INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 4, No 3, 2014

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 4, No 3, 2014 Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0 Research article ISSN Numerical simulation of granular materials behaviour for unbound base layers used in Algerian pavement Tiliouine B, Sandjak K Laboratoire de Génie Sismique et de Dynamique des Structures Ecole Nationale Polytechnique, 10 Avenue H. Badi, 16200, Algiers, Algeria boualem.tiliouine@g.enp.edu.dz doi: /ijcser ABSTRACT Structural analysis of flexible pavements has been and still is currently performed using a multi-layer elastic theory. However, for thinly surfaced pavements subjected to light to medium volumes of traffic, the importance of non-linear stress-strain behaviour of unbound granular materials (UGM s) requires the use of more sophisticated numerical models for structural design and analysis of such pavements. In this work, an axisymmetric finite element model coupled with a shear volumetric strain model is utilized to simulate the nonlinear resilient behaviour of two local unbound granular materials used in Algerian pavements. This model is included herein to adequately incorporate the material non-linearity due to the stress and stiffness dependency of the granular layers in flexible pavement analysis. The numerical performance of the finite element model is examined with regards to its capability of representing adequately the granular material non-linearity under various stress conditions including low values of mean normal stress and large values of shear stress ratio. In addition, the sensitivity of pavement design criteria to the likely variations of asphalt layer thickness and mineralogical nature of unbound granular materials typically used in Algerian pavements are assessed and conclusions of engineering significance are formulated. Keywords: Numerical simulation, mechanical behaviour, material non-linearity, unbound granular materials, flexible pavements. 1. Introduction Low to moderate traffic roads represent more than 80% of the road network in Algeria. They are typically made with flexible pavements in which the behaviour of granular base and subbase layers may significantly influence the structural pavement design and hence the overall structural performance of the flexible pavements. The conventional way of designing a flexible pavement structure in road pavement engineering is to assume a constant stiffness over the granular layer thickness or to derive empirically this stiffness from a rigidity ratio depending on the mechanical properties of the subgrade soil. However, experimental evidence shows clearly that the stiffness modulus of an unbound granular layer is a non-linear function of the stress levels acting at the various points of the aggregates. These stress levels are themselves a function of the traffic loading but also of the various stiffnesses of the pavement structure. Because of this material non-linearity and the sensitivity of the main pavement design criteria to the likely variations of the resilient modulus and Poisson's ratio of the granular layers, a numerical simulation of the non-linear mechanical behaviour of unbound granular materials (UGM s) need be developed. Various constitutive models have been developed to this end (eg. Minkwan et al., 2009: Hornych et al., 1998). These resilient models can be categorized into two main classes: Received on January, 2014 Published on March

2 the resilient modulus models (eg. May and Witczak, 1981: Uzan, 1992) and the shearvolumetric strain models (eg. Lekarp and Dawson, 2000: Hoff et al., 2009), A statistical evaluation of the efficiency of resilient models characterizing coarse granular materials commonly encountered in the field of pavement construction can be found in reference (Ekblad, 2008). Of particular interest in the present study is the mechanical Boyce model with three independent parameters (Boyce, 1980: Jouve and Elhannani, 1994: Sandjak and Tiliouine, 2014). This model, frequently used for granular materials in highway engineering, has the advantage to be in agreement with repeated loading triaxial tests and offers a reasonable compromise between accurate modelling and simplicity. Nevertheless, it should be noted that the computer implementation of this model may lead to some convergence problems for high values of the shear stress ratio and appropriate numerical modifications need to be included into the numerical simulation model (Balay and Kabré, 1996: Tiliouine and Sandjak, 2001). In the present work, an axisymmetric finite element model (Zienkiewicz and Taylor, 2005) coupled with a modified Boyce model is used to simulate the non-linear resilient behaviour of two local unbound granular materials used in Algerian pavements. The modified model is included to adequately incorporate the material non-linearity due to the stress and stiffness dependency of the granular layers. The numerical performance of the finite element model is examined with regards to its capability of representing adequately the granular material nonlinearity, under various stress conditions including low values of mean normal stress and large values of shear stress ratio. In addition, the sensitivity of pavement design criteria to the likely variations of asphalt layer thickness and mineralogical nature of unbound granular materials typically used in Algerian pavements are assessed and conclusions of engineering significance are formulated. 2. Analysed pavement and materials In the following, the pavement analysed, the materials used and their mechanical properties are presented in some details. 2.1 Pavement As the aggregate behaviour is known to be crucial in the overall behaviour of flexible pavements, two pavement with various types of granular materials were studied. A typical pavement structure is illustrated in figure1. Details of the two pavement, each comprising 5 m of subgrade soil over a rigid bottom under two thicknesses of unbound granular layers and a relatively thin asphalt layer, are summarized in Table 1. Table 1: Layer thicknesses of pavement 1 and 2. Structure Asphalt (m) Base (m) Subbase (m) Soil (m) S S The represent, practically, thinly surfaced flexible pavements with structurally significant unbound granular layers commonly used as pavements subjected to low to 420

