-9978/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0965-9978(03)00109-1
Advances in Engineering Software 34 (2003) 657662www.elsevier.com/locate/advengsoft
* Corresponding author. Tel.:61-2-4221-4762; fax:61-2-4221-3238.E-mail address:[email protected] (M.N.S. Hadi).
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Design curves based on road tests became available in
1960s. Road Note 29 was published in 1960 to provide a
guide to the structural design of roads under British
condition of climate, materials, and traffic loading [12].
Similarly, design curves were developed from AASHO road
test during that period[13].
In 1943, Burmister [14] presented a method for
determining stresses and displacements in a two-layer
elastic system. Since then a large number of computer
programs have been developed for calculating stresses,
strains and deflections of layered elastic system. Details of
few such programs can be found in Refs. [1519].
The stresses and strains calculated in these programs are
checked against the defined failure criteria. In all these
programs pavement layers are considered as homogeneous,
linear elastic.In real situations, the assumption of homogeneous, linear
elastic pavement materials becomes invalid. Almost all
pavement materials are not homogeneous. Especially
granular materials are particulate in nature. Even though,
bituminous materialsare mixedin hotmix plants, they arenot
homogeneous. Pavement materials are not linear elastic.
When exposed to stress, pavement materials will exhibit
elastic deformation as well as a number of different
deformations, such as viscous, plastic and visco-elastic
deformations. Since all these deformations are stress
dependent, the materials behave in a non-linear manner [20].
The finite element method for the analysis of
flexible pavements was first applied by Duncan [21].Many computer programs based on this finite element
method were later developed. Details of two well-known
programs developed in 1980s can be found in Refs.[22,23].
The use of finite element method in determining the stresses,
strains and deflections is becoming popular, with the
availability of high-speed computers. Furthermore, this
method can handle structures with non-linear materials.
In all these programs, the traffic loading is considered as
static loading. The incorporation of traffic loading as a
dynamic loading is still in its early stages of research.
3. Flexible pavement analysis
3.1. Static analysis of multi-layered pavement subgrade
systems
In these methods each layer of the multi-layered linear
elastic pavement structure is characterised by its Youngs
Modulus and its Poissons ratio [24]. In some programs
resilient modulus based on the recoverable strain under
repeated loading is used instead of Youngs Modulus.
The stresses, strains and deflections at specified distances
from the load are then theoretically calculated, assuming a
semi-infinite subgrade and infinite lateral boundaries.
These calculated responses are matched with defined failurecriteria. Layer thicknesses and material properties are
adjusted until the computed responses are lower than the
failure criterion.
3.2. Finite element analysis of pavement subgrade systems
In the finite element method the pavement layers are
considered as a solid continuum. The solid continuum
domain of the problem is then divided into sub domains.
These sub domains are then discretised into a number of
finite size elements. Assembly of all these elements will
then represent the problem in the analysis. Finite elements
are interconnected by nodes at their common edges.
This analysis provides an approximate solution for an
engineering structure with various types of boundary
conditions and under various types of loading using a
stiffness or energy formulation[25]. In the derivation of thestiffness matrix for elements, three factors such as the
geometry of elements, the degrees of freedom allowed for
the nodes to displace and the material properties of elements
are considered. This solution yields displacements at the
nodal periods and stresses and strains at integration points.
4. Pavement subgrade model considered for the analysis
The pavement structure selected for this study (site No. 5
of ALF trial at Callington, South Australia) consists of a
45 mm thick asphalt layer (AC14) and a 55 mm thick
asphalt layer (AC20) as the surfacing course, a 85 mm thickgranular (limestone crushed rock) layer as the base course, a
230 mm thick granular (limestone quarry rubble) layer,
a 175 mm thick soil (calcareous clay sand) layer and a
370 mm thick soil (clayey sand) layer as the subbase course,
and a subgrade (siltstone rubble) at the bottom.
The pavement configuration is shown inFig. 1.The material
properties of pavement layers are given in Table 1.
