New Approach for Determining Young Modulus and Poisson Coefficient in the Structural Design of the Pavement

New Approach for Determining Young Modulus and Poisson Coefficient in the Structural Design of the Pavement

Messi A. F., Madjadoumbaye J., Miyo T. F.

Civil Engineering Department National Advanced School of Engineering

University of Yaounde I Cameroon

Koumbe Mbock, Okpwe Mbarga R. African Center of Excellence in Information and Communication Technologies

University of Yaounde I Cameroon

Abstract The French design method integrate two entry parameters that use the Young modulus and Poisson coefficient often useful for pavement structure. The measurements of these parameters are costly in many tropical countries and the change in materials design can change considerably the prescription obtained from the French method. To design the pavement we present an approach that consists of determining an interval of Young modulus and Poisson coefficient in order to deal with the prescribed value and the change in pavement materials in Cameroon. The usefulness of such interval is demonstrated with local data that make use of the French method.

Keywords Pavement structures; French method; Young modulus; Poisson coefficient

1. INTRODUCTION

   Many workers had treated the problematic of roadway and highway design and this can be referred in the literature [5], [8], [18], [19], [24]. The mechanical parameters of both ways play an important role and the estimation of the Young modulus and the Poisson coefficient used in various guidelines ([1], [2], [9], [8], [17], [20], [21], [23]) remains a problem in tropical countries.

   Under the traffic, certain methods used to design the pavement are applied to model the behavior of the pavement structures. Such modeling permits to build relationship that make use of Young modulus E and Poisson coefficient in the design of the pavement structures. In tropical regions, theses parameters often recommended by the literature can change. For example, in Cameroon where the infrastructures are not adapted to determine the desired value.

   To overcome this limitation, we propose an interval of mechanical values E and that take in account the recommended values and give the possibility to design the pavement structures accuratively. To validate this approach, we have used the data found in [3], [11], [12], [13] [14], [15] to characterize the parameters E and useful to obtain the interval of parameter values.

   The section I is the introduction and section II show the procedure accorded to the determination of intervals of parameter values including the materials of the study. In section III, we present these intervals as the main results and

   conclude this work in section IV with some important perspectives.

2. PROPOSED APPROACH

   1. Input parameters

      For this study, we used materials to achieve our goal; namely 18 pavement structures for 3 different types:

      • Type 1 : flexible pavement (bituminous pavement) (Number 1 to 6) ;

      • Type 2 : flexible pavement (bituminous concrete denoted by BC) (Number 7 to 12) ;

      • Type 3: flexible pavement (seal coat denoted by SC) (Number 13 to 18).

        The design of these pavements is done with the help of the software ALIZE 3 taking into account the recommended values E and of [22] as it is showed in table I.

        TABLE I. YOUNG MODULUS AND POISSON COEFFICIENT

        Materials
        	BCa	BGb	CG/Ci/Poc	NLGd
        Ecal (MPa)	2 450	3 500	400	150
        cal	0, 35	0,35	0,35	0,35

        a. BC. : Bituminous concrete.

        b. BG: Bitumen gravel.

        c. CG: Crushed grave Ci : Cinder Po: Pouzzolan.

        d. NLG: Natural laterite gravel.

        Other parameters used in the French method are important in the pavement design namely the traffic and the subgrade parameter. Having the materials defined in the table I above, the additional parameters are showed in the tables II and III below.

        Layer
        	Category	E (MPa)
        Subgrade	S1	25	0,35
        	S2	50	0,35
        	S3	75	0,35
        	S4	150	0,35
        	S5	300	0,35

        Layer
        	Category	E (MPa)
        Subgrade	S1	25	0,35
        	S2	50	0,35
        	S3	75	0,35
        	S4	150	0,35
        	S5	300	0,35

        TABLE II. MECHANICAL CHARACTERISTIC OF THE SUBGRADE [3]

        In this Table, we denote by S1,, S5 the subgrade material with given E and .

