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Constitutive Modeling of Unpaved Flexible Pavement Under Static Loading
Praveen Aggarwal Department. of Civil Engineering, NIT, Kurukshetra, India K. K. Gupta Department of Civil Engineering, IIT Delhi, India and K. G. Sharma Department of Civil Engineering, IIT Delhi, India |
ABSTRACT
An experimental study is carried out to numerically model the behavior of unpaved flexible pavement materials. Behavior of constituting materials, which comprise the Yamuna sand and Water Bound Macadam (WBM) are studied under drained triaxial tests at three different confining pressures of 50, 100 and 200 kPa with specimen size of 100mm diameter and 200mm height. The hierarchical single surface model (HISS), developed by Desai and co-workers is used to model the constituting materials. Predicted stress-strain-volume change behavior of the two materials match closely with the observed results. A model pavement was tested under static loading and load-penetration behavior was obtained. The verified models of the constituting materials are applied to the model pavement using FEM and predicted results are compared with the experimentally obtained results.
Keywords: Flexible pavement, WBM, Yamuna sand, numerical modeling, finite element method.
INTRODUCTION
Pavement is a structure made in between the wheel and the natural ground. The basic aim of a pavement is to provide a hard surface for the movement of wheels without significant deformation of natural ground and to distribute the wheel load effectively to the larger area of natural ground so that the stresses are within bearing capacity of the soil. Temporary or unpaved roads with low volume of traffic are required for construction and access roads, contractors’ haul roads, short-term detour around, for example, bridge replacement construction etc. Further, such roads are also frequently constructed world wide to support resource industry viz. forestry, mining, oil and tar-sand extraction, agriculture and others.
LITERATURE REVIEW
Considering the economic significance of unpaved roads attempts have been made to understand their behavior. A number of experimental and analytical studies have been undertaken by researchers to understand the behavior of pavements (Sheo Gopal, 1993, Dixit, 1994 and Mahmood, 1998). Stress-strain and strength characteristics have been studied mostly by conducting triaxial tests.
Numerical modeling is a technique with the help of which, behavior of any system can be studied, provided the required material properties/ parameters (depending upon the model used) of all the constituting materials of the system are known. Reliability of the system depends upon the suitability of the material models.
Many researchers have carried out numerical modeling of flexible pavements. Some of them assumed the material parameters of constituting materials in absence of triaxial test results (Wathugala et al. 1996) and some used uniaxial test results (Scarpas, et al. 1997) to analyze and predict the pavement behavior.
Zaghloul and White (1993) and Chen et al. (2000) have used the commercial programs ABAQUS and FLAC (1993) respectively, whereas Famiyesin et al. (1998) and Perkins (2001) have used the commercial program CRISP to analyze the flexible pavement.
The yield criteria used to characterize the behavior of constituting materials of the pavements are Drucker-Prager, Mohr-Coulomb and Hierarchical single surface etc.
SCOPE
The scope of the investigation was to conduct triaxial tests on the constituting materials of the unpaved flexible pavement model (the model pavement), to develop a constitutive model using an elasto-plastic theory and to verify the model by comparing the predicted stress-strain-volume change behavior with the experimentally observed behavior with in strain hardening state (up to the peak stress). A plate penetration test was carried out on a model pavement under static loading and the load deformation behavior was observed. The model pavement was analyzed using the finite element (FE) technique, and the predicted load deformation behavior is compared with the experimentally observed behavior.
LABORATORY TESTS
MATERIALS
In the present study, locally available sand from the bank of the river Yamuna, passing from East of Delhi, capital of India (the Yamuna sand), is used as subgrade material and water bound macadam (WBM) as per Indian Road Congress: 19-1977 (1982) is used as the possible base/wearing course in the unpaved flexible pavement. Characteristics of Yamuna sand are shown in Table 1. The WBM consists of a grade of crushed angular coarse aggregate, a fine filler aggregate termed as screening and a binding grade Delhi silt (P.I. = 6). The gradation curves of these materials are shown in Fig. 1. Maximum particle size used for the preparation of WBM in the field is 63.0 mm. Because of practical constraints this has been reduced to 20.0 mm for the present study. The parallel gradation technique (Ramamurthy and Gupta 1986) was used to reduce the size of the aggregate. For the preparation of WBM, coarse aggregate, screening and binding material (Delhi silt) were mixed in the ratio of 1.0:0.28:0.06 by volume in a loose state and in a dry condition.
