ABSTRACT
Water Bound Macadam (WBM) is commonly used as subbase course/base course and even wearing course in India and worldwide. In the present work stressstrainvolume change behaviour of WBM under triaxial loading is studied. Drained triaxial tests were performed at three different confining pressures of 50, 100 and 200 kPa on WBM of specimen size 100 mm diameter and 200 mm height. Hierarchical single surface (HISS) model developed by Desai and coworkers is used to model the behaviour of WBM. A computer program PARA6 is used to calculate the various parameters. Using the calculated parameters stressstrainvolume change behaviour of WBM is predicted by finite element analysis (FEM). The predicted results match closely with the observed results.

* Department of Civil Engineering, NIT Kurukshetra, INDIA

1. INTRODUCTION
Pavement is a structure made in between the wheel and the natural ground. The basic aim of pavement is to provide a hard surface for the movement of wheels without significant deformation and to distribute the wheel load effectively to the larger area of natural ground so that the stresses are within allowable bearing capacity. Water Bound Macadam (WBM) is commonly used as subbase, base and even wearing course in temporary or unpaved roads with low volume of traffic. Behaviour of the pavement depends upon the behaviour of the materials constituting the various layers of the pavement.
Considering the importance of WBM an attempt has been made to study its behaviour. Various factors, which affect the behaviour, are density of the WBM, the confining pressure, the drainage conditions and the stress path followed. Stressstrain relationship of the WBM is important to be studied to analyze loaddeformation problems.
A number of experimental and analytical studies have been undertaken by researchers to understand the behaviour of the WBM. Stressstrain and strength characteristics have been studied mostly by conducting triaxial tests or plate load tests.
2. EXPERIMENTAL PROGRAMME
A very limited study is carried out on pavement materials under triaxial loading. Sheo Gopal (1993) and Dixit (1994) carried out a series of consolidated undrained conventional triaxial tests on Water Bound Macadam (WBM) specimen.
A dynamic finite element program (ABAQUS) was used for the analysis of the flexible pavement by Zaghloul and White (1993). They divided pavement material into three groups:
a. Asphalt mixtures
b. Granular material
c. Cohesive soil
Asphalt mixtures were modeled as viscoelastic material, granular materials were modeled using the Druckerprager model and for the subgrade cohesive soil Camclay model was used.
Wathugala et al. (1996) tried the numerical simulation of geosynthetic reinforced flexible pavement. Hierarchical Single Surface (HISS) models were used for soils, as these models offer better capabilities in capturing the behaviour of granular soil layer than the models available in ABAQUS. In the absence of triaxial tests with volume change behaviour, which are necessary to determine the material parameters for HISS models, typical material parameters have been assumed. DruckerPrager model was used for Hot Mix Asphalt Concrete (HMAC) and crushed limestone and vonMises model was used for the geosynthetic reinforcement.
Bonaquist and Witczak (1997) performed constitutive modeling for granular base, subbase and subgrade soil using HISS model. Triaxial tests were carried out on all the three material. Specimens 101.6 mm diameter by 203.2 mm high were used for the subgrade while the dimensions of subbase and base course samples were 152.4 mm diameter by 304.8 mm high. Constitutive model is a mathematical relation, which reproduces theobserved response of a continuous medium. Constitutive models can be broadly classified into following three categories.
1. Empirical models
2. Elastoplastic models
3. Elastoviscoplastic models
All geological materials show plasticity almost from the beginning. Therefore, stressstrain and volume change response of many geologic media including water bound macadam, bituminous concrete can only be predicted by plasticity models. Some of the yield criteria are
1. MohrCoulomb Criterion
2. DruckerPrager Yield Criterion
3. Critical State Model
4. HISS model
3. HISS MODEL
In the present study HISS model is used. The brief description of the model is as follows. Desai (1980), Desai et al. (1986) developed hierarchical single surface (HISS)0 and 1 models. In these models, a unique and continuous yield function is used that leads to the failure when an ultimate condition is reached. The 0 model is based on associative plasticity and isotropic hardening.
The constitutive equation for elastoplasticity can be written as
. . . Equn. 1 
where, C is the constitutive matrix for elastoplastic approach. 
The yield function for
model is given as
. . . Equn. 2
Sr = Stress ratio = . . . Equn. 3
where, J1 first invariant of stress tensor; J2D second invariant ofdeviatoric stress tensor; J3D third invariant of deviatoric stress tensor; Pa atmospheric pressure; a, b, n and g material constants; m0.5 for geological materials (Desai et al. 1986).
For nonassociative model, plastic potential function Q is defined as a modification of F with replaced by Q, i.e.,
. . . Equn. 4
3.1. Properties of the HISS Yield Function
Some of the features of HISS model are as follows:
1. The model involves only one continous surface, which describe yield or loading surfaces by a single function and also describe the ultimate behaviour. In the model only two parameters and at ultimate are used to define the traditional failure.