3 medium traffic volumes. The two pavement were analysed by varying the mechanical properties of the various unbound granular materials while keeping all other parameters unchanged. The subgrade soil was not considered as a stress dependent material; thus only the stiffness- stress dependency in the unbound granular material was investigated. The two pavement were subjected to a circular load which has radius of 17.5 cm and uniforme pressure of 676 kpa (CTTP, 2000). 2.2 Materials and mechanical properties Asphalt Figure 1: A typical pavement structure Two asphalt layer thicknesses were used (see Table 1). For each asphalt layer, constant values of resilient modulus and Poisson's ratio, respectively equal to 4000 MPa and 0.35, were assumed. These average values are in agreement with those commonly adopted in the Algerian manual for pavement design (CTTP, 2000).The asphalt layer was assumed to behave as a linear elastic material Unbound granular material Modified K-G model was used to model the non-linear resilient behaviour of granular layers used in the present investigation (Balay and Kabré, 1996: Tiliouine and Sandjak, 2001). The resilient modulus and Poisson's ratio expressions are as follows: 421

4 p p a 1 n p q K = K a F p (1) a p G = G a 1 n (2) ' q 1+ γ q p F = 2 P ' q 1+ β p 2 ; with: q σ1 σ 3 = ; σ p = σ 3 3 (3) In the above expressions, p and q represent respectively the volumetric and deviator stresses. The stresses σ1, σ3 are major (axial) and the minor (confining) principal stress respectively. The parameters Ka, Ga, n, β, γ are material constants and Pa atmospheric pressure equal to 100 kpa. This model selected for the sake of simplicity, was chosen for the model. It assumes that the shear and volumetric strains are linked and that the material is isotropic. It offers a reasonable compromise between simplicity and accurate modeling. The parameters of the modified K-G model were determined from repeated loading triaxial tests carried out at the Organisme National du Contrôle Technique des Travaux Publics (CTTP), Algeria. Table 2: Modified K-G model parameters for unbound granular materials used in the present study. UGM Ka (MPa) Ga (MPa) n β γ BBA CAP Subgrade soil The mechanical properties of the subgrade soil were not considered as variables in the present study; the common values of 50 MPa and 0.35 were respectively assumed for the resilient modulus and Poisson's ratio of the subgrade soils of the two. These parameters are of the same order as those corresponding to subgrade soil type S2 (characterised by modulus E soil = 50 MPa) defined in the Algerian manual for pavement design. 422

5 3. Numerical modelling of pavement The continuous trend towards improved computing facilities coupled with increasing knowledge of the mechanical properties of materials has enhanced the crucial need for the development of finite element codes for the non-linear analysis and design of pavements, both in mainframe and personal computers. 3.1 Finite element modelling The finite element method is particularly efficient for modelling the non-linear behaviour of pavement as it can easily accommodate variabilities in material properties, changes in pavement geometry and modifications in applied loading. In pavement engineering, pavement are often modelled as axi-symmetric systems. The finite element domain is modelled using 8-node rectangular ring elements; each node having two degrees of freedom associated with the nodal displacement components in the vertical and the radial directions (Zienkiewicz and Taylor, 2005). The elements each contain four Gaussian points at which stresses and strains are calculated. The mesh is automatically made for a structure by superposed layers of elements, the material parameters being constant for each layer. The heights of these horizontal layers are constant or in geometrical progression in some groups, and the computer code does the same by columns of elements. The are meshed as shown in Tables 3 and 4. The letters R and P respectively stand for a regular mesh and a mesh in geometrical progression. Table 3: Vertical mesh Layers Asphalt Number of elements 2 (for S1) 3 (for S2) Mesh type R Base 5 R Sub-base 5 R Subgrade 6 P Zones Table 4: Horizontal mesh Number of elements Mesh type 1 3 R 2 3 R 3 4 R 4 8 P 423