        TABLE III. TRAFIC CLASSES DEFINED IN TROPICAL COUNTRIES [3]

        EN a
        	Traffic class	Equivalent number of vehicle per day
        < 5×105	T1	< 300
        From 5×105 to 1.5×106	T2	300 to 1 000
        From 1.5×106 to 4×106	T3	1 000 to 3 000
        From 4×106 to 107	T4	3 000 to 6 000
        From 107 to 2×107	T5	6 000 to 12 000

        a. EN: Equivalent number of axles.

   2. States of stress and strain

      In continuum mechanics, the states of stress and strain at a point in cylindric coordinates are determined by:

      • r, t, z : normal stresses;

      • tz, rz, rt : shear stresses ;

      • r, t, z : linear strains;

      • tz, rz, rt : angular strains.

        With known Young modulus and Poisson coefficient given by the formulas:

        Ei = µi.(3.i + µi)/(i + µi) (1)

        And

        i = i/(2.(i + µi)) (2)

        We determine the Lame coefficients µ and . According to the model prescribed by Burmister [16], we have the following graphic:

        Fig. 1. State of stress at a point in cylindric coordinates

        The calculus made with the software ALIZE 3 provides all the information related to the stress and strain tensors for each pavement structure. From these informations, the

        following values are chosen and compared to admissible values in order to design the pavement structure.

        TABLE IV. NATURE OF MAXIMUM STRESS OR STRAIN TO CONSIDERATE [16]

        Material
        	Stress or strain (max)
        Hydrocarbon materials	t
        Concrete and hydraulic binder treated materials	t
        Untreated soil and materials	z

        We note that the admissible values are found in [3].

        TABLE V. ADMISSIBLE DEFLECTIONS IN FUNCTION OF TRAFIC CLASSES [3]

        Traffic class

        Admissible deflections (in 1/100 mm)

        T1

        125

        T2

        90

        T3

        65

        T4

        40

        T5

        35

   3. Description of the procedure

   To determine the interval of value of the parameter E and

   , we followed the steps below:

   • The pavement is designed firstly with the recommended values Ecal and cal;

   • The perturbation of both values does not change the structure of the pavement and the mechanism used to find the interval of admissible parameter values is done with the help of ALIZE 3.

   A way is given to study the sensitivity of the parameter and some operations are done as it is followed:

   X = |Xmax – Xmin| (3)

   SX = X/(2.Xcal) (4)

   Where Xmax and Xmin are respectively the upper bound and the lower bound value of the proposed interval and Xcal is the recommended value.

3. PRESENTATION AND ANALYSIS OF RESULTS

   1. Case of the design of pavement n°1

      In this case, we have the input data:

      • Traffic: Class II

      • Pavement materials and recommended parameters (Ecal and cal) (Table VI)

      • Subgrade (soil) mechanical characteristics (E and ) (Table VI)

        TABLE VI. MECHANICAL CHARACTERISTICS OF THE PAVEMENT N°1

        Layer
        	Material nature/ Category	Ecal (MPa)	cal
        Surface course	BC	2 450	0,35
        Base course	GB	3 500	0,35
        Subgrade	S3	75	0,35

        With these entries, the software ALIZE 3 gives us the following thicknesses:

      • Thickness of the surface course: 5 cm;

      • Thickness of the base course: 32 cm.

        By perturbing the mechanical parameters Ecal and cal, we obtain an interval of admissible values that does not modify the structure of the pavement as we see in the following figures:

        Fig. 2. Thickness of the surface course in function of the Young modulus of the bituminous concrete of the pavement n°1

        Fig. 3. Thickness of the surface course in function of the Poisson coefficient of the bituminous concrete of the pavement n°1

        Fig. 4. Thickness of the base course in function of the Young modulus of the bitumen gravel of the pavement n°1

        Fig. 5. Thickness of the base course in function of the Poisson coefficient of the bitumen gravel of the pavement n°1

        a) Intervals of mechanical parameters E and

        According to the figures 2 to 5, we obtain specific intervals of values that correspond to a specific pavement structure

