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G. Lacey, G. Thenoux, and F. Rodrguez-Roa

April 2008 The Arabian Journal for Science and Engineering, Volume 33, Number 1B 65

THREE-DIMENSIONAL FINITE ELEMENT MODEL FOR

FLEXIBLE PAVEMENT ANALYSES BASED ON FIELDMODULUS MEASUREMENTS

Geraint Lacey*

Civil Engineer - M.Sc.

School of Civil Engineering

Pontificia Universidad Catlica de Chile

Chile

Guillermo Thenoux

Professor

Department of Construction Engineering and Management



Chile

Fernando Rodrguez-Roa

Professor

Department of Structural and Geotechnical Engineering



Chile

:

. . 4

.

.)( . - - )3D FE( .

* Address for Correspondence:


Escuela de Ingeniera

Departamento de Ingeniera y Gestin de la Construccin

Av. Vicua Mackenna, 4860

Santiago

Chile

E-mail: [email protected]

Paper Received 18 May 2006; Revised 27 October 2007; Accepted 4 December 2007

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The Arabian Journal for Science and Engineering, Volume 33, Number 1B April 200866

ABSTRACT

In accordance with the present development of empiricalmechanistic tools, this

paper presents an alternative to traditional analysis methods for flexible pavementsusing a three-dimensional finite element formulation based on a linear-elastic

perfectlyplastic DruckerPrager model for granular soil layers and a linear-elastic

stressstrain law for the asphalt layer. From the sensitivity analysis performed, it wasfound that variations of 4 in the internal friction angle of granular soil layers did

not significantly affect the analyzed pavement response. On the other hand, a null

dilation angle is conservatively proposed for design purposes. The use of a LightFalling Weight Deflectometer is also proposed as an effective and practical tool for

on-site elastic modulus determination of granular soil layers. However, the stiffness

value obtained from the tested layer should be corrected when the measured peak

deflection and the peak force do not occur at the same time. In addition, some

practical observations are given to achieve successful field measurements. Theimportance of using a 3D FE analysis to predict the maximum tensile strain at the

bottom of the asphalt layer (related to pavement fatigue) and the maximum vertical

compressive strain transmitted to the top of the granular soil layers (related to rutting)is also shown.

Key words: civil engineering, pavement analysis, finite element method, light fallingweight deflectometer.

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THREE-DIMENSIONAL FINITE ELEMENT MODEL FOR FLEXIBLE PAVEMENT

ANALYSES BASED ON FIELD MODULUS MEASUREMENTS

1. INTRODUCTION

Structural analysis in pavements has been greatly developed since the initial studies carried out by Boussinesq [1] inwhich soils were modeled as a linear-elastic material. Boussinesqs theory was then extended to a multilayer elastic

model due to the work of Burmister [2, 3] and Schiffman [4]. This theory has been quite popular in pavement analysesand served as a base for programs such as BISAR [5]. The shortcoming of linear elasticity is that it is not a good model

for the actual stressstrain behavior of soils, except at low stress levels and small strains. Because the most appropriate

values of Youngs modulus,E, and Poissons ratio, , depend on confining pressure and deviator stress, there is no fullyrational way of selecting simple values of E and for a linear-elastic analysis. This resulted in the formulation of

numerous nonlinear constitutive models over the past forty years, some more rigorous than others, some based on

experimental evidence, and others based on theoretical principles.

Duncan and Chang [6] proposed the hyperbolic model based on the generalized Hookes Law to relate strain

increments to stress increments, and on the Coulomb failure criterion. This well known nonlinear-elastic model was laterimproved by Rodrguez-Roa [7] who showed that it predicts with sufficient accuracy the nonlinear response of granular

soils in most practical cases, as long as the soil mass is not close to failure.

Elastoplastic stressstrain relationships are more complex than those discussed previously, but they model in a more

realistic manner the coupling between shear and volumetric strains, and the behavior of soils at high stress levels [8, 9].At high stress levels, strains that result from an increment of stress are strongly affected by the stresses existing in the

soil before the new stress increment is applied. Thus, the greatest differences between nonlinear-elastic and elastoplastic

behavior occur close to failure, at failure, and after failure [7].

A general description and evaluation of the different constitutive models available today was made by Lade [10].