This pavement is subjected to a cyclic loading equal to
80 kN, applied through a dual wheel assembly, whichsimulates the loading pattern of the ALF machine.
The interval at which the cyclic loading is applied, is
considered as 5 s, to simulate the unidirectional trafficking
speed of 20 km/h of the ALF machine.A pavement structure having the layer configuration as
shown in Fig. 1 and having layer thicknesses and Elastic
Material properties as given in Table 1, is modelled as a
finite element model, using the finite element computer
package ABAQUS/STANDARD[26].
The results of laboratory and field tests carried out during
the ALF trial at Callington, South Australia, are given in
Ref. [27]. In estimating the linear properties of pavement
materials those test results are used together with
the AASHO Road Guide[28]. In estimating the non-linear
properties of granular materials, results published in the
research report Stabilisation of Pavement Soils from South
Australia, are used[29].This report presents the results ofrepeated load triaxial tests carried out on soils collected
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from two South Australian borrow pits. Nataatmadja[30],
has suggested a method of modelling non-linear
characteristics of granular materials to suit Australian
conditions. However, due to unavailability of exact
parameters required for such modelling, the ktheta
model, is used in this study with assumed k-values (k1andk2) as given inTable 2[31].In estimating k1; k2 and u
values, Refs[28,29]are used.
The 40 kN wheel load is assumed to be uniformlydistributed over the contact area between tyre and
pavement. The size of contact area depends on the contact
pressure. The contact pressure is assumed as equal to the
tyre pressure. Tyre pressure is equal to 700 kPa, as given in
ARR198[6].
The contact area can be represented by two semicircles
and a rectangle as shown in Fig. 2.
Further, this shape of two semicircles and a rectangle is
converted to a rectangle as suggested by Huang, having
an area of 0.5227L2 andawidthof0.6L; asshown inFig. 3(a)
[32]. Since L0:330 m; L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi{40=700=0:5227}p thecontact area has the dimensions of 0.288 m 0.198 m as
shown inFig. 3(b).
Due to symmetry, the pavement under a half wheel load
is considered in the analysis. A pavement block under half
wheel load, having a length of 1.3 m, a width of 1.5 m and a
depth of 3.16 m, is considered for the analysis.
This pavement structure is loaded in an area of
0.144 m 0.198 m which represents the half wheel load
as shown inFig. 4.
In ABAQUS[26]this pavement block is modelled with
C3D27R (Continuum 3-Dimensional 27 node elements with
reduced integration) brick elements. C3D27R element type
is quadratic. Quadratic elements yield better solution than
linear interpolation elements[33].
Since the cracked asphalt surface has been removed and
replaced with a new asphalt layer of 100 mm thick (45 mm
thick layer and 55 mm thick layer) at the ALF site,
the stress strain relationships of the two asphalt layers are
assumed to be in the elastic region. The stressstrain
relationships of granular layers are assumed to be in theplastic region. Therefore two asphalt layers in the pavement
structure are considered as homogeneous, linear elastic and
isotropic, while granular layers and the subgrade are
considered as linear initially and later as non-linear.
In this study the top surface is considered as free
from any discontinuities (no cracks) or unevenness.
Fig. 2. Contact area between tyre and pavement surface.
Fig. 1. Pavement configuration.
Table 2
k-Values used for non-linear modellingMk1uk2
Layer k1 k2
Base 6000 0.62
Sub-base 5800 0.56
Fill 5000 0.53
Rockfill 4600 0.52
Subgrade 4500 0.5
Table 1
Layer thickness and elastic material properties
Layer Thickness
(mm)
Modulus of elasticity
(kPa)
Poissons ratio
Asphalt (AC14) 45 1,800,000 0.3
Asphalt (AC20) 55 1,725,000 0.3
Base 85 138,000 0.35
Sub-base 230 96,600 0.35
Fill 175 72,450 0.35
Rock fill 370 62,100 0.35Subgrade Infinite 55,200 0.35
Fig. 3. Equivalent contact area.