Figure 1. Gradation of Yamuna Sand, Delhi Silt, Coarse Aggregate and Screening
Table 1. Characteristics of Yamuna Sand
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| Property | Value | |||
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| % Sand | 94 | |||
| % Silt | 6 | |||
| Specific Gravity (Gs) | 2.67 | |||
| Coefficient of Uniformity Cu | 2.24 | |||
| Coefficient of Curvature Cc | 1.14 | |||
| Maximum Dry Density (gd max) | 16 kN/m3 | |||
| Minimum Dry Density (gd min) | 13 kN/m3 | |||
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EXPERIMENTAL PROGRAM
A Proctor compaction test as per IS: 2720 (Part VII) was carried out on WBM to determine the optimum moisture content (OMC) and maximum dry density (MDD).
A series of consolidated drained triaxial tests were conducted on the Yamuna sand and WBM at three confining pressures of 50, 100 and 200 kPa. All the triaxial tests were performed on cylindrical specimens of size 100 mm diameter and 200 mm height.
A plate penetration test was performed on the model pavement, consisting of a tank, 700 mm wide, 700 mm long and 600 mm deep. The diameter of the loading plate used was 100 mm. The model pavement consisted of 100 mm thick WBM over 500 mm thick the Yamuna sand.
EXPERIMENTAL PROCEDURE
A standard Proctor compaction test was carried out on the WBM by varying the moisture content and giving the same compactive energy for each sample. The results of the Proctor compaction tests are shown in Fig. 2. Optimum moisture content (OMC) and corresponding maximum dry density (MDD) are 6.8% and 22.3 kN/m3 respectively. In all other tests on the WBM, it is compacted at OMC to achieve maximum dry density.
Figure 2. Proctor compaction test on Water Bound Macadam
The specimens of WBM for triaxial tests were prepared using a three-piece split sampler (Aggarwal 2005). The specimens of Yamuna sand for triaxial tests were prepared using a two-piece split sampler. The specimens were prepared on the pedestal of a triaxial cell itself using the Yamuna sand (boiled in water to make it free from any entrapped air). A rubber pad was used for compaction.
The specimens were saturated by allowing water to pass through the base of the triaxial cell, and using a top drainage system for removing air voids. The specimens were first subjected to the required confining pressure and were then sheared by applying axial loading using a Hounsfield loading frame. The rate of shearing is kept as 2.0 mm/minute. The readings of vertical displacement, volume change and axial load were taken at periodic intervals.
For the preparation of the model pavement a steel tank was placed, axially centered on the frame of a MTS machine (250 kN capacity, servo-controlled, with a cross head travel of 100 mm). After placing the tank in this position, the model pavement was prepared by filling subgrade and WBM in layers (Fig. 3). Each layer was compacted uniformly to the required density (same as that of triaxial samples). Necessary precautions were taken to reduce the friction from the sides of the steel tank. The loading ram of the machine (to which the loading plate of 100mm diameter is attached) was lowered in such a manner that the loading plate just touched the prepared finished surface of the model pavement.
Figure 3. Unpaved Pavement Model
The static compression test was carried out at a constant rate of deformation of 2.0 mm/minute, various loads corresponding to the penetration of the loading plate was observed.
EXPERIMENTAL RESULTS
The stress-strain-volume change behaviors of the Yamuna sand and WBM at three confining pressures of 50, 100 and 200 kPa are shown in Figs. 4 -7. Both the materials showed an increase in volumetric strain with increase in confining pressure. The Yamuna sand results, indicate that the deviator stress at failure at a confining pressure of 50 kPa is 215 kPa and increases to 835 kPa for a confining pressure of 200 kPa. The axial strain at failure (at peak deviator stress) ranges from 6.50 % to 7.15 %. A strain-softening behavior is observed beyond these failure strains. Volume of the specimen reduces up to an axial strain of 1.00 % at a confining pressure of 50 kPa and up to an axial strain of 2.00 % at a confining pressure of 200 kPa. Dilation is observed with the increase in axial strain thereafter.
WBM results indicate that deviator stress at failure is 380 kPa at a confining pressure of 50 kPa and increases to 1100 kPa at a confining pressure of 200 kPa. The failure occurs at an axial strain of 6.50 % for confining pressure of 50 kPa and at 5.00 % for confining pressure of 200 kPa. Beyond these strains the deviator stress decreases significantly, thus exhibiting brittle, strain-softening behavior. Volume of the specimens reduces initially and contraction in volume continues up to an axial strain of 2.00 % for confining pressure of 50 kPa and up to 2.50 % for confining pressure of 200 kPa and thereafter dilation takes place for higher axial strains.
