17.ijaest vol no 6 issue no 1 assessment and improvement of the accuracy of the odemark...

8/6/2019 17.IJAEST Vol No 6 Issue No 1 Assessment and Improvement of the Accuracy of the Odemark Transformation Metho

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Assessment and Improvement of the Accuracy of the

Odemark Transformation Method

Sherif M. El-Badawy*, Ph.DAssistant Professor, Public Works Department

Faculty of Engineering, Mansoura University

Mansoura, [email protected]

Mostafa A. Kamel, Ph.D.Assistant Professor, Public Works Department

Faculty of Engineering, Mansoura University

Mansoura, [email protected]

Abstract Flexible pavement structures are very complex

systems usually consist of multi-layers with each layer having

different properties (elastic modulus and Poissons ratio). In

order to simplify these complex systems for stress and strain

calculations, Odemark has developed a method to transform

these multi-layer systems into an equivalent one-layer system

with equivalent thicknesses but one elastic modulus. This methodhas been used in many advanced research studies and design

methods including the newly developed Mechanistic-Empirical

Pavement Design Guide (MEPDG). This paper investigates the

accuracy of the Odemark method and presents a methodology to

increase its accuracy. A two-layer system with different modular

ratios and thicknesses was extensively analyzed. The results

showed that a correction factor must be used with Odemarks

method in order to produce highly accurate stress and strain

results. This correction factor is not constant and depends not

only on the modular ratio and the thickness of the layer but also

on the depth of interest.

Keywords; Odemark; stress; stiffness; elastic modulus, MEPDG

I. INTRODUCTIONOdemark has developed an approximate method to

calculate stresses and strains in multiplayer pavement systems by transforming this structure into an equivalent one-layersystem with equivalent thicknesses but one elastic modulus.This concept is known as the method of equivalent thickness(MET) or Odemarks method. MET assumes that the stressesand strains below a layer depend only on the stiffness of thatlayer. If the thickness, modulus and Poissons ratio of a layer ischanged, but the stiffness remains unchanged, the stresses andstrains below the layer should also remain (relatively)unchanged. According to Odemark, the stiffness of a layer is

proportional to the following term [1]:

2

3

1 v

Eh

(1)

where:

h = thickness of the layerE= elastic modulus

v = Poissons ratio

For the transformed section shown in Fig. 1 the equivalentthickness he can be calculated as follows:

32

12

2

21

1

2

2

2

3

2

1

1

3

1

1

1

;11

uE

uEhh

orv

Eh

v

Eh

e

e

(2)

For the case of a two-layer system with equal Poissonsratio, the equivalent thickness can be calculated using thefollowing formula:

3

2

1

1E

Ehhe (3)

Comparing stresses and strain calculated using the

Odemarks method with those from the elastic theory led to theconclusion that they are relatively different. In order to achievea better agreement between Odemarks method and the elastictheory, a correction factor f was applied to the aboveequation as follows.

3

2

1

1E

Ehfhe (4)

Researchers reported that the value of the correction factorf depends on the layer thicknesses, modular ratios, and thenumber of layers in the pavement structure. Furthermore, theymentioned that using a value of 0.8 to 0.9 for f leads to areasonably good agreement between the two methods [2].

h1 E1 1 he E2 2

E2 2 E2 2

Figure 1. Odemarks Transformation of a Layered System

Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIESVol No. 6, Issue No. 1, 105 - 110

ISSN: 2230-7818 @ 2011 http://www.ijaest.iserp.org. All rights Reserved. Page 105
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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For a multi-layer system the equivalent thickness of theupper n-1 layers with respect to the modulus of layer n, may becalculated as follows:

3

1

1

,

n

in

i

ineE

Ehfh

Where:he,n = equivalent thickness of the layer of interest

(layer n).

f = correction factor

hi = thickness of layer i

Ei,,En= elastic moduli of layers i and n, respectively.