6 Near the load where the stress and strain gradients are large, the mesh should be fine whereas at greater distances from the applied load, the mesh can be coarser. Because of axi-symmetry, both geometrical and material, only the region to one side of the load centre line was considered. The mesh is fixed at the bottom allowing no lateral movement and rollers on the sides allow vertical displacement to take place. For illustration purposes, the mesh of pavement structure 1 is presented in Figure 2. Figure 2: Typical mesh for pavement structure 1 The resilient modulus and Poisson's ratio can then be computed from the classical relationships: 9KG E = 3K + G (4) 3K 2G υ = 6K + 2G (5) When using the modified K-G model, a limit value (υlim) for Poisson's ratio and a minimal value for the mean normal stress Pmin were introduced to ensure a fast convergence of iterative process. In the present study, the following values were used: υlim=0.40 and Pmin=0.01MPa. It is to be noted that the selected upper limit (υlim=0.40) is in agreement with the values of Poisson s ratio found from a series of RLT VCP tests conducted by Allen and Thompson (Bruce, 2005). Their results showed that the values of Poisson s ratio for different unbound aggregates were in the range of 0.35 to 0.40 for shear stress ratios that varied from 2 to

7 3.3 Method of resolution The system of nonlinear equations is solved by the secant method. Finite element formulation is based on the following matrix relations (Zienkiewicz and Taylor, 2005): { ε} = [ B]{ U} { σ} = [ D]{ ε} (6) where: T { F} [ B] { σ}dv = (7) e V { ε } Resilient strain tensor { σ }: Resilient stress tensor { U }: Nodal displacement vector [ B ]: Matrix of linear operators (derivatives of shape functions) [ D ]: Elasticity matrix { F }: Nodal force vector, which is given by the load due to the half-axle of a vehicle. V : volume of the element which leads to { } [ K]{ U} F = (8) where the global stiffness matrix can be written as T [ K] [ B] [ D] [ B]dv V = (9) e In these equations [ K ] is the secant stiffness matrix, which depends on the stresses. To calculate nodal displacement vector { U }, the direct iteration method was used, as follows: n n 1 { U } [ K ] 1 { F} = (10) and: { σ } [ ][ ]{ } n = D n 1 B U n (11) The initial values are initial stresses due to self-weight. The program continues to iterate until sequential displacement computations agree with some specified tolerance. 425

8 4. Results of non-linear analyses The main results of the numerical analyses were summarized in terms of values of the design criteria generally used in pavement engineering: 1. The horizontal strain at the bottom of the bituminous layer usually related to risks of asphalt layer cracking by tensile fatigue failure. 2. The vertical strain at the top of the soil usually related to risks of rutting of pavement. 3. The deflection at the surface, which is, to some extent, an indication of the structure ability to bear repeated traffic loads. In Table 5, the values of the three design criteria obtained from the non-linear analysis for pavement structure 1, are presented. Only the mechanical properties of the unbound granular materials were varied while all the other geometrical and mechanical parameters of the pavement layers were kept unchanged. Table 5: Variation of design criteria with granular type for structure 1 UGM type W(mm) εr(10-6 ) εz (10-6 ) UGM BBA UGM CAP It is observed that the three design criteria are very sensitive to changes in granular type. In particular, the horizontal strain at the bottom of the asphalt layer is the most sensitive design parameter to variations in the mechanical characteristics of the unbound granular materials. This clearly illustrates the need and the importance of correctly incorporating non-linear aggregate behaviour in flexible pavement analysis. Another interesting feature of the analysis is shown in Table 6. Here, only the asphalt layer thickness was changed (from 6 cm (pavement structure 1) to 12 cm (pavement structure 2). Results are presented for the unbound granular material characteristics corresponding to UGM BBA. Table 6: Variation of design criteria with asphalt layer thicknesses Structures W(mm) εr(10-6 ) εz (10-6 ) It is seen that the parameters values used in pavement design might be strongly affected by changes in the asphalt layer thickness. Thus, it is confirmed from the results of Tables 5 and 6, that pavement design criteria are very sensitive to variations both in the granular material and in the bituminous layer thickness. In Figures 3a, 3b, 4a and 4b values of the resilient modulus E and Poisson's ratio υ are plotted, on the load axis of pavement structure 1 and 2 respectively, in the aggregate layer for the two types of unbound granular materials UGM BBA and UGM CAP. From figures 3 and 4 it is observed that values of the mechanical characteristics (resilient modulus and Poisson s ratio) are different when using UGM BBA or UGM CAP in the same pavement structure. This shows the effect of the mineralogical nature on the mechanical characteristics of UGM. One can also notice that the curves representing the variation of the mechanical properties of the two UGMs shows a nonlinear form especially in the case of structure