        TABLE VII. INTERVAL OF VALUES FOR MATERIAL USED IN THE PAVEMENT N°1

        Layer
        	Material nature	Thickness (cm)	Ecal (MPa)	L.E.Va (MPa)
        Surface course	BC	5	2 450	[1 950, 3 250]
        Base course	BG	31	3 500	[3 450, 3 600]

        Layer
        	SE (%)	cal	L..Vb	S (%)
        Surface course	26.5	0.35	[0.29, 0.43]	20
        Base course	2.1	0.35	[0.29, 0.41]	17.1

        a. I.E.V : Interval of E values.

        b. I..V : Interval of values.

   2. Application on a finite number of pavement structures

      We applied the procedure described in the previous section to design 18 pavement structures as it is show in the next table.

      TABLE VIII. DESIGN OF THE 18 STRUCTURES PAVEMENT

      Type of Pave ment	Numbered pavement
      		Surface course		Base course
      		Material	Thickness (cm)	Material	Thickness (cm)
      1	1	BC	5	BG	31
      	2	BC	5	BG	27
      	3	BC	6	BG	36
      	4	BC	5	BG	32
      	5	BC	7	BG	40
      	6	BC	6	BG	36
      2	7	BC	4	CG/ Ci/ Po	21
      	8	BC	4	CG/ Ci/ Po	11
      	9	BC	4	CG/ Ci/ Po	39
      	10	BC	4	CG/ Ci/ Po	23
      	11	BC	4	CG/ Ci/ Po	47
      	12	BC	4	CG/ Ci/ Po	33
      3a	13	SC	2	CG/ Ci/ Po	26
      	14	SC	2	CG/ Ci/ Po	12
      	15	SC	3	CG/ Ci/ Po	33
      	16	SC	3	CG/ Ci/ Po	18
      	17	SC	3	CG/ Ci/ Po	47
      	18	SC	3	CG/ Ci/ Po	34

      a. The seal coat does not take part in the structural design of the

      pavement.

      Type of pavement	Numbered pavement
      		Subbase		Subgrade		Trafic class
      		Matériau	Thickness (cm)	Category	Thickness (cm)
      1	1	/	/	S3		T2
      	2	/	/	S4		T2
      	3	/	/	S3		T3
      	4	/	/	S4		T3
      	5	/	/	S3		T4
      	6	/	/	S4		T4
      2	7	NLG	25	S3		T2
      	8	NLG	20	S4		T2
      	9	NLG		24	S3		T3
      	10	NLG	24	S4		T3
      	11	NLG	25	S3		T4
      	12	NLG	21	S4		T4
      3	13	NLG	19	S3		T1
      	14	NLG	19	S4		T1
      	15	NLG	20	S3		T2
      	16	NLG	20	S4		T2
      	17	NLG	25	S3		T3
      	18	NLG	22	S4		T3

      Numbered pavement
      	cal	I. .V		S (%)
      1	0.35	[0.29, 0.41]	0.12	17
      2	0.35	[0.00, 0.43]	0.43	61.4
      3	0.35	[0.27, 0.41]	0.14	20
      4	0.35	[0.33, 0.47]	0.14	20
      5	0.35	[0.25, 0.37]	0.12	17
      6	0.35	[0.27, 0.43]	0.16	22.8
      Average				26.4

      Numbered pavement
      	cal	I. .V		S (%)
      1	0.35	[0.29, 0.41]	0.12	17
      2	0.35	[0.00, 0.43]	0.43	61.4
      3	0.35	[0.27, 0.41]	0.14	20
      4	0.35	[0.33, 0.47]	0.14	20
      5	0.35	[0.25, 0.37]	0.12	17
      6	0.35	[0.27, 0.43]	0.16	22.8
      Average				26.4

      The next tables show the interval of mechanical parameters values for the 18 pavement structures including recommended values.

