Highway Research Bulletin

YEAR 2004-2005
Bulletin No. 73

NONLINEAR ANALYSIS OF MULTILAYERED FLEXIBLE PAVEMENT BY finite element method
By A.U. Ravi Shankar*, P.C. Naveena** & Krishnamoorthy***


ABSTRACT

The use of Finite Element Method (FEM) in the analysis of pavements, especially flexible pavements, which are multilayered, is gaining much importance in recent years. The technique has proved to be a useful tool in the pavement design. Most of the methods adopted by practicing engineers are empirical. The theoretical analysis based on material properties, stresses and strains will definitely suffice the empirical method and it will result in economical and scientific solutions. Therefore, in the present study, an analytical work is carried out to determine the stresses and deflections under the centre line of loading and also away from the centre line of loading. The pavement layers considered in the analysis are (i) subgrade, (ii) Water Bound Macadam (WBM) and (in) bituminous concrete. The behaviour of the flexible three layered pavement model is analysed by calculating the stresses and deflections by non-linear finite element technique using a computer program written in ‘C’ language in ‘Linux’ operating system The pavement model considered for the FEM analysis consists of 1416 four noded quadrilateral elements and 1533 nodes. The loading material is descritised into 16 numbers of elements. By varying the different properties of materials suitably, the stresses and deflections are observed and the results obtained are compared with those obtained from the three layered theory and linear FEM technique. A pavement model of higher thickness consisting of 1816 elements with 1836 four noded quadrilateral elements is analysed using NISA (Numerically Integrated Elements For System Analysis). The reduction in the pavement thickness is obtained by comparing the results of NISA with that of FEM analysis used for earlier pavement model.
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* Asst. Prof. Dept. of Civil Engg, National Institute of Technology Kanataka, Surathkal P.O. Srinivasnagar -575025
** Lecturer, Dept. of Civil Engg., MSRIT, Bangalore
*** Professor, Dept. of Civil Engg, Manipal Institute of Technology, Manipal -576119
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1. INTRODUCTION
The techniques of designing and constructing flexible asphalt pavements have changed rather significantly in the past several years. Pavements could be easily classified into one of two categories, namely “flexible” or “rigid”. Flexible pavements were (and still are) classified by a pavement structure having a relatively thin asphalt-wearing course with layers of granular base and sub base being used to protect the subgrade from being overstressed. This type of pavement design was primarily based upon empiricism of experience with theory playing only a subordinate role in the procedure.

Flexible pavement layers distribute the wheel loads at the top of the pavement strucure to the bottom by grain-to-grain transfer through the points of contact in the granular structure. A rational method of pavement design requires a through knowledge of the stresses and deflections induced by the imposed traffic loads. Hence, it becomes necessary to compute or measure the stresses and deflections at any point within the flexible pavement layers under wheel load and away from the wheel load (Yoder et al. 1975).

2. OBJECTIVES OF THE PRESENT STUDY
The main objectives of the present study are:

1. To model a suitable pavement layer, which consists of three layers i.e., bituminous concrete layer, WBM layer, and subgrade layer of different thickness.

2. To find the stresses and deflections by FEM linear-analysis and FEM non-linear analysis.

3. Comparision of the results of three-layer theory with FEM linear-analysis and FEM non-linear analysis.

4. To evaluate the reduction in thickness of the pavement by comparing the results obtained from FEM analysis and analysis using NISA software package.

5. To study the variations of deflections and stresses by changing the properties of WBM layer and bituminous concrete surface layer.

3. PRESENT INVESTIGATION
In the present investigation a flexible pavement system consisting of three layers was analysed. The stresses and displacements at salient points are determined by non-linear finite element analysis and are compared with three-layer theory and analysis using NISA software package. The reduction in the thickness of the pavement is determined by comparing the stresses and deflections obtained from FEM analysis and analysis using NISA software by considering the pavement model of different thickness. The pavement structure considered for the analysis consists of subgrade, WBM as base course and bituminous concrete as surface course. The elastic modulus values of WBM, and bituminous concrete are varied and the stresses and displacements at the required locations are determined for all combinations. The normal wheel load is (P) / tyre pressure (p) is applied over a rectangular area of size 1.0m x 0.3m.

Following assumptions are made in the analysis:

1.The component layers of pavement are of finite thickness, finite width but are assumed to be of infinite extent in length direction.

2.The component material in each layer is assumed to be homogeneous and isotropic .

3.The bituminous surface layer and WBM layer are considered as elasto-plastic in stress-strain behavior.