Some of the most advanced models require between 11 and 21 soil parameters to be obtained from laboratory tests. Inthis paper, a typical flexible pavement structure is analyzed by means of the commercial finite element (FE) code

ANSYS, version 8.1 [11]. A linear-elastic perfectlyplastic DruckerPrager model was used for simulating the stress

strain behavior of granular soil layers, and a linear-elastic law for the asphalt layer. Elastic moduli for granular soil layerswere measured on-site using a Light Falling Weight Deflectometer (LFWD). For the asphalt layer, the empirical

expression proposed by Pellinen and Witczak [12] for the modulus was adopted in conjunction with the NCHRP Guide

for MechanisticEmpirical Design recommendations [13]. The effects on the studied pavement structure derived fromtraffic loads are analyzed for different soil parameters and loading conditions.

2. ASPHALT CONCRETE CHARACTERIZATION

The National Cooperative Highway Research Program (NCHRP) Project 1-37A was responsible for developing the2002 Guide for the Design of Pavement Structures. This document recommends the use of the complex modulus as a

design parameter in mechanistic design [14]. The dynamic modulus is the absolute value of the complex modulus, and is

basically an elastic modulus obtained from a viscoelastic model which incorporates factors such as temperature, loading

rate, bitumen viscosity, and grain-size characteristics of the mix, among others. Pellinen and Witczak [12] studied the

behavior of 205 asphalt mixes and 23 binders in a temperature range between 18C and 54C with a loading frequencyrange of 0.1 to 25 Hz, calibrating a Dynamic Modulus Predictive Model using a sigmoidal function adjustment of the so-

called Master Curve. According to Garca and Thompson [15], this model gives accurate results compared with

laboratory results. The elastic modulus obtained from such a model may range from values lower than 2000 MPa up to

values higher than 10 000 MPa only combining the asphalt mix characteristic properties, temperature, and loading rate.For the FE analysis it was assumed a value of 4000 MPa based on recommendations given by the NCHRP Guide [13].

This value corresponds to a continuously graded mix, in new conditions, and temperature range between 15 to 20 C.

3. GRANULAR MATERIAL CHARACTERIZATION

A linear-elastic perfectly-plastic DruckerPrager model was chosen to simulate the stressstrain behavior of

cohesionless soil layers because of its simplicity. If compressive stresses are defined as negative values, the DruckerPrager failure criterion may be written as [9]:

021 =+= JIf (1)

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Figure 1. DruckerPrager failure criterion for a granular soilwhere,

I1= first invariant of stress tensor

J2= second invariant of deviatoric stress tensor

)sin3(3

sin2

=

= internal friction angle.

In the principal stress space, Equation (1) represents a right-circular cone with symmetry about the hydrostatic axis

(Figure 1). This cone circumscribes the outer vertices of the irregular hexagonal pyramid of the Coulomb failure surface.For a linearelastic perfectly-plastic material, the yield surface is fixed in stress space, and therefore plastic deformation

occurs only when the stress path moves on the yield surface. On the other hand, elastic behavior occurs if, afterincremental changes of stresses, the new state of stresses is within the elastic domain, i.e., if f< 0.

When the flow rule is associated, the plastic flow is normal to the failure surface that is the plastic potential surface

and the failure surface are the same, and the dilation angle, , is equal to the internal friction angle. Experimental

evidence shows that this assumption leads to a significant overestimation of the bearing capacity of granular soils [9].Thus, a non-associated flow rule is most often applied. Therefore, the following potential function was herein selected:

021 =+= JIg (2)

where,

)sin3(3

sin2

=

Empirical relationships based on CBR have been historically used for the determination of the elastic modulus (orresilient modulus) of granular soils. Some of these relationships are [16]:

10.34r

CBR= (Heukelom and Klomp, 1962)

0.7137.29r CBR= (Green and Hall, 1975)

0.6417.58rM CBR= (Transportation and Road Research Laboratory, 1984)

whereMris the resilient modulus in MPa.

1

3

2

Hydrostatic Axis

(1 = 2= 3)

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Despite these studies, Thompson and Robnett [17] showed that there is no evident correlation betweenMrand CBR.

It is important to bear in mind that the CBR test was not conceived for granular materials with large particle sizes andthat the use of CBR correlations for this type of materials can lead to significant errors.