      TABLE IX. INTERVAL OF VALUES FOR BITUMINOUS CONCRETE USED IN THE SURFACE COURSE

      Numbered pavement
      	Ecal (MPa)	I.E.V (MPa)	E (MPa)	SE (%)
      1	2450	[1950, 3250]	1300	26.5
      2	2450	[1950, 3050]	1100	22.4
      3	2450	[1850, 2850]	1000	20.4
      4	2450	[2350, 4250]	1900	38.7
      5	2450	[1850, 2550]	700	14.2
      6	2450	[2050, 3150]	1100	22.4
      7	2450	[1850, 2850]	1000	20.4
      8	2450	[1650, 3650]	2000	40.8
      9	2450	[1750, 2550]	800	16.3
      10	2450	[1850, 3250]	1400	28.5
      11	2450	[1750, 2550]	800	16.3
      12	2450	[1550, 2550]	1000	20.4
      Average				24

      Numbered pavement
      	cal	I. .V		S (%)
      1	0.35	[0.31, 0.43]	0.14	20
      2	0.35	[0.27, 0.41]	0.14	20
      3	0.35	[0.27, 0.39]	0.12	17
      4	0.35	[0.33, 0.47]	0.14	20
      5	0.35	[0.25, 0.37]	0.12	17
      6	0.35	[0.31, 0.41]	0.1	14.3
      7	0.35	[0.05, 0.43]	0.38	54.3
      8	0.35	[0.05, 0.49]	0.44	63
      9	0.35	]0.00, 0.37]	0.37	53
      10	0.35	[0.19, 0.45]	0.26	37.9
      11	0.35	]0.00, 0.37]	0.37	53
      12	0.35	]0.00, 0.37]	0.37	53
      Average				35

      TABLE X. INTERVAL OF VALUES FOR BITUMEN GRAVEL USED IN THE BASE COURSE

      Numbered pavement
      	Ecal (MPa)	I.E.V (MPa)	E (MPa)	SE (%)
      1	3500	[3450, 3600]	150	2
      2	3500	[3400, 3600]	200	2.8
      3	3500	[3400, 3550]	150	2
      4	3500	[3450, 3700]	250	3.6
      5	3500	[3400, 3550]	150	2
      6	3500	[3450, 3600]	150	2
      Average				2.4

      TABLE XI. INTERVAL OF VALUES FOR CRUSHED GRAVEL/ CINDER/ POUZZOLAN USED IN THE BASE COURSE

      Numbered pavement
      	Ecal (MPa)	I.E.V (MPa)	E (MPa)	SE (%)
      7	400	[370, 420]	50	6.2
      8	400	[330, 460]	130	16.2
      9	400	[350, 410]	60	7.5
      10	400	[370, 430]	60	7.5
      11	400	[340, 410]	70	8.7
      12	400	[330, 410]	80	10
      13	400	[390, 500]	110	13.7
      14	400	[360, 470]	110	13.7
      15	400	[350, 410]	60	7.5
      16	400	[330, 450]	120	15
      17	400	[350, 460]	110	13.7
      18	400	[320, 410]	90	11.2
      Average				11

      Numbered pavement
      	cal	I. .V		S (%)
      7	0.35	[0.29, 0.45]	0.16	22.8
      8	0.35	[0.25, 0.45]	0.2	28.6
      9	0.35	[0.33, 0.5[	0.17	24.3
      10	0.35	[0.29, 0.39]	0.1	14.3
      11	0.35	[0.31, 0.5[	0.19	27.1
      12	0.35	[0.33, 0.5[	0.17	24.3
      13	0.35	[0.33, 0.43]	0.1	14.3
      14	0.35	[0.29, 0.43]	0.14	20
      15	0.35	[0.29, 0.37]	0.08	11.5
      16	0.35	[0.31, 0.37]	0.06	8.6
      17	0.35	[0.33, 0.37]	0.04	5.7
      18	0.35	[0.29, 0.37]	0.08	11.5
      Average				17.8