4.The subgrade soil is considered as non-linear in stress-strain behaviour.

5. The pavement is assumed to be symmetric about the axis of loading.

6. Subgrade layer is assumed to be of finite thickness (150mm).

3.1. Pavement Model
A flexible pavement model consists of subgrade soil, WBM as base course and bituminous concrete layer as a surface course. The pavement structure will be analysed by restraining the pavement from both the ends and at the subgrade level. So now the elements at the ends of the pavement and the bottom subgrade layer are restrained to move in X-direction, and the bottom subgrade layer only restrained to move in Y-direction.

1. Loading material (tyre) descritized into 16 elements and 27 nodes of each element 37.5 mm × 75 mm (b × d) each.

2. Layer 1 consists of bituminous concrete, descritized into 200 elements and 303 nodes of each element 37.5 mm × 37.5 mm (b × d) each.

3. Layer 2 consists of WBM descritized into 800 elements and 909 nodes of each element 37.5 mm × 37.5 mm (b × d) each.

4. Layer 3 consists of subgrade layer descritized into 400 elements and 505 nodes of each element 37.5 mm × 37.5 mm (b × d) each.

5. Total depth of the pavement considered for the analysis is 525 mm.

6. Total width of the pavement considered for the analysis is 3750 mm.

3.2. Analysis and Material Properties

Stresses and deflections are calculated for various combinations of material properites. The analysis is carried out in following steps.

1. Selection of material properties such as elastic modulus, poisson’s ratio and thickness values of the three layers of the pavement system considered for the analysis.

2. Determination of the stresses and deflections at each elements and nodes, respectively.

3. Calculation of strains at nodes using single stress strain theory of elasticity.

4. Comparison of the results obtained by FEM linear and non-linear analysis to that of elastic three-layer theory and NISA.

5. Reduction in the thickness of the pavement from the results obtained.

The material properties i.e., elastic modulus and poisson’s ratio for each layer has to be selected for the analysis of stresses and deflections. A range of the elastic moduli and poisson’s ratios values for the suberade, WBM and bituminous concrete are selected from triaxial test results (Mohan 1988) in which tests have been carried out with a confining pressure of 0.15 N/mm2 at a strain rate of 1.25mm/min. The material properties considered for the analysis are tabulated in Table 1.


The analysis is carried out by considering the stress-strain curve as “elasto-plast’c” in behaviour for both the bituminous concrete and WBM material The stress-strain curve of both the materials is as shown in the Fig. 1 The stress-strain curve is considered as linear upto the yield point where the Young’s modulus value becomes very less (» 00005 k-N/m2). The stress corresponding to this yield point is taken as the yield stress for the analysis.


The bituminous concrete elements and WBM elements considered in the FEM analysis did not fail because the stress developed in the element due to application of load is lower than the yield stress considered for the analysis.

The ideal stress-strain curve of subgrade soil is non-linear throughout the curve. Therefore the subgrade soil is considered as non-linear in stress-strain characteristic for the analysis. The Young’s modulus value will change at every point on the curve. Hence to calculate Et (at any stage of loading, non-linear soil model equation (James and Chang 1970) has been adopted for the analysis. The stress-strain curve adopted for the nonlinear soil model is hyperbolic in nature and is as shown in the Fig.2.

3 .3. Determination of Deflections and Stresses
The analysis is carried out, using finite element method for the various input parameters as given in Table 1.

The ‘Non-Linear FEM Package’ is developed by using the FEM program written by Krishnamoorhty (2002). The Non-Linear FEM Program was run for different combinations of elastic moduli and Poisson’s ratios of WBM layer and bituminous concrete layers. The following parameters are estimated



1. Horizontal deflection at each node, Ux
2. Vertical deflection at each node, Vy
3. Horizontal stress in each element, sx
4. Vertical stress in each element, sy
5. Shear stresses in each element, txy

The pavement layers considered for the initial stage of analysis using NISA and FEM are as follows:
1. Two layer of bituminous concrete layer 3 7.5mm thick.
2. Twelve layers of WBM of 37.5mm thick each (Total thickness 450 mm).
3. Four layers of subgrade of 37.5mm thick each (Total thickness 150mm).

The pavement layers considered for the next stage of analysis using three Layer Theory and FEM are as follows:
1. Two layer of bituminous concrete layer 37.5mm thick.
2. Eight layers of WBM of 37.5mm thick each (Total thickness 300 mm).
3. Four layers of subgrade of 37.5mm thick each (Total thickness 150mm).