On the other hand, the determination of the elastic modulus of granular materials by means of triaxial compression

tests and/or in-situ plate loading tests are time consuming and may imply significant costs. Added to this is the limited

number of samples or measurements that can be taken to control on-site technical specifications for pavements.To simplify and improve soil parameter evaluation and to complement compaction control, it is herein proposed to

take advantage of portable non-destructive devices that have been developed in order to measure the in-situ elastic

modulus. Some of the devices available in the market are: the Prima 100 LFWD and the German Dynamic Plate Bearing

Test [18, 19]; the Geogauge [20]; and the Briaud Compaction Device (BCD) [21]. Studies conducted by Fleming [18,

19] and Abu-Farsakh et al. [20] have compared some of these devices with standard tests such as the DCP (DynamicCone Penetrometer), Static Plate Loading Test, and CBR. Both studies agree that these devices are appropriate for use.

According to Fleming [19], the Prima 100 LFWD presents the most promising alternative due to its capability of

recording the load pulse (taken from the internal load cell) and also the deflection history of the underlying structure bythe integration of the geophones speed output.

Based on those studies, the Prima 100 device was herein chosen to measure the elastic modulus of the analyzed

granular materials. However, it was found that this device has a theoretical problem in the algorithm for determining the

elastic modulus. The stiffness value of the tested layer is calculated as the ratio between the peak force and the peak

deflection. This method is inaccurate when the peak deflection and the peak force do not occur at the same time, as

shown in Figure 2.

Figure 2. Lag between the peak force and peak displacement recorded by the Prima 100 device

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Hoffman et al. [22] showed that the peak method can induce to significant errors. To overcome this problem they

suggested performing a spectral analysis approach assuming that the soil structure responds as a linear single degree offreedom (SDOF) system [23]. By analyzing the input (force) and output (speed or deflection) signals in the frequency

() domain the dynamic stiffness as a function of ,K(), is obtained. By doing this, the value of K(0) is extrapolated

for a null frequency. The value of K(0) corresponds to the static stiffness k. Then, the elastic modulus, E, can be

obtained from Boussinesqs half-space equation for a circular footing [22] as:

2 (1 )kE

a

2

=

(3)

in which a is the radius of the circular plate, is the Poissons ratio of the soil, and a non-dimensional shapefactor that depends on the contact-pressure distribution at the interface soil/footing. When the contact pressures are

uniformly distributed (flexible footing), = , and when the plate is rigid, = 4.

In the field work carried out for this study, the corrective method proposed by Hoffman et al. [22] was successfullyused. In addition, surface irregularities were avoided to obtain accurate measurements. It was also observed that if more

people than the operator were standing very close to the device, the elastic modulus could be affected. Hence, it is

suggested that only the operator stays close to the device and at a minimum distance of one plate diameter.

4. MECHANISTICEMPIRICAL ANALYSIS

The pavement structure profile analyzed is illustrated in Figure 3. It consists of a flexible pavement with variableasphalt layer thickness. The granular base and subbase correspond to poorly graded gravel, GP, with approximately 60%

crushed aggregate and 40% rounded aggregate. The subgrade is dense, well-graded sand (SW). The angle of internal

friction and the dilation angle of the granular layers were also varied in order to determine their relevance in the

structural behavior of the pavement. The parameters adopted for the FE analyses are summarized in Table 1. The values

for the elastic modulus of base, subbase and subgrade included in Table 1 were obtained from Prima 100 in-situ tests.

Figure 3. Pavement structure analyzed

50 to 120 mm

200 mm

200 mm

800 kPa

Soil GP, Maximum particle size = 40 mm

Soil GP, Maximum particle size = 25 mm

Soil SW, Maximum particle size = 4.75 mm

Thickness

ASPHALT

BASE

SUBBASE

SUBGRADE

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Table 1. Material Properties of Analyzed Pavement Structure

Asphalt Layer Base Layer Subbase Layer Subgrade

Elastic Modulus

E, MPa4000 241 139 89

Poissons Ratio

0.3 0.3 0.3 0.3

Internal Friction

Angle Range

44 48 40 44 36 40

Dilation Angle

0 ;

2

; 0 ;

2

; 0 ;

2

;

5. SENSITIVITY ANALYSESInternal friction angles of 44, 40, and 36 for the base, subbase and subgrade, respectively, were firstly assumed. A

high uniform pressure of 800 kPa applied over a circular area of 100 mm radius was assumed. This traffic loading

condition corresponds to the effect of a typical heavy truck on the pavement. The axisymmetric FE mesh used was

composed of 20 mm 20 mm rectangular 4-node elements. After preliminary analyses, the final extension adopted forthe mesh was 1500 mm in the vertical direction and 1200 mm in the radial direction. The numerical results obtained for

different asphalt layer thicknesses are shown in Figure 4 for linear-elastic and DruckerPrager elastoplastic granular soil

layers. Regarding the dilation angles, they were assumed equal to 0, 2, and , for each soil layer.