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The interfaces between layers are considered as fully
bonded (no gaps) and rough, at this stage.
The three-dimensional view of the finite element model
considered for the analysis, using computer package
ABAQUS/STANDARD is shown inFig. 5.
5. Boundary conditions
Since brick elements are considered in the finite element
modelling, rotation is not allowed for at all nodes.
Therefore, only three degrees of freedom have to be
considered in defining the boundary conditions.
The following conditions are applied with reference to
Fig. 4, when defining the boundary conditions.
The vertical displacements of the nodes on the bottom
plane (plane ABCD) of the model are fixed.
The plane ADHE is considered as plane of symmetry
between the two wheels, thus the orthogonal
displacements to the plane are prevented.
The plane ABFE is considered as vertical plane passing
through midway of one wheel, thus the orthogonal
displacements to the plane are prevented.
6. Results
Displacements computed in the vertical direction of some
selected nodes, by the finite element package ABAQUS/
STANDARD and the deflections measured by the Multi-
Depth Deflection Gauge (MDDG) at similar locations duringALF trial are presented graphically inFig. 6.
The MDDG has measured deflections at depths of 100,
200, 600 and 2030 mm and at distances of 0, 200, 250, 300,
500, 600, 900, 1200 and 1500 mm from the point of load
application, after 28, 52.6, 77.3, 103.1, 135.6, 160.6, 185.9,
211.0, 232.2 and 257.4 kcycles of load application.
Since, asphalt layers are considered as linear elastic in
this study, the deflections measured at a depth of 100 mm
are not considered for comparison. The deflections
measured at a depth of 600 mm are considered for
comparison, as they are the closest to the top of subgrade,
which is at a depth of 960 mm. The computer analysis is
being limited to five cycles, in this study.
Fig. 6shows that, if all pavement layers are considered as
linear elastic, the deflections are similar for both static and
cyclic loading. The deflections increase when non-linear
materials are present. The deflections computed when
non-linear materials subjected to a cyclic loading are the
closest to the field measured values.
Fig. 6shows further that, the deflections measured at the
point of load application increase slightly with the increase
in the number of cycles of load application, while
deflections measured beyond 200 mm from the point of
load application, do not show any significant increase.
If the analysis is carried out with exact material propertiesand extended loading cycles, an agreement between
displacements computed by ABAQUS/STANDARD and
deflections measured in the field could be achieved.
7. Summary and conclusion
A pavement structure consisting of two asphalt layers, a
granular base layer and a subbase layer on top of subgrade,
is modelled as a 3D finite element model using the finite
element computer package ABAQUS/STANDARD.
The analysis is carried out considering linear and
non-linear behaviour of pavement materials when they aresubjected to static and cyclic loading. Quadratic brick
Fig. 4. Pavement considered in the analysis.
Fig. 5. Three-dimensional view of the finite element model.
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elements are used in finite element modelling as they yield
results with greater accuracy.
The computer analysis is limited to five cycles, while
deflections measurements during the ALF trial are being
carried out after 28, 52.6, 77.3, 103.1, 135.6, 160.6,
185.9, 211.0, 232.2 and 257.4 kcycles of load application.
Although it is not possible to compare the results at this
stage, all deflection bowls show a similar trend.
Even though, the cyclic loading in this study islimited to five cycles, it can be seen that, cyclic loading
which could be simulated to wheel loading together with
non-linear pavement materials produces a higher
defl ecti on at t he t op of t he s ubgr ade, t han t he
static loading together with either linear or non-linear
pavement materials.
Based on the work in this preliminary study, it can be
predicted that, if pavement designs are carried out assuming
static loading and linear pavement materials, the deflections
at top of subgrade are higher than the expected values,
when pavement sections with non-linear materials are
subjected to the moving load. These higher deflections on
top of subgrade can cause the pavement sections to failbefore the end of design life.
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