Odemark transformation method has been utilized andimplemented in many applications [2], [3], [4], [5], [6], [7], [8].Subagio et al used this method to calculate the residual life andoverlay thickness required based on deflection data measuredusing the falling weight deflectometer (FWD) [2]. Senseney

and Mooney also used this method in the characterization of atwo-layer soil system using lightweight deflectometer [3]. Thenewly developed mechanistic empirical pavement design guide(MEPDG) implemented this method to transform a multi-layer

pavement system into an equivalent one layer system. Thisequivalent system is used to determine the frequency of loading

based on the effective length of the stress pulse and vehiclevelocity using the following relationship [6].

eff

sl

L

v

t

6.171

where:

fl = frequency of load, Hz

t = time of load, sec

vs = velocity (mph)

Leff = effective length of the stress pulse, inch

The effective length concept which has been employed inMEPDG defines the stress pulse at a specific depth within the

pavement system as shown in Fig. 2. In this Figure, the line AAshows the length of the stress pulse at the mid-depth of the AClayer, whereas line BB shows the length of the stress pulse inthe granular base layer. The sloped lines along with the depthof interest define the effective length of the stress pulse.

Because the slope of the stress distribution shown in Fig. 2

is a function of the stiffness of the layer and since there is nopresent relationship exists to relate them together, a multi-layer pavement system is transformed into an equivalent one layersystem in order to estimate the effective length. Thetransformed section using MET is shown in Fig. 3. Thetransformed section has the modulus of the subgrade and hasan equivalent thickness of he. In MEPDG, for simplicity, thestress distribution for a typical subgrade soil is assumed to be at45 degree as shown in Fig. 4. Using this stress distribution, theeffective length can be calculated at any depth within thetransformed pavement system.

Figure 2. Effective Length Concept in MEPDG [6]

Figure 3. Equivaent Thicnkess Calculation

Figure 4. Effective Length Calculation using the Transformed Section [6]

In MEPDG, for any pavement layer, the effective length ofthe stress pulse is computed at the effective depth (Zeff). Theeffective depth is the transformed depth at which the loadingfrequency is needed. The effective depth for the transformedsection (as shown in Fig. 4) is calculated with the help ofEquation. 8 [6]:

1

1

33

n

i SG

nn

SG

iieff

E

Eh

E

EhZ (8)




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The computed loading frequency at the effective depth isthen used to determine the complex modulus (E*) of theasphalt layer. Thus the accuracy of the MET affects theaccuracy of the E* which in turn influences the MEPDG

predicted rutting, load associated cracking and roughness.

II. OBJECTIVESThis paper has two primary objectives. The first objective isto investigate the accuracy of the Odemark transformation

method (MET). The second objective of this research is toimprove the accuracy of the MET, if warranted.

III. STUDY METHODOLOGYAn extensive study is introduced to quantify the influence of

layer thickness, depth, and modular ratios on the correctionfactor f of the Odemarks transformation method. A two-layer system with the first layer thickness (h) values of 2, 6, 10,and 15 inches was used in the analysis. A total of 5 differentmodular ratios of E1/E2 = 3.33, 16.67, 33.33, 50.00, and 66.67for each thickness were analyzed. A Poissons ratio of 0.35 was

assumed in all computations. Fig. 5 shows the applied load andthe properties of the two layer system used in the analysis.

Figure 5. Two-Layer Pavement System used for the Analysis

First a linear elastic solution was performed on the two-layer structure using the KENPAVE software to calculate thevertical and radial stresses at different depths measured fromthe surface of the upper layer under the centerline of the load.Then Odemark transformation concept was used to convert thetwo-layer problem into one layer with equivalent thicknessesand one modulus. A comparison between stresses calculatedfrom both systems was made. The influence of the correctionfactor term on the computed stresses of the transformed systemusing MET method was studied.

IV. ANALYSIS AND RESULTSComparing Odemark solution, without using a correction

factor (f=1), to KENPAVE solution yielded different stressvalues at the points of interest. This is clearly shown in Fig. 6.This figure only shows the stresses calculated at differentdepths underneath the centerline of the load (radial distance=0).