Figure 4. Pavement response for different constitutive models of granular soil layers

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Table 2. Sensitivity Analysis for Internal Friction Angle

Values for granular soil layers

Upper Boundof Shear Strength

Lower Boundof Shear Strength

Maximum Deflection (mm) 0.896 0.988

Maximum Tensile Strain atBottom of Asphalt Layer

0.00120 0.00135

It can be seen from Figure 4 that the maximum deflection of the pavement structure can be significantly higher whenusing a DruckerPrager constitutive model with =0. When the dilation angles are increased, maintaining the values of

, the maximum deflections decrease. In particular, when the dilation angle is equal to the internal friction angle, i.e.,

when the flow rule is associated, the resulting pavement response is closer to the linear elastic model. Therefore, it is

conservatively recommended for design purposes, to use small values for the dilation angle.

An internal friction angle sensitivity analysis showed that small variations in its value do not affect the pavements

response significantly. In this analysis the dilation angles were assumed null, and two extreme situations were considered

for the internal friction angles. The first one assumed that is equal to the highest value of the expected range for each

granular layer shown in Table 1 (upper bound of shear strength); the other considered as the lowest value of the

expected range for each layer (lower bound of shear strength). For this sensitivity analysis the following pessimistictheoretical condition was supposed: A circular loading area of 100 mm radius with an 800 kPa pressure, and a 50 mm

asphalt layer with an elastic modulus of 1000 MPa.

From the results shown in Table 2, it can be concluded that even for a pessimistic condition, variations of 4 in the

angle of internal friction for all granular soil layers cause small changes in tensile strains and deflections induced on theasphalt layer. This means that for a greater thickness and elastic modulus of the asphalt layer, the effects of small

changes in the internal friction angles would be even less relevant. Therefore, it is concluded that this parameter can beestimated from previous local geotechnical experience obtained with similar materials. Then, triaxial tests would not be

strictly needed.

6. THE 3D FE ANALYSES

A 3D FE analysis was performed for both single and dual wheel models. In the single wheel model, as shown in

Figure 5(a), a uniform pressure of 800 kPa was assumed over a square tire footprint of side 175 mm, i.e., a total load of

24.5 kN. In the second analyzed case, the same applied pressure was assumed acting over two isolated square footprintsof side 175 mm. For this case, a dual spacing of 50 mm was assumed, as shown in Figure 5(b).

For both analyses the adopted parameters were the values listed in Table 1. The lowest values and null dilatancy

were used for each granular layer. An asphalt layer of 50 mm thick was assumed for the analyses. Because of

symmetrical conditions, only of the whole problem needs to be modeled. The size of the model was a cubical block of

side 1500 mm. The block was subdivided into 8-node hexahedral elements of size 25 mm. Figure 6 shows the vertical

strain distribution obtained for both single and dual wheel loads. Because of the higher load the dual wheel modelpresents higher vertical compressive strains (compressive strains are negative in ANSYS) than the single wheel model at

the top of the granular layers.

The single wheel modeled as a square tire footprint transmits half the load of the dual wheel model, however, it was

found that it presents higher transverse horizontal strains (tensile strains are positive) at the bottom of the asphalt layerthan the dual wheel model (Figures 7(a) and 7(b). On the contrary, longitudinal horizontal strains induced by the dual

model are higher than those induced by the single wheel (Figures 7(a) and 7(c). These results can be relevant for

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pavement design since failure can take place due to fatigue in the asphalt layer (if high tensile strains are induced at the

bottom of the asphalt layer) or to rutting (due to high vertical compressive strains in the granular layers).