The results obtained from both initial analysis and next stage analysis is compared. The critical stresses and strains in case of the initial stage analysis using NISA and FEM with 450mm thick WBM layer are not decreasing significantly, when compared to the critical stresses and strains obtained in the next stage analysis using three layer theory and FEM with 300mm thick WBM layer. Hence, the thickness of the WBM layer is decreased from 450 mm to 300 mm. to optimize the design of the pavement. The pavement of total thickness 525 mm with 300 mm thick WBM layer is considered for the future analysis.

3.4. Proposed Method of Analysis
The finite element method is essentially a process by which a continuum with finite degree of freedom can be approximated by an assemblage of sub regions (or elements), each having a finite number of unknowns. Each element interconnects with other through the element boundary nodes. Thus any continuum can be descritized into a finite number of regions. Since this descritization process can fit geometry of any complexity, it has become the most powerful and popular numerical technique for analyzing problem of structural and continuum mechanics. Turner, Clough, Martin and Top introduced the concept in 1956 (Desai and Abel 1987) for structural analysis. Since, then various researchers have made much progress and its application embraces variety that the same general technique is applicable in analyzing the deflection and stresses in any type of continuum and both loading and boundary conditions may be arbitrary.

The following six steps summarize the procedure of finite element analysis:

(i) Descritization of the Continuum: The continuum is the physical body, structure, of solid being analyzed. Discretization may be simply described as the process in which the given body is given subdivided into an equivalent system of finite number of elements.

(ii) Selection of the Displacement Models: The assumed displacement functions of models represent only approximately the actual distribution of the displacements. Obviously, it is generally not possible to select a displacement function that can represent actual variation of displacement in the element. Hence the basic approximation of finite element method is introduced at this stage.

(iii) Derivation of the Element Stiffness Matrix: The stiffness matrix consists of the coefficients of the equilibrium equations derived from the material and geometric properties of an element and obtained by the use of principal of minimum potential energy. The stiffness matrix relates the displacements at the nodal points (the nodal displacements) with the applied forces at the nodal points (the nodal forces). The distributed forces applied to the structure are converted into equivalent concentrated forces at the nodes. The equilibrium relation between the stiffness matrix [K], the nodal forces vector {Q} and the nodal displacement vector displacement vector {d} is expressed as the set of simultaneous linear algebraic equations in the form of

[K] {d}={Q} . . . Equn. 1

(iv) Assembly of the Equations for the Overall Discretization Continuum: This process includes the assembly of overall or global stiffness matrix for the entire body from the individual element stiffness matrices, and the overall or global force or load vector from the element nodal force vectors.

(v) Solution for the Unknown Displacements: The algebraic equations assembled in step (iv) are solved for the unknown displacements after imposing the appropriate boundary conditions.

(vi) Computation of the Element Strains and Stresses from the Nodal Displacement: The stresses are proportional to the derivatives of the displacements and in the domain of each element meaningful! values of the required quantities are calculated. This “meaningful values” are calculated at the gauss points for achieving better accuracy. In the present study, four noded iso-parametric plane strain element with two degrees of freedom at each node, has been chosen for the analysis. The element stiffness matrices have been evaluated using gauss quadrature (3x3) numerical integration.

3.4.1. Non-linear Solution Technique Adopted for the Analysis :
The solution of non-linear problems by the finite element method is usually attempted by one of the three basic techniques: incremental or stepwise procedure, iterative or Newton method and step-iterative or mixed procedure.

In the present analysis “incremental or stepwise procedure” has been adopted. The basis of the incremental or stepwise procedure is the subdivision of the load into many small partial loads or increments. Usually these load increments are equal magnitude but in general they need not be equal. The load is applied in one increment at a time, and during the application of each increment the equations are assumed to be linear. In other words, a fixed value of [k] is assumed throughout each increment but [k] may take different values during different load increments. The solution for each step of loading is obtained as an increment of the displacements {q}. These displacement increments are accumulated to give the total displacement at any stage of loading and the incremental process is repeated until the total load has been reached.