2. Entire hardening and ultimate behaviour is defined by only one function.
3. The plot of yield function F is continous and convex in the stress space for geological material. However it intersects the J1 axis at right angles, as a result it can be implemented in the context of the classical theory of plasticity.
4. As the intersection of two or more surfaces and comer plane are avoided, the model is easier to implement in numerical analysis.
5. A single parameter growth function can simulate hardening and include the effect of stress path, volume change and coupling of shear and volumetric responses.
4. DETERMINATION OF MATERIAL CONSTANTS
The HISS model requires nine parameters for the constitutive modeling of any material, which can be classified into five categories.
1. Elastic constants (E, v)
2. Ultimate parameters ()
3. Phase change parameter (n)
4. Hardening parameters ()
5. Nonassociative parameter ()
The procedure for the determination of material parameters has been described in detail in various references (Varadarajan & Desai 1993, Desai 1994). It is briefly presented herein.
1. Elastic constants (E, v)
The two elastic constants for an isotropic material, Young’s modulus, E and Poisson’s Ratio, v are determined from the average slopes of the initial part of the stressstrain curves and the ratio of lateral and axial strains, respectively. Janbu’s relation is used to correlate Young’s modulus with confining pressure.
. . . Equn. 6
where, k and N are constants.
2. Ultimate parameters()
For many geological materials m is found to be  0.5 (Desai et al. 1986). Therefore, in the present work, m is considered as  0.5. The procedure adopted for the calculation of and from the laboratory results is described below.
At the ultimate state, the value of tends to zero thus, the yield surface degenerates to an open surfaceintersecting J_{1} , axis at infinity. Applying the condition to the yield function, Equation 2, the slope of the ultimate line is derived as:
. . . Equn. 7
where, S_{r} = 1 for compression path and S_{r} = 1 for extension path. The ultimate parameters can be found out by conducting least square fitting procedure on Equation 7 for at least two triaxial tests on J_{1}  plane.
3. Phase change parameter (n)
The phase change parameter, n, is calculated using the zero plastic volume change condition, . An average of n values for different tests is taken as an overall value of n for the material.
4. Hardening parameters ()
In the present study, growth function a is assumed as the function of as
. . . Equn. 8
Taking natural log on both sides of Equation 8 gives,
. . . Eqin. 9
are determined from the least square fitting procedure for each test. The average value of from various tests are taken as overall values of the hardening parameters.
5. Nonassociative parameter ()
Nonassociative parameter , in the plastic potential formation, Q is assumed as constant and is determined from the conditions near the ultimate. Basic steps in evaluating are given below.
From the flow rule,
. . . Equn. 10
we get,
. . . Equn. 11
or . . . Equn. 12
. . . Equn. 13
Then from Equation 5,
5. TEST PROGRAM
In order to model the behaviour of WBM a series of drained triaxial test is carried out on WBM at three confining pressures of 50, 100 and 200 kPa.
5.1 Materials
(i) WBM
(ii) Aggregates
Generally large sized aggregates are used while constructing WBM in the field (maximum particle size up to 63 mm). Because of the practical constraints of performing triaxial tests on the WBM specimen made from this large size aggregate, some procedure was looked for scaling down the maximum particle size for testing.
There are four techniques available for scaling down the material size namely (i) scalping technique, (ii) parallel gradation technique, (iii) quadratic grain size distribution curve technique, and (iv) replacement technique. Out of all the above techniques, parallel gradation technique is considered to be more appropriate and as such widely used (Ramamurthy and Gupta, 1986). The same technique is used here in the present study.
Two grades of crushed stone aggregates were used for the preparation of the Water Bound Macadam (WBM) mix. The particle size distributions of the two grades of aggregates used are shown in Fig. 1. The grade A is used as coarse aggregates and grade B as screening. The two grades of crushed stone aggregates A and B were mixed with binder having grade C (P.I. = 6), in the ratio of 1.0:0.16:0.15 by volume in loose state to obtain the required gradation for the WBM mix (IRC: 19  1977).
Water Bound Macadam (WBM) Mix Design:Water Bound Macadam mix was designed as per (IRC: 19  1977) specification, for possible use as surfacing course. Delhi Silt (P.I = 6 per cent) was used as a binding material. Required quantity of both the grades A and B (Coarse aggregate and Screening) was mixed with binding material in the ratio of 1.0: 0.16: 0.15 by volume in loose state in dry condition. Then Proctor Compaction Test was carried out to find out the Optimum Moisture Content (OMC) and Maximum Dry Density (MDD) for further use. OMC for WBM is 6.8 per cent and MDD achieved at this moisture content is 22.30 kN/m3 (Fig. 2).

