Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 6, Issue No. 1, 105 - 110
Assessment and Improvement of the Accuracy of the Odemark Transformation Method Sherif M. El-Badawy*, Ph.D
Mostafa A. Kamel, Ph.D.
Assistant Professor, Public Works Department Faculty of Engineering, Mansoura University Mansoura, Egypt sbadawy@mans.edu.eg
Assistant Professor, Public Works Department Faculty of Engineering, Mansoura University Mansoura, Egypt Mostafakamel2000@yahoo.com
Abstract— Flexible pavement structures are very complex systems usually consist of multi-layers with each layer having different properties (elastic modulus and Poisson’s 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 method has 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 Odemark’s 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.
For the transformed section shown in Fig. 1 the equivalent thickness “he” can be calculated as follows:
Keywords; Odemark; stress; stiffness; elastic modulus, MEPDG
Comparing stresses and strain calculated using the Odemark’s method with those from the elastic theory led to the conclusion that they are relatively different. In order to achieve a better agreement between Odemark’s method and the elastic theory, a correction factor “f” was applied to the above equation as follows.
INTRODUCTION
T he
h1 3
IJ A
Odemark has developed an approximate method to calculate stresses and strains in multiplayer pavement systems by transforming this structure into an equivalent one-layer system with equivalent thicknesses but one elastic modulus. This concept is known as the method of equivalent thickness (MET) or Odemark’s method. MET assumes that the stresses and strains below a layer depend only on the stiffness of that layer. If the thickness, modulus and Poisson’s ratio of a layer is changed, but the stiffness remains unchanged, the stresses and strains below the layer should also remain (relatively) unchanged. According to Odemark, the stiffness of a layer is proportional to the following term [1]:
(2)
E1 1 u 22 E 2 1 u12
ES
I.
h13 E1 he3 E 2 ; or 1 v12 1 v 22
For the case of a two-layer system with equal Poisson’s ratio, the equivalent thickness can be calculated using the following formula:
he
he
h1 3
E1 E2
f h1 3
(3)
E1 E2
(4)
Researchers reported that the value of the correction factor “f” depends on the layer thicknesses, modular ratios, and the number of layers in the pavement structure. Furthermore, they mentioned that using a value of 0.8 to 0.9 for “f” leads to a reasonably good agreement between the two methods [2].
h3 E
1 v2
where:
h = thickness of the layer E = elastic modulus v = Poisson’s ratio
ISSN: 2230-7818
(1)
h1 E1 E2
1
2
he E2
2
E2
2
Figure 1. Odemark’s Transformation of a Layered System
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Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 6, Issue No. 1, 105 - 110
For a multi-layer system the equivalent thickness of the upper n-1 layers with respect to the modulus of layer n, may be calculated as follows: n 1
f hi 3 i 1
Ei En
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.
fl t vs Leff
Figure 3.
Equivaent Thicnkess Calculation
= frequency of load, Hz = time of load, sec = velocity (mph) = effective length of the stress pulse, inch
IJ A
where:
1 17.6vs fl t Leff
ES
Odemark transformation method has been utilized and implemented in many applications [2], [3], [4], [5], [6], [7], [8]. Subagio et al used this method to calculate the residual life and overlay thickness required based on deflection data measured using the falling weight deflectometer (FWD) [2]. Senseney and Mooney also used this method in the characterization of a two-layer soil system using lightweight deflectometer [3]. The newly developed mechanistic empirical pavement design guide (MEPDG) implemented this method to transform a multi-layer pavement system into an equivalent one layer system. This equivalent system is used to determine the frequency of loading based on the effective length of the stress pulse and vehicle velocity using the following relationship [6].
Figure 2. Effective Length Concept in MEPDG [6]
T
he,n
The effective length concept which has been employed in MEPDG defines the stress pulse at a specific depth within the pavement system as shown in Fig. 2. In this Figure, the line AA shows the length of the stress pulse at the mid-depth of the AC layer, whereas line BB shows the length of the stress pulse in the granular base layer. The sloped lines along with the depth of 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 no present relationship exists to relate them together, a multi-layer pavement system is transformed into an equivalent one layer system in order to estimate the effective length. The transformed section using MET is shown in Fig. 3. The transformed section has the modulus of the subgrade and has an equivalent thickness of he. In MEPDG, for simplicity, the stress distribution for a typical subgrade soil is assumed to be at 45 degree as shown in Fig. 4. Using this stress distribution, the effective length can be calculated at any depth within the transformed pavement system.
