Modeling Resilient Modulus of Granular Subgrade Soil   Abdulrahman Al-Suhaibani College of engineering, King Saud University, Riyadh, Saudi Arabia

ABSTRACT

Since resilient modulus (M_R) of unbound pavement materials is highly dependent on stress conditions, mathematical modeling of M_R is usually based on bulk stress, as recommended by AASHTO 294-92I test. One major limitation of the bulk stress model is that it provides identical resilient properties for identical bulk stresses obtained either from high confining and low deviator stresses or from low confining and high deviator stresses. Thus, this model does not distinguish between the effect of confining and deviator stresses. However, other models that could be used to model M_R have been reported in the literature. In this paper, several linear and non-linear mathematical models were compared based on their ability to model M_R results. The model that contains both confining and deviator stresses was found to be the best among the investigated models. Ranking of other models was, also, reported.

Keywords: Resilient Modulus, Subgrade, Granular Soil.

INTRODUCTION

Pavements are subjected to continuous dynamic loading by motor vehicles. Empirical design procedures were based on static loading properties of subgrade soil such as soil support value (S) and California Bearing Ratio (CBR). These tests do not simulate the dynamic loading that the pavement is subjected to. The latest version of AASHTO (1993) and the Asphalt institute methods for pavement design adapted the resilient modulus (M_R) as the subgrade property to be used for pavement design.

M_R is a dynamic response of materials defined as the ratio of the repeated axial deviator stress to the recoverable axial strain. M_R could be determined in the laboratory by means of a triaxial test at different confining and deviator stresses. The magnitude and sequence of these stresses are different depending on whether the material is granular or fine-grained soil. AASHTO 294-92I test classifies soil as type I and type II for granular and fine-grained soils, respectively. For the purpose of mathematically modeling M_R of granular soils, AASHTO test method recommends the use of a model that based solely on bulk stress as the independent variable. One major limitation of the bulk stress model is that it provides identical resilient properties for identical bulk stresses obtained either from high confining and low deviator stresses or from low confining and high deviator stresses. Thus, this model does not distinguish between the effect of confining and deviator stresses. In this paper, several linear and non-linear models are compared based on their ability to model M_R results.

MODELING RESILIENT MODULUS

Results of resilient modulus tesst are a series of M_R values corresponding to a combination of deviator and confining stresses used during testing process. The past few decades have witnessed the development and use of several constitutive equations that model M_R behavior. These models provide powerful tools for research and design engineers to conduct pavement analysis in a more realistic manner. However, for the stress and deformation to be useful, the constitutive model should correctly describe the actual behavior of material that has been used in the analysis.

The resilient modulus of granular soils increases with increasing confining stresses (Witczak and Uzan 1988; Barksdale 1972). Several relationships have been used to describe the non-linear stress- strain behavior of granular materials. AASHTO 294-92I test method uses bulk stress (q) to model M_R as follows:

(1)

where

      k₁, k₂ = model constants and

      q = the bulk stress (s₁ + s₂ + s₃)

Another form of this equation is used by most pavement engineers (Hicks and Monismith, 1971; Shook et al., 1982; Santha, 1994) and can be obtained by dividing both bulk stress and resilient modulus by the atmospheric pressure to make the resulting regression constants dimensionless. The equation is as follows:

(2)

where

      s_atm = atmospheric pressure, in units same as those for M_R and q is the bulk stress

The main disadvantage of the bulk stress is that it does not account for shear stresses and shear strains developed during loading (Louay et al. 1999; Uzan 1985; Witczak and Uzan 1988). This model does not properly handle volumetric strains of soils (Brown and Pappin 1981). Moreover, it can not adequately explain the non-linear behavior of granular soils (Uzan 1985).

Another model using octahedral stress was proposed by Louay et al. (1999). The model can be used for various soil types without altering model attributes, octahedral normal and shear stresses. It gives results in octahedral stress environments, which are assumed to represent realistic stress states occurring in the field (Houston et al. 1992). They stated that octahedral normal and shear stresses, on which M_R properties depend, provide a better explanation for stress states of a material in which stresses change during loading. The octahedral model is as follows:

(3)

where

      s_atm = atmospheric pressure

      s_oct = octahedral normal stress =1/3 (s₁ + 2 s₃) = 1/3 (s_d + 3 s₃)

      t_oct = octahedral shear stress = (2/3)^1/2 (s₁ - s₃) = (2/3)^1/2 s_d

      k₁, k₂, k₃ are regression constants

A universal model as proposed in 1985 by Uzan takes the following form after modification by dividing M_R and stresses by the standard atmospheric pressure:

(4)

This model is practically the same as equation 3 above since for triaxial setup octahedral normal stress, (oct, equals to 1/3 q and octahedral shear stress, t_oct, equals to (2/3)^1/2 s_d

Rafael Pezo (1993) suggested a pavement engineer-oriented model that contains separate terms for both deviator stress and confining stress. It is a general model that suits both granular and fine-grained soils. The suggested model was as follows:

(5)

All terms were as defined earlier.