9 (a) (b) Figure 3: Evolution of granular layer characteristics with depth for structure 1: (a) resilient modulus; (b) Poisson's ratio. (a) (b) Figure 4: Evolution of granular layer characteristics with depth for structure 2: (a) resilient modulus; (b) Poisson's ratio. 5. Conclusion In the present study, a numerical investigation has been carried out to simulate the mechanical behaviour for two local granular materials typically used in Algeria for flexible road pavements subjected to low to moderate of volumes traffic. 427

10 As regards to the sensitivity of pavement design criteria to the likely variations of UGM s type and asphalt layer thickness, the simulation results show that 1. A change in the stiffness of the unbound granular materials can affect significantly the tensile strain at the bottom of the asphalt layer and hence the fatigue life of the pavement. 2. The stiffness of the local UGM s remain relatively constant when using non-linear analysis of pavement under thick asphalt layers. This suggests the possibility of using multi-layer linear elastic theory to the structure analysis and design for such pavements. From an economical perspective and with regards to the important production level of the two tested aggregates, it may be concluded that the utilization of the two tested local unbound granular materials as significant structural layers, combined with more efficient use of bituminous materials, could reduce the construction cost of road Algerian pavements. This is especially important in flexible road pavements subjected to low to moderate volumes of traffic. However, a better understanding of the nonlinear behaviour of the local granular materials for large values of the shear stress ratio is needed. Acknowledgement The authors would like to thank D. Bensilem and A. Boudjellali from the Organisme National du Contrôle Technique des Travaux Publics (CTTP), Algeria, for their kind cooperation. 6. References 1. Balay, J. and Kabré, H., (1996) Modelization of flexible pavements with César-LCPC FEM program. Flexible Pavements, Gomes Correia, Belkama, Rotterdam. 2. Boyce H.R., (1980), A Non-linear model for the elastic behaviour of granular materials under repeated loading, International. Symposium on Soils Under Cyclic and Transient Loading, Swansea, U.K, pp Bruce, D.S., (2005) The development and verification of a pavement response and performance model for unbound granular pavements. Ph.D thesis, University of Canterbury. 4. CTTP-Direction des Routes. (2000) Catalogue de Dimensionnement des Chaussées Neuves, Algérie. 5. Ekblad J, (2008), Statistical evaluation of resilient models characterizing coarse granular materials, Materials and Structures, 41, pp Hoff I., Wichtman T., Triantafyllidis Th. and Lizcano A, (2009), Comparison of cyclic triaxial behavior of unbound granular material under constant and variable confining pressure, Journal of Transportation Engineering, ASCE,135(7), pp Hornych P.,Kazai A, and Piau J-M, (1998), Study of the resilient behaviour of unbound granular materials, In: Nordal RS, Refsdal G(eds) Proceedings of BCRA 98,3, Trondheim, pp

11 8. Jouve P., and Elhannani M., (1994), Application des modèles non linéaires au calcul des chaussées souples. Bulletin of liaison Labo.P.et Ch-190-, pp Lekarp, F. and Dawson, A. (2000) State of the art I: resilient response of unbound aggregates, Journal of Transportation Engineering, 126(1), pp May, R.W. and Witczak, M.W. (1981) Effective granular modulus to model pavement responses, Journal of the Transportation research record, 810, pp Minkwan K., Tutumluer E., and Kwon J., (2009), Nonlinear Pavement Foundation Modeling for Three-Dimensional Finite-Element Analysis of Flexible Pavements, International Journal of Geomechanics, 9(5), pp Sandjak, K. and Tiliouine, B. (2014) Experimental evaluation of non-linear resilient deformations of some Algerian aggregates under cyclic loading. Arabian Journal of Science and Engineering, 39, pp Tiliouine B., and Sandjak K., (2001), Non-linear finite element modelling of unbound granular materials in flexible pavement analysis, European Conference on Computational Mechanics, Crakow, Poland. 14. Uzan J, (1992), Resilient characterization of pavement materials, International Journal of Numerical Annals of Meth Geomechanical, 16, pp Zienkiewicz, O.C., and Taylor, R.L. (2005), The Finite Element Method, Oxford, 6th edition. 429