ISSN: 2230-7818
Figure 4. Effective Length Calculation using the Transformed Section [6]
In MEPDG, for any pavement layer, the effective length of the stress pulse is computed at the effective depth (Zeff). The effective depth is the transformed depth at which the loading frequency is needed. The effective depth for the transformed section (as shown in Fig. 4) is calculated with the help of Equation. 8 [6]: n 1 E E Z eff hi 3 i hn 3 n E SG E SG i 1
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(8)
The computed loading frequency at the effective depth is then used to determine the complex modulus (E*) of the asphalt layer. Thus the accuracy of the MET affects the accuracy of the E* which in turn influences the MEPDG predicted rutting, load associated cracking and roughness. II.
OBJECTIVES
This paper has two primary objectives. The first objective is to investigate the accuracy of the Odemark transformation method (MET). The second objective of this research is to improve the accuracy of the MET, if warranted.
125 100 75 50
25 Line of Equality
0 0
STUDY METHODOLOGY
9000 lb 120 psi
2a = 9.772 in E1,
150
Figure 6. Comparison of the Two-Layer and the Equivalent One-Layer Pavements Computed Stresses
A correction factor f was then introduced into the equation to calculate the corrected equivalent depth. First, a unique f value was applied to all points of interest for each modular ratio. The results showed good agreement only for the vertical stresses calculated at the interface between the two layers when using f of 0.8 to 0.9. However, at any depth other than the interface between the two layers the results showed a significant difference between the two solutions. This means that the correction factor f is also dependent on the depth. In order to verify that, the “Solver” function in Excel spreadsheet was used for each modular ratio, to calculate the f factor at each depth such that: vzi – vzti = 0
h
(9)
where:
vz
E2,
vzti
IJ A
Figure 5. Two-Layer Pavement System used for the Analysis
First a linear elastic solution was performed on the twolayer structure using the KENPAVE software to calculate the vertical and radial stresses at different depths measured from the surface of the upper layer under the centerline of the load. Then Odemark transformation concept was used to convert the two-layer problem into one layer with equivalent thicknesses and one modulus. A comparison between stresses calculated from both systems was made. The influence of the correction factor term on the computed stresses of the transformed system using MET method was studied. IV.
25 50 75 100 125 Two-Layer Calculated Stress using KENPAVE, psi
ES
An extensive study is introduced to quantify the influence of layer thickness, depth, and modular ratios on the correction factor “f” of the Odemark’s transformation method. A twolayer system with the first layer thickness (h) values of 2, 6, 10, and 15 inches was used in the analysis. A total of 5 different modular ratios of E1/E2 = 3.33, 16.67, 33.33, 50.00, and 66.67 for each thickness were analyzed. A Poisson’s ratio of 0.35 was assumed in all computations. Fig. 5 shows the applied load and the properties of the two layer system used in the analysis.
Z
150
T
III.
Equivalent Transformed Section Calculated Stress, psi
Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 6, Issue No. 1, 105 - 110
ANALYSIS AND RESULTS
Comparing Odemark solution, without using a correction factor (f=1), to KENPAVE solution yielded different stress values at the points of interest. This is clearly shown in Fig. 6. This figure only shows the stresses calculated at different depths underneath the centerline of the load (radial distance =0).
ISSN: 2230-7818
= vertical stress calculated from a two-layer system at depth zi using KENPAVE. = vertical stress calculated from Odemark transformed depth zti (One layer system) using Boussinesq or KENPAVE.
The results showed that f depends not only on the modular ratio 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 the correction factor “f” and depth at different modular ratios for the investigated two layer system with h1 = 6, 10, and 15 inches respectively. For the points (Z values) in the first layer the relationship between f and Z was found to be a 3rd degree polynomial. The values of the R2 were 0.99+ for all investigated modular ratios as well as the different structures considered in the analysis. For the points (Z values) in the second layer the relationship between f and Z was found to be a 2rd degree polynomial. The values of the R2 were also found to be 0.99+ for all different modular ratios and the different structures considered in the analysis. The relationships between f and Z for the two layers are shown in Fig. 9 for the pavement system with h1=15 inch. Examples of these relationships are shown in Fig. 10 and Fig. 11 for the two layer system with h1=6 inch and a modular ratio of 50.
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Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 6, Issue No. 1, 105 - 110
1.00 0.90 E1/E2 = 3.33
0.80
E1/E2 = 10.00 E1/E2 = 16.67 E1/E2 = 33.33 E1/E2 = 50.00 E1/E2 = 66.67
Interface between layers 1 and 2
Correctin Factor, f
E1/E2 = 6.66
0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0
2
4
6
8
10
12
Depth, Z (in)
T
Figure 7. Relationships between the Correction f and Depth (Z) for the system with (h1 = 6 in.) 1.00 0.90
E1/E2 = 3.33
0.70
E1/E2 = 16.66 E1/E2 = 33.33
ES
0.60
Interface between layers 1 & 2
Correction Factor, f
0.80
0.50 0.40 0.30 0.20 0.10 0.00 0
10
E1/E2 = 66.66
Depth, Z (in)
15
20
25
Relationships between the Correction f and Depth (Z) for the system with (h1 = 10 in.)