Figure 5. Top view (units in mm) of: (a) single wheel model; (b) Dual wheel model

7. CONCLUSIONS

Several investigations have shown that the use of linear-elastic models to predict stresses and strains in pavement

structures can lead to significant errors. To overcome this problem a linear-elasticperfectly-plastic DruckerPrager

model is proposed for granular soil layers, assuming conservatively a null dilation angle for design purposes. Hence, this

simplified elastoplastic model only depends on the elastic modulus and the internal friction angle. From the internal

friction angle sensitivity analysis, it was concluded that this parameter can be estimated from previous local geotechnicalexperience with similar materials, because changes of 4 did not significantly affect the analyzed pavement response.

The use of a Light Falling Weight Deflectometer (LFWD) is also proposed to perform in-situ measurements of the

elastic moduli for granular soil layers. At the same time, the use of this non-destructive device can be a valuable

alternative to dry density in compaction control.

However, the stiffness value obtained from the tested granular layer should be corrected when the measured peakdeflection and the peak force do not occur at the same time.

According to our field experience with the use of the LFWD, it was seen that people standing very close to the devicecan affect the accuracy of the measurements. Hence, it is suggested that a single operator handles the device, standing at

a minimum distance of one plate diameter from the equipment. In addition, it must be pointed out that measurements

should be taken on a flat surface in order to minimize distortions in the LFWD signals.

It was concluded from the 3D FE analyses performed, that a single wheel can induce higher transverse tensile strains

at the bottom of the asphalt layer than a dual wheel which transmits twice the load. On the other hand, longitudinal

tensile strains induced by the analyzed dual wheel were higher than those induced by the single wheel. In addition, as

expected, the dual wheel induces higher vertical compressive strains than the single wheel at the top of the granular

layers.

Therefore, the maximum horizontal tensile strains at the bottom of the asphalt layer, and the maximum vertical

compressive strains in the granular soil layers, must be evaluated by means of a 3D FE analysis for all different possible

loading conditions. These aspects are relevant for rutting and fatigue prediction.

Loading area

87.5

87.5

Symmetrical plane

Symmetricalplane

1500

1500

1500

1500

Loading area

175

25

87.5

(a) (b)

Symmetrical plane

Symmetricalplane

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Figure 6. Vertical strains for: (a) single wheel load; (b) dual wheel load (transverse section);

(c) dual wheel load (longitudinal section)

-0.001414

-0.001212

-0.00101

-0.808E-03

-0.606E-03

-0.404E-03

0.142E-03

0.202E-03

0.404E.03

ASPHALT

BASE

SUBBASE

SUBGRADE

800 kPa

-0.002065

-0.001755

-0.001485

-0.001195

-0.955E-03

-0.615E-03

-0.325E-03

-0.347E-04

0.255E-03

0.545E-03

ASPHALT

BASE

SUBBASE

SUBGRADE

800 kPa

(a)

(b)

(c)

-0.002065

-0.001755

-0.001485

-0.001195

-0.955E-03

-0.615E-03

-0.325E-03

-0.347E-04

0.255E-03

0.545E-03

800 kPa

ASPHALT

BASE

SUBBASE

SUBGRADE

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Figure 7. Horizontal strains for: (a) single wheel load; (b) dual wheel load (transverse section);

(c) dual wheel load (longitudinal section)

(a)

(b)

(c)

-0.439E-03

-0.326E-03

-0.214E-03

-0.102E-03

0.105E-03

0.123E-03

0.235E-03

0.347E-03

0.460E-03

0.572E-03

800 kPa

ASPHALT

BASE

SUBBASE

SUBGRADE

-0.617E-03

-0.417E-03

-0.325E-03

-0.179E-03

-0.330E-04

0.113E-03

0.259E-03

0.405E-03

0.551E-03

0.607E-03

800 kPa

ASPHALT

BASE

SUBBASE

SUBGRADE

-0.817E-03

-0.610E-03

-0.403E-03

-0.197E-03

0.101E-04

0.217E-03

0.423E-03

0.630E-03

0.837E-03

0.001043

800 kPa

ASPHALT

BASE

SUBBASE

SUBGRADE

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REFERENCES

[1] J Boussinesq,Application des potentiels l'tude de l'quilibre et du mouvement des solides lastique. Paris: Gauthier-Villard, 1885.