0

25

50

75

100

125

150

0 25 50 75 100 125 150

Equival

entTransformedSection

Ca

lculatedStress,

psi

Two-Layer Calculated Stress using KENPAVE, psi

Line of Equality

Figure 6. Comparison of the Two-Layer and the Equivalent One-Layer

Pavements Computed Stresses

A correction factorfwas then introduced into the equationto calculate the corrected equivalent depth. First, a unique fvalue was applied to all points of interest for each modularratio. The results showed good agreement only for the verticalstresses calculated at the interface between the two layers when

using f of 0.8 to 0.9. However, at any depth other than theinterface between the two layers the results showed asignificant difference between the two solutions. This meansthat the correction factorfis also dependent on the depth.

In order to verify that, the Solver function in Excelspreadsheet was used for each modular ratio, to calculate the ffactor at each depth such that:

vzi vzti = 0 (9)

where:

vz = vertical stress calculated from a two-layer

system at depth zi using KENPAVE.

vzti = vertical stress calculated from Odemarktransformed depth zti (One layer system) using

Boussinesq or KENPAVE.

The results showed that fdepends not only on the modularratio and the thickness of the upper layer in the two-layer

pavement system but also on the depth of interest.

Figs. 7 through 9 depict the relationship between thecorrection factor f and depth at different modular ratios forthe investigated two layer system with h1 = 6, 10, and 15 inchesrespectively. For the points (Z values) in the first layer therelationship between f and Z was found to be a 3

rddegree

polynomial. The values of the R2

were 0.99+

for all investigatedmodular ratios as well as the different structures considered inthe analysis. For the points (Z values) in the second layer therelationship between f and Z was found to be a 2

rddegree

polynomial. The values of the R2

were also found to be 0.99+

for all different modular ratios and the different structuresconsidered in the analysis. The relationships between f and Zfor the two layers are shown in Fig. 9 for the pavement systemwith h1=15 inch. Examples of these relationships are shown inFig. 10 and Fig. 11 for the two layer system with h1=6 inch anda modular ratio of 50.

9000 lb

120 psi

2a = 9.772 in h

E1,

E2,

ZZ




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0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 2 4 6 8 10 12

Depth, Z (in)

Correctin

Factor,

f

E1/E2 = 3.33

E1/E2 = 66.67

E1/E2 = 50.00

E1/E2 = 33.33

E1/E2 = 10.00

E1/E2 = 6.66

Interfacebetweenlayers1and2

E1/E2 = 16.67

Figure 7. Relationships between the Correction f and Depth (Z) for the system with (h1 = 6 in.)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 5 10 15 20 25Depth, Z (in)

CorrectionFactor,f

E1/E2 = 3.33

E1/E2 = 16.66

E1/E2 = 33.33

E1/E2 = 50.00

E1/E2 = 66.66

Interfacebetweenlayers1&

2

Figure 8. Relationships between the Correction f and Depth (Z) for the system with (h1 = 10 in.)

y = -5E-05x3 + 0.0024x2 - 0.0146x + 0.7033R = 0.995

y = 0.0001x3 + 0.0003x2 - 0.0074x + 0.4241R = 0.9975

y = 0.0003x3 - 0.0028x2 + 0.0075x + 0.3256R = 0.9989

y = 0.0004x3 - 0.005x2 + 0.0185x + 0.2728R = 0.9985

y = 0.0005x3 - 0.0066x2 + 0.0271x + 0.2375R = 0.9973

y = 0.0001x2 - 0.0096x + 0.9712R = 1

y = 0.0005x2 - 0.0323x + 1.17

R = 1

y = 0.0007x2 - 0.0432x + 1.2841R = 1

y = 0.0008x2 - 0.0497x + 1.3542R = 1

y = 0.0009x2 - 0.0539x + 1.3996R = 1

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 5 10 15 20

CorrectionFactor(f)

Depth (Z), in.