The model pavement test response under monotonic loading is shown in terms of load versus settlement in Fig. 8. From the result it is observed that load increases very sharply with the increase in settlement especially in the initial portion. At a load of 2000 N, the settlement in the model pavement is 13.0 mm. Further increase in settlement increases the load but at a comparatively slower rate. At a load of 3000 N, settlement in the model pavement is 34.0 mm. After a settlement of 36.0 mm (corresponding load of 3070 N) in the model pavement, load increases at very slow rate with settlement. At a settlement of 56.0 mm the corresponding load is 3250 N.
Figure 4. Variation of deviator stress with axial strain for Yamuna sand
Figure 5. Variation of volumetric stress with axial strain for Yamuna sand
Figure 6. Variation of volumetric stress with axial strain for water bound Macadam
Figure 7. Variation of volumetric strain with axial strain for water bound Macadam
Figure 8. Variation of settlement with load under monolithic loading in Pavement model
CONSTITUTIVE MODEL
In the present study the hierarchical single surface (HISS) model based on the theory of elasto-plasticity, is used to characterize the Yamuna sand and WBM. Desai et al. (1986) developed hierarchical single surface (HISS) models d0 and d1. In these models, a unique and continuous yield function is used that leads to the failure when an ultimate condition is reached. The d0 model is based on associative plasticity and isotropic hardening.
The constitutive equation for elasto-plasticity can be written as
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(1) |
where 
is the constitutive matrix for this elasto-plastic approach
The yield function for the d0 model is given as
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(2) |
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(3) |
where J1 is first invariant of stress tensor;
J2D and J3D are second and third invariants of deviatoric stress tensor respectively;
Pa is atmospheric pressure;
a, b and n are material constants;
m = -0.5 for most soils (Desai et al. 1986).
For the non-associative model d1, the plastic potential function Q is defined as a modification of F with a replaced by aQ, i.e.,
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(4) |
where
|
(5) |
in which k is a non-associative parameter, a0 is a at the beginning of shear loading and 
, xv is volumetric part of
x (plastic strain trajectory).
The model involves only one continous surface which describes yield or loading surfaces by a single function and also describes the ultimate behavior. In the model only two parameters g and b are used to define the traditional failure. Entire hardening and ultimate behavior is defined by only one function. The plots of yield function F are continuous and convex in the stress space for geological materials. However the yield surface intersects the J1 axis at right angles, and as a result it can be implemented in the context of the classical theory of plasticity. As the intersection of two or more surfaces and corners in the p plane are avoided, the model is easier to implement in numerical analysis. A single parameter growth function a can simulate hardening and include the effect of stress path, volume change and coupling of shear and volumetric responses. As a result, the model is simplified significantly.
DETERMINATION OF MATERIAL PARAMETERS
The procedure for determination of material parameters required in the model has been described in detail in many references (for example Varadarajan and Desai 1993; Desai 1994; Soni 1995; Sharma et al. 2001, Aggarwal 2002). The procedure is briefly presented herein.
ELASTIC PARAMETERS (E, n)
The two elastic parameters for an isotropic material, Young's modulus, E and Poisson's Ratio, n are determined from the average slopes of the initial part of the stress-strain curves and the ratio of lateral and axial strains respectively. The value of E is expressed as a function of confining pressure, s3, using Janbu’s (1963) relationship as
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(6) |
where K and n’ are constants.
ULTIMATE PARAMETERS (g, b, m)
For many geological materials m is found to be approximately -0.5 (Desai et al. 1986). Therefore, in the present work, m is considered as -0.5. The procedure adopted for the calculation of g and b from the laboratory results is described below.
The ultimate parameters g and b can be related to the angle of internal friction in compression, fC and the angle of internal friction in extension, fE as follows, (Gupta 2000)
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(7) |
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(8) |
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(9) |
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(10) |
After determining the value of b from Eq. (10), the value of g is calculated using Eq. (7) or Eq. (8).
As only conventional triaxial compression tests have been conducted during the present study the angles of internal friction, fC in compression and fE in extension sides of the yield surface are assumed to be the same, i.e. fC = fE.