      Numbered pavement
      	Ecal (MPa)	I.E.V (MPa)	E (MPa)	SE (%)
      7	150	[145, 180]	35	11.6
      8	150	[130, 195]	65	21.6
      9	150	[145, 205]	60	20
      10	150	[105, 155]	40	13.3
      11	150	[120, 170]	50	16.6
      12	150	[130, 185]	55	18.3
      13	150	[145, 180]	35	11.6
      14	150	[110, 220]	110	36.6
      15	150	[120, 170]	50	16.6
      16	150	[120, 190]	70	23.3
      17	150	[120, 175]	55	18.3
      18	150	[110, 165]	55	18.3
      Average				18.8

      Numbered pavement
      	Ecal (MPa)	I.E.V (MPa)	E (MPa)	SE (%)
      7	150	[145, 180]	35	11.6
      8	150	[130, 195]	65	21.6
      9	150	[145, 205]	60	20
      10	150	[105, 155]	40	13.3
      11	150	[120, 170]	50	16.6
      12	150	[130, 185]	55	18.3
      13	150	[145, 180]	35	11.6
      14	150	[110, 220]	110	36.6
      15	150	[120, 170]	50	16.6
      16	150	[120, 190]	70	23.3
      17	150	[120, 175]	55	18.3
      18	150	[110, 165]	55	18.3
      Average				18.8

      TABLE XII. INTERVAL OF VALUES FOR NATURAL LATERITE USED IN THE SUBBASE

      Numbered pavement
      	cal	I. .V		S (%)
      7	0.35	[0.23, 0.37]	0.14	20
      8	0.35	[0.31, 0.37]	0.06	8.5
      9	0.35	[0.00, 0.37]	0.37	52.8
      10	0.35	[0.29, 0.37]	0.08	11.4
      11	0.35	[0.15, 0.50]	0.35	50
      12	0.35	[0.33, 0.39]	0.06	8.5
      13	0.35	[0.17, 0.37]	0.2	28.5
      14	0.35	[0.17, 0.37]	0.2	28.5
      15	0.35	[0.19, 0.41]	0.22	31.4
      16	0.35	[0.33, 0.27]	0.06	8.5
      17	0.35	[0.19, 0.41]	0.22	31.4
      18	0.35	[0.31, 0.37]	0.06	8.5
      Average				24

   3. Summary of the desired sensitivity

   The small change of the parameters E and provide with the help of ALIZE 3 a good level of sensitivity of the Young modulus taken between 14% and 41% in the case of a bituminous concrete in surface course while the sensitivity of the Poisson coefficient can be taken between 17% and 54%.

   We observe also that the sensitivity of change from 17% to 61% in the case of the bituminous gravel in base course while it decrease from 9% to 53% in the case of laterite gravel in subbase. Another observation is made for the sensitivity of E between 1% and 37% for the natural laterite gravel in subbase.

   TABLE XIII. SYNTHESIS OF THE SENSITIVITY OF MECHANICAL PARAMETERS E AND

   Layer
   	Material	SE (min, %)	SE (max, %)	S (min, %)	S (max, %)
   Surface course	BC	14	41	17	54
   Base course	CG/ Ci/ Po	8	16	6	29
   	BG	2	4	17	61
   Subbase	NLG	12	37	9	53

   These levels of sensitivity are useful in particular for the countries where the infrastructure to design the pavement fails and give an alternative to recommended design parameters.

4. CONCLUSION

This work has been based on the fact that determining an interval of mechanical values E and permits to decide on the desired design of the pavement in tropical regions, for example in Cameroon. Instead of using the recommended values, these intervals open a large zone of application that can involve different kind of materials. This approach opens a new perspective in the design of the pavement structures in tropical countries.

ACKNOWLEDGMENT

I would like to thank the Center of Excellence in Information of Communication Technology at the University of Yaounde I for their support and collaboration.

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