In writing equations for the incremental method, let the initial or reference state of the body be given by the initial loads and displacements {Qo} and {qo}. Usually, {Qo} and {qo} are null vectors because we start from the undeformed state of the body. We can however, specify any initial equilibrium state of {Qo} and {qo}. We divide the total load into M increments, so the effective load is

{Q}-{Qo}+ . . . Equn. 2

where, the notation is used to indicate a finite increment. Hence, after the application of the ith increment, the load is given by

. . . Equn. 3

We adopt a similar notation for the displacements, so that after the ith step the displacements are

. . . Equn. 4

To compute the increment of displacements, we use a fixed value of the stiffness, which is evaluated at the end of the previous increment. The initial value of the stiffness, [k0] is computed from material constants derived from the given stress strain curve at the start of the loading. The incremental procedure is schematically indicated as shown in Fig. 3.



Usually, in the incremental procedure the tangent moduli Et are used to formulate [C()] and to compute the stiffness matrix [k] in the equation. This matrix is often referred to as the tangent stiffness matrix.

The material nonlinearity of soil is modelled using a constitutive model proposed by James and Chang (1970). The principal advantage of linear static analysis is the simplicity. In this approach, only two parameters, namely Young’s modulus (E) and Poisons ratio are needed to characterize the stress-strain behaviour of soil. According to James-Chang model, tangent modulus Et derived from triaxial tests can be expressed as follows :

. . . Equn. 5

Where, i is the stage of incremental loading, then the expression for the tangent modulus is obtained as :

. . . Equn. 6




The computer program consists of a main function and sub-functions. The program of main function and sub-function are depicted in Fig. 4.


4. RESULTS AND DISCUSSION
The finite element analysis (both linear and non-linear) has been carried out for the various cases and results are analysed. The variation of horizontal deflection, vertical deflection, horizontal stress, vertical stress and shear stress under various conditions are studied and discussed. By changing properties of WBM material, the variation of horizontal deflection, vertical deflection, horizontal stress, vertical stress and shear stress below the centre line and at 1.05m (for r/z = 2.0) away from the centre line of loading are analysed and discussed in subsequent topics.

4.1. Below Centre Line of Loading
By changing Youngs modulus value of WBM layer from 7.5 x 104 kN/m2 to 10 x 104 kN/m2, the following observations were made




i) Referring to Table 3 and Fig. 5. it is clear that the horizontal deflections (tensile in nature) are reducing in case of FEM linear analysis and the horizontal deflections (compressive in nature) are reducing in case of FEM non-linear analysis. The horizontal deflections obtained from FEM non-linear analysis are more than the horizontal deflections obtained from FEM-linear analysis for the same material. This is because of the non-linear stress-strain behaviour of the subgrade soil considered for the analysis.

ii) Referring to Table 4 and Fig. 6 it is evident that the vertical deflections (compressive in nature) are reducing both in FEM linear and non-linear analysis. The vertical deflections obtained from FEM non-linear analysis are more than the vertical deflections obtained from FEM-linear analysis for the same material. But the vertical deflections below the W.B.M. layer are reducing drastically.









iii) Referring to Table 5 and Fig. 7 it is clearly observed the horizontal stresses reduced both in FEM linear and non-linear analysis. The horizontal stresses have reached their peak values at the interface of WBM and subgrade layer in case of FEM non-linear analysis.

iv) Referring to Table 6 and Fig. 8 it is clearly observed that the vertical stresses are reducing both in FEM linear analysis and FEM non-linear analysis. The vertical stresses obtained from FEM non-linear analysis are less than the vertical stresses obtained from FEM-linear analysis for the same material. This is because of the re stress distribution due to the non-linear stress-strain behaviour of the subgrade soil considered for the analysis.









v) Referring to Table 7 and Fig. 9 it is clear that the shear stresses are slightly reducing both in FEM linear and non-linear analysis. The shear stresses obtained from FEM non-linear analysis are more than the shear stresses obtained from FEM-linear analysis. This is because of the non-linear stress-strain behaviour of the subgrade soil considered for the analysis.

4.2. At 1.05m (r/z=2) Away From the Centre Line of Loading

By changing E value of WBM layer from 7.5 x 104 kN/m2 tolO x 104 kN/m2, the following observations were made.

i) Referring to Table 8 and Fig. 10, it is clearly observed that the horizontal deflections (tensile in nature) are reducing in case of FEM linear analysis and the horizontal deflections (tensile in nature) are slightly reducing in case of FEM non-linear analysis. But the horizontal deflections have reached their peak values (minimum and maximum) at the interface of pavement layers in case of FEM non-linear analysis.









ii) Referring to Table 9 and Fig. 11 it is clearly observed that the vertical deflections (tensile in nature) are reducing both in FEM linear and non-linear analysis. Replacement of combination 1 to combination 2, reduces the heaving. The vertical deflections obtained from FEM non-linear analysis are more than the vertical deflections obtained from FEM-linear analysis for the same material. This is because of the non-linear stress-strain behaviour of the subgrade soil considered for the analysis.

