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

M.N.S. Hadi, B.C. Bodhinayake / Advances in Engineering Software 34 (2003) 657662658


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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.

M.N.S. Hadi, B.C. Bodhinayake / Advances in Engineering Software 34 (2003) 657662 659


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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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Fig. 6. Deflections at a depth of 600 mm from the surface.

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2003 Hadi FEM Pavements