5.2 Testing
Triaxial Test: Conventional triaxial apparatus (Bishop & Henkel 1962) was used for the triaxial tests. A Perspex triaxial cell capable of withstanding cell pressure more than 1 MPa and with the facility of accommodating 100 mm diameter and 200 mm height specimen was used. Axial strains, deviator stresses and volumetric strains were observed during the tests. The experimentally observed behaviour is presented in Fig. 3 to 6. Details about the preparation of specimen, testing procedure, etc. are given in Aggarwal (2002).
6. MODELLING
All the nine constants for HISS model are calculated for WBM and are presented in Table 1.
7. PREDICTION
In the present study the stressstrain behaviour of WBM is modeled up to strain hardening stage and prediction beyond strain hardening (strain softening) is beyond the scope of this Paper. Finite Element Method (FEM) is used for the prediction of stressstrainvolume change behaviour of the WBM. Considering the symmetry, only a half part of the triaxial test specimen was discretised into eight, 8noded axisymmetric solid elements (Fig. 7). Boundary conditions are applied in such a way that nodes on centerline are free to move in vertical direction only and nodes on the base are free to move in horizontal direction only as shown in Fig 7. All other nodes are free in both horizontal and vertical directions.

The incremental constitutive relation (Equn. 1) has been used to predict the stressstrainvolume change response. The equation is integrated starting from the initial hydrostatic state. The prediction is made using the nine parameters calculated for WBM under strain control condition. Both predicted and experimentally observed variation of deviatoric stress and volumetric strain with axial strain are presented in Fig. 3 and Fig. 4 for WBM. The observed and predicted behaviour matches closely.
8. CONCLUSIONS
From the results it is observed that deviator stress at failure increases with increase in confining pressure. From the stressstrain behaviour, it is concluded that WBM behave like a brittle material. Modeled behaviour matches closely with the observed results.
REFERENCES
1. Aggarwal Praveen (2002), Numerical Modeling of Geogrid Reinforced Unpaved Flexible Pavement, Ph.D. Thesis, Indian Institute of Technology Delhi, New Delhi, India.
2. Bishop, A.W. and Henkel, D.J. (1962), The Measurement of Soil Properties in the Triaxial Test, Edward Arnold Publishers ltd., ‘London.
3. Bonaquist, Ramon F. and Witczak, Matthew W. (1997), A Comprehensive Constitutive Model for Granular Materials in Flexible Pavement Structures, Proc. 8th Int. Conf. on Asphalt Pavements, University of Washington, Seattle, Washington, U.S.A.
4. Desai, C.S. (1980), A General Basis for Yield, Failure and Potential Functions in Plasticity. Int. Journ. Num. Anal. Meth. Geomecch., 15(9), 649680.
5. Desai, C.S. (1994), Hierarchical Single Surface and the Disturbed State Constitutive Models with Emphasis on Geotechnical ; Applications, Geotech. Eng. Emerging Trends in Design and Practice, Chap. 5, K.R. Saxena (editor). New Delhi, India. Oxford & IBH Pub. Co. Pvt. Ltd
6. Desai, C.S., Somasundaram, S and Frantziskonis, G. (1986), A Hierarchical Approach for Constitutive Modeling of Geologic Materials. Int. J. Num. and Analytical Methods in Geomech., 10, 225257.
7. Dixit, A. (1994), Behaviour of Model Pavements with Geosynthetics, Ph.D. Thesis, Indian Institute of Technology Delhi, New Delhi, India.
8. IRC: 191977 (1982), Standard Specification and Code of Practice for Water Bound Macadam. The Indian Roads Congress, New Delhi, India.
9. Ramamurthy, T. and Gupta, K.K. (1986). “ Response Paper to How Ought One to Determine Soil Parameters to be Used in the Design of Earth and Rockfill Dams”, Proc. Indian Geotech. Conf. New Delhi, 2, 1519.
10. Sheo Gopal (1993), Behaviour of Model Pavements with Geosynthetics Under Repeatitive Loading, Ph.D. Thesis, Indian Institute of Technology Delhi, New Delhi, India.
11. Varadarajan, A. & Desai, C.S. (1993), Material Constants of a Constitutive Model Determination and Use, Indian Geotech. J., 23 (3), 291313.
12. Wathugala, G.W., Huang, B. and Pal, S. (1996), Numerical Simulation of Geosynthetic Reinforced Flexible Pavements, Transpotation Research Record, 1534, 5865.
13. Zaghloul, S. and White, T. (1993), Use of ThreeDimensional, Dynamic Finite Element Program for Analysis of Flexible Pavement, Transpotation Research Record, 1388, 6069. 