iii) Referring to Table 10 and Fig. 12 it is clearly observed that the horizontal stresses are increasing both in FEM linear analysis and FEM non-linear analysis. The horizontal stresses obtained from FEM non-linear analysis are more than that obtained from FEM-linear analysis for the same material at the interface of W.B.M. and subgrade layer. This is because of the re stress distribution due to non-linear stress-strain behaviour of the subgrade soil considered for the analysis.





iv) Referring to Table 11 and Fig. 13 it is clearly noticed that the vertical stresses are increasing both in FEM linear and non-linear analysis. The vertical stresses obtained from FEM non-linear analysis are slightly more than the vertical stresses obtained from FEM-linear analysis.









v) Referring to Table 12 and Fig. 14 it is clear that the shear stresses are increasing both in FEM linear and non-linear analysis within middle thickness of the pavement and the shear stresses at top and bottom layers are almost same. The shear stresses obtained from FEM non-linear analysis are more than the shear stresses obtained from the FEM-linear analysis. This is because of the non-linear stress-strain behaviour of the subgrade soil considered for the analysis.

The ‘non-linear finite element analysis’ has been carried out after changing the combination 3 (E value of WBM = 7.5 x 104 kN/m2 and E value of Bituminous concrete = 45 x 104 kN/m2) by combination 4 (E value of WBM= 10 x 104 kN/m2 and E value of Bituminous concrete = 45 x 104 kN/m2), the variation of horizontal deflection, vertical deflection, horizontal stress, vertical stress and shear stress below the centre line and at 1.05m (for r/z = 2.0) away from the centre line of loading are analysed and the observations are as follows:

i) Referring to Table 13 and Fig. 15 it is clearly observed that the horizontal deflections are reducing below the centre line of loading and at 1.05m away from the centre line of loading. The horizontal deflections are changing from tensile to compressive in nature.





ii) Referring to Table 14 and Fig. 16 it is clearly observed that the vertical deflections are reducing below the centre line of loading but at 1.05m away from the centre line of loading; the vertical deflections are almost same. The vertical deflections below the W.B.M. layer are reducing drastically at the centre line of loading.





iii) Referring to Table 15 and Fig. 16 it is clearly observed that the horizontal stresses are reducing below the centre line of loading and the horizontal stresses are increasing at 1.05m away from the centre line of loading.





iv) Referring to Table 16 and Fig. 17 it is clearly observed that the vertical stresses are reducing below the centre line of loading and the vertical stresses are slightly increasing at 1.05m away from the centre line of loading. But there is considerable difference in the vertical stresses obtained at the centre line of loading and at 1.05m away from the centre line of loading.





v) Referring to Table 17 and Fig. 18 it is clearly observed that the shear stresses are reducing below the centre line of loading and the shear stresses are increasing at 1.05m away from the centre line of loading.





4.3. Comparison of the stresses and deflections calculated by finite element method and NISA (Numerically Integrated Elements For System Analysis) software package.


The comparison of deflections and stresses calculated by Finite Element Method (both linear and non-linear analysis) and NISA is given in Table 18. The results obtained from NISA are enclosed in the appendix. The NISA model consists of mesh in which four noded quadrilateral elements are considered. The linear analysis is carried out using NISA. The results of NISA are nearly matching the FEM analysis results. The analysis is carried out in both FEM and NISA by considering the following material properties of the individual layer of the pavement:

The Young’s moduli of elasticity considered for the analysis are 150N/mm2, 75N/mm2 and 30N/mm2 for bituminous concrete, WBM and subgrade soil, respectively. The poisson’s ratios considered for the analysis are 0.35, 0.40 and 0.40, respectively. Bituminous surface layer of thickness 75mm, WBM layer of thickness 450mm and subgrade of thickness 150mm are considered for the analysis. The Young’s modulus of elasticity of 2.45x104 N/mm.2 and poisson’s ratio of 0.3 is considered for the loading material to make it ‘rigid’. The same elastic moduli and Poisson’s ratios are used for the analysis of stresses and deflections using three-layer theory.



In three-layer theory the stresses and deflections are calculated only at the interface of the pavement layers, whereas, in case of FEM method and using NISA package, the deflections and stresses can be calculated throughout the pavement layers. Therefore, the FEM method of analysis and analysis using NISA are more scientific.