IJ A
Figure 8.
5
E1/E2 = 50.00
1.00
Correction Factor (f)
0.90
y = -5E-05x3 + 0.0024x2 - 0.0146x + 0.7033 R² = 0.995
0.80 0.70
0.60 0.50
Interface Between Layers 1&2
y = 0.0001x3 + 0.0003x2 - 0.0074x + 0.4241 R² = 0.9975
y = 0.0001x2 - 0.0096x + 0.9712 R² = 1
y = 0.0003x3 - 0.0028x2 + 0.0075x + 0.3256 R² = 0.9989
0.40 0.30 0.20
y = 0.0004x3 - 0.005x2 + 0.0185x + 0.2728 R² = 0.9985
0.10
y = 0.0005x3 - 0.0066x2 + 0.0271x + 0.2375 R² = 0.9973
0.00 0
5
10 Depth (Z), in.
E1/E2 = 3.33
y = 0.0005x2 - 0.0323x + 1.17 R² = 1
E1/E2 = 16.66
y = 0.0007x2 - 0.0432x + 1.2841 R² = 1
E1/E2 = 33.33
y = 0.0008x2 - 0.0497x + 1.3542 R² = 1
E1/E2 = 50.00
y = 0.0009x2 - 0.0539x + 1.3996 R² = 1
E1/E2 = 66.66
15
20
Figure 9. Relationships between the Correction f and Depth (Z) for the system with (h1 = 15 in)
ISSN: 2230-7818
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Corretion factor (f) for layer-1
Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 6, Issue No. 1, 105 - 110
Unfortunately, normalizing the depth values (Z) by the thickness of the upper layer did not eliminate the effect of layer thickness. However, there seems to be a general relationship that relates the f value for each layer, in a two-layer system, to the modular ration and the depth.
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
y = 0.0012x3 + 0.0225x2 - 0.1252x + 0.6244 R² = 0.9992
0
1
2
3 4 Depth, Z (in)
5
6
T
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
Fig. 13 presents an example of the relationship between the vertical stresses calculated at different radial distances, for different depth values, for the two-layer system with (h1 = 10 in) and a modular ratio of 16.67. Theses vertical stresses were calculated for a two-layer problem using KENPAVE. This two-layer system was then transformed using Odemark’s method with the correction factor f as a function of depth and the vertical stresses at the transformed depths (Zt) were calculated. This is shown on Fig. 14. Comparing the values of the vertical stresses from both methods resulted in agreement as shown in Fig. 15.
y = 0.0068x2 - 0.1653x + 1.5915 R² = 0.9997
5
6
7 8 9 Depth, Z (in.)
ES
Corretion factor for layer-2
Figure 10. Relationship between Depth and Correction Factor for Layer-1 (h1=6 in., E1/E2=50)
Figures 7 thru 9 show also that for the cases with modular ratios higher than 3.33 the value of f asymptotes to 0.85, 0.8, and 0.79 for the pavements systems with h1 = 6, 10, and 15 in respectively. For the 3.33 modular ratio, the f at the interface is 0.89, 0.87, and 0.85 for the systems with h1 = 6, 10, and 15 in respectively. It can be concluded from these results that, in a two-layer system, for different modular ratios, f in the range of 0.8 to 0.9 yields vertical stresses that are relatively close to the ones from theory of elasticity at the interface between the two layers.
10
120
11
110 100
80
60
Line of Equality
0
0
25
50
75
100
125
150
Two-Layer Calculated Stress using KENPAVE, psi
Z = 9 in 19.772 in
Leff @ Z = 5 in
50
23.772 in
Leff @ Z = 7 in
40
27.772 in
Leff @ Z = 9 in
10 0 -24
-20
-16
-12
-8
-4
0
4
8
12
16
20
24
Radial Distance, in
Figure 13. Relationship between Vertical Stresses and Radial Distances at Different Depths (E1/E2 = 16.67), Two-Layer Solution 120 110
Zt = 1.24 in
12.252 in
Leff @ Zt = 1.24 in
Zt = 3.47 in
100
75
25
Z = 7 in
70
20
100
50
Z = 5 in
Leff @ Z = 3 in
150 125
Z = 3 in 15.772 in
30
Zt = 6.15 in
Leff @ Zt = 3.47 in
90 Vertical Stress, psi
Equivalent Transformed Section Calculated Stress, psi
IJ A
Using the developed relationships between the correction factor and depth for each layer, an excellent agreement between vertical stresses computed at different depths underneath the centerline of the load for the transformed system and the twolayer system is achieved. This is shown in Fig. 12.