Interface BetweenLayers 1&2

E1/E2 = 3.33

E1/E2 = 16.66

E1/E2 = 33.33

E1/E2 = 50.00

E1/E2 = 66.66

Figure 9. Relationships between the Correction f and Depth (Z) for the system with (h1 = 15 in)




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y = 0.0012x3 + 0.0225x2 - 0.1252x + 0.6244R = 0.9992

0.00

0.100.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 1 2 3 4 5 6

Corre

tionfactor(f)forlayer-1

Depth, Z (in)

Figure 10. Relationship between Depth and Correction Factor for Layer-1

(h1=6 in., E1/E2=50)

y = 0.0068x2 - 0.1653x + 1.5915R = 0.9997

0.00

0.10

0.20

0.30

0.40

0.500.60

0.70

0.80

0.90

1.00

5 6 7 8 9 10 11Corretionfactor

forlayer-2

Depth, Z (in.)

Figure 11. Relationship between Depth and Correction Factor for Layer-2

(h1=6 in., E1/E2=50)

Using the developed relationships between the correction factor

and depth for each layer, an excellent agreement betweenvertical stresses computed at different depths underneath the

centerline of the load for the transformed system and the two-

layer system is achieved. This is shown in Fig. 12.

0

25

50

75

100

125

150

0 25 50 75 100 125 150

Equiva

lentTransformedSection

C

alculatedStress,

psi

Two-Layer Calculated Stress using KENPAVE, psi

Line of Equality

Figure 12. Comparison of the Two-Layer and the Equivalent One-Layer

Pavements Computed Stresses after Appling the Developed Correction

Factors (fas a Function of Depth).

Unfortunately, normalizing the depth values (Z) by thethickness of the upper layer did not eliminate the effect of layerthickness. However, there seems to be a general relationshipthat relates the

fvalue for each layer, in a two-layer system, to

the modular ration and the depth.

Figures 7 thru 9 show also that for the cases with modularratios higher than 3.33 the value offasymptotes to 0.85, 0.8,and 0.79 for the pavements systems with h1 = 6, 10, and 15 inrespectively. For the 3.33 modular ratio, thefat the interface is0.89, 0.87, and 0.85 for the systems with h1 = 6, 10, and 15 inrespectively. It can be concluded from these results that, in atwo-layer system, for different modular ratios,fin the range of0.8 to 0.9 yields vertical stresses that are relatively close to theones from theory of elasticity at the interface between the twolayers.

Fig. 13 presents an example of the relationship between thevertical stresses calculated at different radial distances, fordifferent depth values, for the two-layer system with (h1 = 10in) and a modular ratio of 16.67. Theses vertical stresses werecalculated for a two-layer problem using KENPAVE. Thistwo-layer system was then transformed using Odemarksmethod with the correction factorfas a function of depth andthe vertical stresses at the transformed depths (Zt) werecalculated. This is shown on Fig. 14. Comparing the values ofthe vertical stresses from both methods resulted in agreementas shown in Fig. 15.

0

10

20

30

40

50

60

70

80

90

100

110

120

-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

VerticalStress,psi

Z = 1 in

Z = 3 in

Z = 5 in

Z = 7 in

Z = 9 in

Leff @ Z = 3 in

Leff @ Z = 9 in

Leff @ Z = 7 in

Leff @ Z = 5 in

Leff @ Z = 1 in11.772 in

27.772 in

23.772 in

19.772 in

15.772 in

Figure 13. Relationship between Vertical Stresses and Radial Distances at

Different Depths (E1/E2 = 16.67), Two-Layer Solution

0

10

20

30

40

50

60

70

80

90

100

110

120

-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

VerticalStress,psi

Zt = 1.24 in

Zt = 3.47 in

Zt = 6.15 in

Zt = 10.11 in

Zt = 17.27 in

Leff @ Zt = 3.47 in

Leff @ Zt = 17.27 in

Leff @ Zt = 10.11 in

Leff @ Zt = 6.15 in

Leff @ Zt = 1.24 in 12.252 in

44.312 in

29.992 in

22.072 in

16.712 in

Figure 14. Relationship between Vertical Stresses and Radial Distances at

Different Depths (E1/E2 = 16.67), Transformed Section




6/6

-20

0

20

40

60

80

100

120

140

-20 0 20 40 60 80 100 120 140

VerticalStress,psi(O

demarkOne-Layer

Solution)