There is some difference in values of stresses and deflections obtained from FEM linear analysis and FEM non-linear analysis. This is because in FEM non-linear analysis the subgrade soil is considered as non-linear and bituminous surface layer and WBM layer are considered as elasto-plastic in stress strain behavior.

4.4. Comparison of stresses and deflections calculated by Finite Element Method and Three Layer Theory
The comparison of deflections and stresses calculated by Finite Element Method (both linear and non-linear analysis) and Three Layer Theory is given in Table 19. The material properties and thickness of different layers of the three layered flexible pavement considered for the analysis are as follows:


The elastic moduli and poisson’s ratios of the individual layers in the pavement system are same as considered in the analysis using NISA. The thickness of WBM layer is reduced from 450mm to 300mm and the thickness of other layers are kept same as considered in the analysis using NISA.
By comparing the stresses and strains given in Table 18 and Table 19, it is clear that the critical stresses and critical strains are not decreasing significantly in case of analysis using NISA with 450mm thick W.B.M. Hence, it is decided to reduce the thickness of


WBM from 450mm to 300m. m for optimizing the design of pavement. The results of three-layer theory are nearly matching the FEM analysis results. The variations are not high. In FEM method, the poisson’s ratio can not be considered as 0.5, because it is observed that for Poisson’s ratio greater than 0.45, the results obtained by FEM will not be in conformity with those obtained from the three layer theory. The boundary condition introduced in the FEM analysis may also slightly affect the stress or strain in the pavement, where as, in actual pavement is of finite thickness and infinite in lateral direction. In case of three-layer theory the Poisson’s ratio is constant and is taken as 0.5. Because of the above reasons, there is a small discrepancy (5 to 8 per cent) in the values of stresses and deflections obtained from FEM analysis and three-layer theory analysis.

5. CONCLUSIONS
The following conclusions are drawn after running the FEM program with different material compositions.
(i) When Young’s modulus of WBM material is changed from 7.5 x 104 kN/m2 to 10x104 kN/m2, horizontal deflections and vertical deflections are decreasing in case of FEM non-linear analysis. Decrease in horizontal deflection is about 63 to 9 per cent and decrease in vertical deflection is about 35 to 17 per cent respectively along the centre line of loading. Whereas decrease in horizontal deflection and vertical deflection are about 49 to 6 per cent and 33 to 11 per cent respectively at 1.05m away from the centre line of loading.

(ii)
When Young’s modulus of WBM material is changed from 7.5 x 104 kN/m2 to 10x104 kN/m2, horizontal stresses, vertical stresses and shear stresses are decreasing in case of FEM non-linear analysis. Decrease in horizontal stress is about 42 to 18 per cent decreases in vertical stresses is about 8 to 3 per cent and shear stresses are about 57 to 4 per cent . Whereas horizontal stresses are increasing about 5 to 36 per cent, vertical stresses are increasing about 11 to 72 per cent and shear stresses are increasing about 2 to 10 per cent at 1.05m away from the centre line of the loading.

(iii)
When properties of WBM material changed the horizontal deflections and vertical deflections are decreasing in case of FEM non-linear analysis. Decrease in horizontal deflection is about 73 to 6 per cent and decrease in vertical deflection is about 37 to 16 per cent respectively, along the centre line of loading. Whereas, decrease in horizontal deflection and vertical deflection are about 53 to 15 per cent and 24.45 to 19 per cent respectively at 1.05m away from the centre line of loading.

(iv) When properties of WBM material changed horizontal stresses, vertical stresses and shear stresses are decreasing in case of FEM non-linear analysis. Decrease in horizontal stress is about 42 to 6 per cent decreases in vertical stresses is about 12 to 3 per cent and decrease in shear stresses is about 72 to 9 per cent. Whereas horizontal stresses are increasing about 13 to 48 per cent, vertical stresses are increasing about 33 to 68 per cent and shear stresses are increasing about 2 to 9 per cent at 1.05m away from the centre line of the loading.

(v) Three-layer theory gives higher values of stresses and strains compared to FEM analysis.

(vi) By changing the Young’s modulus of WBM material from 7.5 x 104 kN/m2 to 10x104 kN/m2, a reduction of 150mm has been achieved in the thickness of WBM layer.

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2.
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3. Bunnister, D. M (1943), “ The Theory of Stresses and Displacements in Layered System and Application to the Design of Airport Runways,” H.R.B. proceedings. Vol. 23, pp 123-133. (Extracted from Yoder E.J. and Witzak. M.W.1975).

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