Vertical Stress, psi
Figure 11. Relationship between Depth and Correction Factor for Layer-2 (h1=6 in., E1/E2=50)
Z = 1 in
11.772 in
Leff @ Z = 1 in
90
16.712 in
Zt = 10.11 in
80
Zt = 17.27 in
Leff @ Zt = 6.15 in
70
22.072 in
60 Leff @ Zt = 10.11 in
50
29.992 in
40 Leff @ Zt = 17.27 in
30
44.312 in
20 10 0
Figure 12. Comparison of the Two-Layer and the Equivalent One-Layer Pavements Computed Stresses after Appling the Developed Correction Factors (f as a Function of Depth).
ISSN: 2230-7818
-24
-20
-16
-12
-8
-4
0
4
8
12
16
20
Radial Distance, in
Figure 14. Relationship between Vertical Stresses and Radial Distances at Different Depths (E1/E2 = 16.67), Transformed Section
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Sherif M. El-Badawy et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 6, Issue No. 1, 105 - 110
REFERENCES Vertical Stress, psi (Odemark One- Layer Solution)
[2]
120 100 80
[3]
60 40
[4]
Z=1 Z=3 Z=5 Z=7 Z=9
20 0
[5]
-20 -20
0
20
40
60
80
100
120
140
Vertical Stress, psi (Two-Layer Solution)
Figure 15. Comparison between Odemark Solution and Two-Layer Solution for Vertical Stresses at Different Depths and Radial Distances using Correction Factor (f as a Function of Depth) for the system with h1 =10 in. and E1/E2 = 16.67
V.
CONCLUSIONS
[7]
[8]
ES
Based on the analyses conducted in this research, the following conclusions were highlighted:
[6]
Ullidtz, P., (1987), Pavement Analysis, Development in Civil Engineering, Vol.19, Amsterdam, the Netherlands. Subagio, B., Cahyanto H., Rachman, A., and Mardiyah, S., “Multi-Layer Pavement Structural Analysis Using Method of Equivalent Thickness, Case Study: Jakarta-Cikampek Toll Road,” Journal of the Eastern Asia Society for Transportation Studies, Vol. 6, pp. 55 - 65, 2005. Senseney, C., and Mooney, M., “Characterization of a Two-Layer Soil System Using a Lightweight Deflectometer with Radial Sensor,” Transportation Research Record, Journal of the Transportation Research Board, 2186, Washington DC, pp. 21-28, 2010. Crowder, J., Shalaby, A., Cauwenberghe, R., and Clayton, A., “Assessing Spring Load Restrictions Using Climate Change and Mechanistic-Empirical Models,” Cafiso, S., and Graziano, A., “Evaluation of Flexible Reinforced Pavement Performance by NDT,” In Transportation Research Record, TRB Annual Meeting CD ROM, 2003. ARA, Inc., ERES Consultants Division. “Guide for MechanisticEmpirical Design of New and Rehabilitated Pavement Structures.” NCHRP 1-37A Final Report, Transportation Research Board, National Research Council, Washington, DC, 2004. El-Badawy, S., Jeong, M., and El-Basyouny M., “Methodology to Predict Alligator Fatigue Cracking Distress based on AC Dynamic Modulus,” In Transportation Research Record, Journal of the Transportation Research Board, No. 2095, Transportation Research board of the National Academies, Washington, DC, 2009, pp. 115-124. Sotil, A., “Use of the Dynamic Modulus E* Test as Permanent Deformation Performance Criteria for Asphalt Pavement Systems”, Ph.D. Dissertation. Department of Civil and Environmental Engineering, Arizona State University, Tempe, AZ, December 2005.
T
[1]
140
In order to get a good agreement between the stresses and strains calculated using Odemark’s concept and those from theory of elasticity; a correction factor f has to be introduced. This correction factor was found to be a function of the layer thickness, depth and modular ratio.
The study showed a good agreement between the vertical stresses at the interface between the two layers, in a two-layer system, calculated using the theory of elasticity and Odemark’s concept when using a correction factor (f) in the range of of 0.8 to 0.9 which agrees with the other literature studies.
IJ A
However, at any other depth within each layer, this correction factor is not a constant value. It was found that this correction factor varies with the change in the depth of interest. The study showed that, the points (Z values) in the first layer follow a 3rd degree polynomial relationship with the correction factor (f) for each modular ratio and thickness. On the other hand, the points (Z values) in the second layer follow a 2rd degree polynomial relationship with the correction factor for each modular ratio and thickness.
Unfortunately, normalizing the depth values (Z) by the thickness of the upper layer did not eliminate the effect of layer thickness.
MEPDG should consider introducing a correction factor as a function of depth, layer thickness, and modular for an accurate calculation of the effective length and depth required for E* computations.
ISSN: 2230-7818
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