[2] D. M. Burmister, "The Theory of Stresses and Displacements in Layered Systems and Applications to the Design ofAirport Runways",Proceedings, Highway Research Board, Vol. 23. Washington, D.C., USA, 1943.

[3] D. M. Burmister , "The General Theory of Stresses and Displacements in Layered Soil Systems", Journal of AppliedPhysics, 16(2)(1945), pp. 8996; 16(3), pp. 296127; 16(5), pp. 296302.

[4] R. Schiffman, "General Solution of Stresses and Displacements in Layered Elastic Systems",International Conferenceon the Structural Design of Asphalt Pavement, Proceedings, University of Michigan, Michigan, USA, 1962.

[5] G. Hildebrand, Verification of Flexible Pavement Response From a Field Test, Danish Road Institute, Report 121.Ministry of Transport, Denmark, 2002.

[6] J. M. Duncan and C. Y. Chang, Nonlinear Analysis of Stress and Strain in Soils,ASCE Journal of the Soil Mechanicsand Foundations Division, 96(1970), p. 1629.

[7] F. Rodrguez-Roa, Observed and Predicted Behavior of Maipo River Sand, Soils and Foundations (JapaneseGeotechnical Society), 43(2003), p. 1.

[8] D. R. J. Owen and E. Hinton, Finite Elements in Plasticity: Theory and Practice. Swansea, UK: Pineridge Press

Limited, 1980.

[9] W. F Chen and E. Mizuno, Nonlinear Analysis in Soil Mechanics: Theory and Implementation. New York: Elsevier,1990.

[10] P. V. Lade, Overview of Constitutive Models for Soils,ASCE Geotechnical Special Publication, N 128, USA, p.1.

[11] Ansys Inc,Release 8.1 Documentation for Ansys.Ansys CivilFEM, Ingeciber S.A., 1999.

[12] T. K. Pellinen and M. W. Witczak, Stress Dependent Master Curve Construction for Dynamic (Complex) Modulus,Annual Meeting Association of Asphalt Paving Technologists, Colorado Springs, Colorado, USA, March 2002.

[13] NCHRP, Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures. Final Report, Part 3,Design Analysis, 2004.

[14] T. Clyne, X. Li, M. Marasteanu, and E. Skok,Dynamic and Resilient Modulus of Mn/DOT Asphalt Mixtures. Minnesota

Department of Transportation, USA, 2003.[15] G. Garca and M. Thompson, Hot Mix Asphalt (HMA) Dynamic Modulus Prediction, State of the Art of Pavement

Structural Design and the New AASHTO Interim Guide: CONICYT Project. Santiago: Pontificia Universidad Catlica

de Chile, 2005.

[16] B. Sukumaran, N. Chamala, M. Willis, J. Davis, S. Jurewicz, and V. Kyatham, Three Dimensional Finite ElementModeling of Flexible Pavements, 2004 FAA Worldwide Airport Technology Transfer Conference. Atlanta , New

Jersey, 2004.

[17] M. R. Thompson and Q. L. Robnett, Resilient Properties of Subgrade Soils, ASCE Journal of TransportationEngineering, 105(1979), p. 77.

[18] P. R. Fleming, M. W. Frost, and C.D.F. Rogers, A Comparison of Devices for Measuring Stiffness In-situ,Proceedings of the Fifth International Conference onUnbound Aggregate In Roads,Nottingham, UK, 2000.

[19] P. R. Fleming, Field Measurement of Stiffness Modulus for Pavement Foundations, 79th Annual Meeting of the

Transportation Research Board, Washington DC, 2001.

[20] M. Abu-Farsakh, K. Alshibli, M. Nazzal, and E. Seyman, Assessment of In-Situ Test Technology for ConstructionControl of Base Courses and Embankments. Louisiana Transportation Research Center, USA, 2004.

[21] J. Briaud, Y. Li, and K. Rhee, BCD: A Soil Modulus Device for Compaction Control,ASCE Journal of Geotechnicaland Geoenvironmental Engineering, 132(2006), p.108.

[22] O. Hoffmann, B. Guzina and A. Drescher, Enhancements and Verification Tests for Portable Deflectometers.Minnesota Department of Transportation, USA, 2003.

[23] J. Bendat and A. Piersol,Random Data: Analysis and Measurement Procedures. New York: Wiley-Interscience, 1971.

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