Vertical Stress, psi (Two-Layer Solution)

Z = 1Z = 3Z = 5Z = 7Z = 9

Figure 15. Comparison between Odemark Solution and Two-Layer Solution

for Vertical Stresses at Different Depths and Radial Distances using

Correction Factor (fas a Function of Depth) for the system with h1 =10 in. andE1/E2 = 16.67

V. CONCLUSIONSBased on the analyses conducted in this research, the

following conclusions were highlighted:

In order to get a good agreement between the stressesand strains calculated using Odemarks concept andthose from theory of elasticity; a correction factorfhasto be introduced. This correction factor was found to

be a function of the layer thickness, depth and modularratio.

The study showed a good agreement between thevertical stresses at the interface between the two layers,in a two-layer system, calculated using the theory ofelasticity and Odemarks concept when using acorrection factor (f) in the range of of 0.8 to 0.9 whichagrees with the other literature studies.

However, at any other depth within each layer, thiscorrection factor is not a constant value. It was foundthat this correction factor varies with the change in thedepth of interest. The study showed that, the points (Zvalues) in the first layer follow a 3

rddegree polynomial

relationship with the correction factor (f) for eachmodular ratio and thickness. On the other hand, the

points (Z values) in the second layer follow a 2rd

degree polynomial relationship with the correctionfactor for each modular ratio and thickness.

Unfortunately, normalizing the depth values (Z) by thethickness of the upper layer did not eliminate the effectof layer thickness.

MEPDG should consider introducing a correctionfactor as a function of depth, layer thickness, andmodular for an accurate calculation of the effectivelength and depth required for E* computations.

REFERENCES

[1] Ullidtz, P., (1987), Pavement Analysis, Development in CivilEngineering, Vol.19, Amsterdam, the Netherlands.

[2] Subagio, B., Cahyanto H., Rachman, A., and Mardiyah, S., Multi-LayerPavement Structural Analysis Using Method of Equivalent Thickness,Case Study: Jakarta-Cikampek Toll Road, Journal of the Eastern AsiaSociety for Transportation Studies, Vol. 6, pp. 55 - 65, 2005.

[3] Senseney, C., and Mooney, M., Characterization of a Two-Layer SoilSystem Using a Lightweight Deflectometer with Radial Sensor,

Transportation Research Record, Journal of the Transportation ResearchBoard, 2186, Washington DC, pp. 21-28, 2010.

[4] Crowder, J., Shalaby, A., Cauwenberghe, R., and Clayton, A.,Assessing Spring Load Restrictions Using Climate Change andMechanistic-Empirical Models,

[5] Cafiso, S., and Graziano, A., Evaluation of Flexible ReinforcedPavement Performance by NDT, In Transportation Research Record,TRB Annual Meeting CD ROM, 2003.

[6] ARA, Inc., ERES Consultants Division. Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures.NCHRP 1-37A Final Report, Transportation Research Board, NationalResearch Council, Washington, DC, 2004.

[7] El-Badawy, S., Jeong, M., and El-Basyouny M., Methodology toPredict Alligator Fatigue Cracking Distress based on AC DynamicModulus, In Transportation Research Record, Journal of the

Transportation Research Board, No. 2095, Transportation Researchboard of the National Academies, Washington, DC, 2009, pp. 115-124.

[8] Sotil, A., Use of the Dynamic Modulus E* Test as PermanentDeformation Performance Criteria for Asphalt Pavement Systems,Ph.D. Dissertation. Department of Civil and Environmental Engineering,Arizona State University, Tempe, AZ, December 2005.



17.ijaest vol no 6 issue no 1 assessment and improvement of the accuracy of the odemark...

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