PHASE CHANGE PARAMETER (n)
The phase change parameter, n, is calculated using the zero plastic volume change condition, 
. Based on Eq.(2), the expression for n can be obtained as
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(11) |
The value of n is calculated using this equation for each test. An average of n values for different tests is taken as an overall value of n for the material.
HARDENING PARAMETERS (a1 and h1)
In the present study, the growth function a is assumed as function of x as
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(12) |
where a1 and h1 are material parameters; and x is the trajectory of plastic strain given by
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(13) |
For each test, at all the observed points of the stress-strain curve, the value of x is known. The value of the growth function for the observed points was calculated using the yield function i.e. Eq. (2). Substituting the values for a and x in Eq. (12) and conducting a least squares analysis, the hardening parameters a1 and h1 were obtained for each test. The average value of a1 and h1 from various tests are taken as overall values of the hardening parameters.
NON-ASSOCIATIVE PARAMETER (k)
The non-associative parameter, k in the plastic potential formation, Q is assumed to be constant and is determined from the conditions near the ultimate. Basic steps in evaluating k are given below.
From the flow rule,
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(14) |
where 
is the axial plastic strain increment, and 
is the axial stress.
The volumetric plastic strain 
is given by
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(15) |
or
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(16) |
Taking the ratio of Eqs. (16 and 14), one gets
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(17) |
The ratio of 
can be obtained from the slope of the observed 
vs 
response by choosing a point in the ultimate state. The value of aQ which is represented on the right hand side of Eq.(17) can then be found since the left hand side is now known. Using this value along with k and rv at the ultimate condition, the average value of k is determined.
The material parameters were determined by using the results from the triaxial tests on the Yamuna sand and WBM at three confining pressures of 50, 100 and 200 kPa. The material parameters obtained for two constituting materials of the model pavement are shown in Table 2.
Table 2. The Material Parameters for Yamuna Sand and WBM
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| Constant | Yamuna sand | WBM |
| ||
| K | 133.30 | 197.70 |
| N' | 0.986 | 0.922 |
| g | 0.0600 | 0.0907 |
| b | 0.739 | 0.740 |
| m | -0.5 | -0.5 |
| n | 2.900 | 2.900 |
| a1 | 0.00002268 | .00016000 |
| h1 | 1.0800 | 0.6687 |
| k | 0.236 | 0.150 |
| fc | 42.0o | 49.5o |
| ||
From the parameters in Table 2 the following observations are made:
VERIFICATION OF MODEL
The incremental constitutive relation has been used to predict the stress-strain-volume change response. Eq. (1) is integrated starting from the initial hydrostatic state. The prediction is made using the nine parameters calculated for the Yamuna sand and WBM under strain control conditions. Both predicted and experimentally observed variation of deviator stress and volumetric strain with axial strain are also presented in Figs. 4-7 for the Yamuna sand and WBM. The observed and predicted behaviors match closely, hence verifying the model.
PREDICTION OF THE MODEL PAVEMENT BEHAVIOR
Verified models and calculated material parameters of constituting materials of the model pavement were used for FE analysis of the model pavement. The top 100 mm thick layer of WBM in the model pavement is discretized into 133 elements. Beneath this layer, the 500 mm thick subgrade (the Yamuna sand) is discretized into 171 elements. The 20 mm thick rigid steel loading plate (100 mm diameter) (E = 2000000 kN/m2, n = 0) is discretized into 4 elements. So the model pavement as a whole is discretized into 308, 8-noded, solid elements and 997 nodes (Fig. 9). The analyses have been conducted assuming axisymmetric conditions. The sides and bottom of the model pavement are assumed to be rigid and incompressible.
Figure 9. Discretization of Pavement Model
Boundary conditions are taken in such a way that all the nodes along the centerline of the pavement model and along the vertical wall of the model pavement tank are restrained, so they cannot move in the horizontal direction and are allowed to move in the vertical direction only. Whereas all the nodes along the base of the model pavement tank are restrained against movement in the vertical direction and are allowed moving in the horizontal direction. All other nodes were free to move in both horizontal and vertical directions as shown in Fig. 9.
ANALYSIS
Finite element analysis of the model pavement under monotonic loading has been carried out using the computer program code DSC-SST-2D (Desai 1997). The material parameters for the Yamuna sand and WBM as presented in Table 2 are used. In-situ stresses are also considered in the analysis. The value of coefficient of lateral earth pressure at rest has been calculated using following relationship
 |
(18) |
where K0 is the coefficient of lateral earth pressure at rest, and fc is the angle of shearing resistance in compression.
The effect of overburden pressure is considered for selecting values of modulus of elasticity.
Nodes representing the loading plate are subjected to total vertical load of 2360 N in 58 increments in the pavement model. Beyond these loads numerical convergence was not achieved. Tolerance for yield function convergence and tolerance for unbalanced load convergence were kept equal to 1%. A maximum of 50 iterations has been used for each increment of loading to ensure convergence.
The predicted load penetration behavior using the FE technique is plotted in Fig. 8. The predicted response of the model pavement matches closely with the experimentally observed behavior up to the plate load of 1250 N. However, for the plate load more than 1250 N, prediction shows less settlement than the experimental observations, in the model pavement. Deviations in the predicted behavior from the experimentally observed behavior, in the model pavement are because of a number of elements that enter into the strain softening state (subjected to the maximum strain, higher than the axial strain corresponding to the peak deviator stress in triaxial tests).
In the present study constitutive model was developed for the materials of the model pavement up to the peak stress. Beyond the peak stress, experimental results show decrease in the stress with increase in strain resulting in strain softening for both the materials (Figs. 4 and 6). As the constituting materials are modeled up to peak stresses, the load-deformation behavior of these materials can be predicted accurately as long as the strains in these materials are less than the strains corresponding to peak stresses. From the analysis it was observed that at the plate load of 1250 N, only three elements (near the edge of loading plate) are subjected to the maximum strains more than 6% (strain corresponding to peak stress in the experimental results, Figs. 4 and 6), and rest of the elements were at strains below 6%, which indicates the start of strain softening in these three elements (subjected to higher strain). Whereas at the plate load of 2360 N, fifty-two elements were subjected to maximum strains more than 6%. So during FE analysis of the model pavement, for the plate load more than 1250 N, elements start entering into the strain softening state. But these elements were modeled up to peak stresses (strain hardening portion) only. This is the reason for the deviation in the prediction from the observed load-settlement behavior, for the plate load of more than 1250 N, in the model pavement.
Figure 10. Variation of vertical displacement with depth along centerline in Pavement Model
From the results of FE analysis of the model pavement, variation of vertical displacements with depth along the centerline of the loading plate at 400 N, 1000 N and 2000 N of monotonic load on the loading plate are shown in Fig. 10. From the figure it is observed that vertical displacement along the centerline of the loading plate decreases with the depth in the pavement model at these plate loads. Variation of vertical displacements along the top surface, at the plate loads of 400 N, 1000 N and 2000 N, in the model pavement is shown in Fig. 11. From the figure it is depicted that vertical displacement is maximum beneath the loading plate and it decreases with increase in distance from the centerline of the loading plate, for all these loads. This is as per expectation and previous quoted results in the literature. The deformed shape of the model pavement (using NISA package) at the plate loads of 1250 N and 2360 N are shown in Figs. 12 and 13 respectively.
Figure 11. Variation of vertical displacement along top surface in Pavement Model
Figure 12. Deformed Shape of Pavement Model at Plate Load of 1250 N
Figure 13. Deformed Shape of Pavement Model at Plate Load of 2360 N
Figure 14. Variation of vertical stress with depth along centerline in Pavement model
From the finite element analysis of the model pavement, variation of vertical stresses with depth along the centerline of the loading plate at 400 N, 1000 N and 2000 N of monotonic load on the loading plate are shown in Fig. 14. From the figure it is observed that vertical stresses are maximum at the surface and decreases with depth. After achieving a minimum value, the vertical stress increases gradually with the depth, at all the plate loads in the model pavement. This is because of the in-situ stresses.
CONCLUSIONS
Drained triaxial tests have been conducted on the Yamuna sand and WBM (constituting materials of an unpaved flexible pavement) at three confining pressures. Stress-strain-volume change behavior of the materials were presented and discussed.
A HISS, elasto-plastic model was used to depict the behavior of the Yamuna sand and WBM. The constitutive model predicts the behavior of both the materials satisfactorily, under drained triaxial compression condition.
A plate penetration test was performed on the model pavement under monotonic loading and load versus displacement is obtained. Using the constitutive model of the Yamuna sand and WBM, finite element analysis of the model pavement was conducted.
Modeled behavior matches closely with the observed results up to the plate load of 1250 N. However, at the plate loads more than 1250 N, there is a deviation in the predicted and observed behavior because of strain softening in a number of elements.
The present study can be extended to following directions:
REFERENCES
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