The model can be modified by dividing M_R, s_d and s₃ by the standard atmospheric pressure, s_atm, which equals approximately 100 KPa. In this form of the model the resulting constants will be dimensionless, especially, k1. The modified model takes the following form:

(6)

Johnson et al. (1986) suggested using second stress invariant and octahedral shear stress ratio to describe M_R behavior. The model accounts for both confining pressure and principal stress ratio in a manner appropriate for many granular materials. The model is as follows:

(7)

where

Drumm et al. (1990) reported a hyperbolic model that could be used to represent the resilient behavior of soils. A problem associated with the use of this model is that it predicts excessive values of M_R when the values of (d are close to zero. However, this problem is easily overcome when a lower limit to the calculated M_R is enforced as is done with bilinear and exponential models. The model is of the form:

(8)

EXPERIMENTAL WORK

The data for this study was obtained as part of a research project sponsored by King Abdalziz City for Science and Technology (KACST) (Al-Suhaibani, et al. 1997) Soil samples were collected from all over the Kingdom of Saudi Arabia. Samples were collected from approximately 50-cm deep hole located about 20 m from the pavement edge. Most major highways in the Kingdom were covered in this study. Soil samples were subjected to routine soil tests. In the field, field density was measured whenever possible. In addition, field (natural) moisture content was determined. In the laboratory, maximum dry density, optimum moisture content, Atterberg limits and grain size distribution were determined. Using soil properties obtained in the laboratory, soils were classified using AASHTO, Unified and SHRP classification systems. The later system was necessary for M_R testing, since the method of testing is somewhat different for different the SHRP classes, especially, stress magnitudes and sequences. In addition, California Bearing Ratio (CBR) and unconfined compressive strength were determined for each sample. In this paper, properties of soil samples that are classified as type I by SHRP system, totaling 28 samples are analyzed and discussed. The properties of these soil samples are shown in Table 1.

Resilient modulus (M_R) was measured for each soil sample at the optimum moisture content and maximum dry density. The M_R test was conducted according to AASHTO 294-92I test procedure. Test samples, 100 mm in diameter and 200 mm in height, were prepared using static compaction.

Table 1. Properties of Soil Samples

MDD = max. dry density, OMC = optimum moisture content, CBR = California
Bearing Ratio, LL = liquid limit, PL = plastic limit, PI = plasticity index.

DATA ANALYSIS AND DISCUSSION:

M_R values for each soil sample were modeled using models presented in Equations No. 2, 3, 6, 7 and 8 as presented above. Models 1, 4 and 5 are equivalent to 2, 3 and 6, respectively, therefore, they will be excluded from any further discussion. Each model was used to fit M_R data, from which regression constants were obtained. Average correlation coefficients corresponding to each model as it fits the experimental data are shown in Table 2. In addition maximum, minimum and standard deviation are also shown in the table. Predicted values using each model as well as residuals were also recorded. The predicted values using each model were compared with the original M_R values, named here, “measured” M_R. Plots of measured vs. predicted M_R values for models 2, 3, 6 and 7 are shown in Figures 1 through 4. The least scattered plot is that of model 6 followed by model 3 then model 2 and the most scattered one is that for model 7. Model 6 is the universal model including separate terms for both deviator and confining stresses. Model 6 has the highest average R² among all models. Model 3 comes second in fitting M_R data and is similar to model 6 except it includes bulk stress instead of confining stress. Models 2 and 7 include only one term and their scatters are about the same. However, model 2 includes only bulk stress while model 7 includes second stress invariant divided by octahedral shear stress. Model 3 also comes second in average M_R value while model 2 comes third followed by model 7 as seen in Table 2. Thus, the level of scatter is consistent with the average R² values. Ranking these four models by the standard deviation of R² values puts them in the same order as that obtained using average R². For each model, the trend line (passing through the origin) as well as its equation and its coefficient of determination (R²) are shown on each plot. Ranking models by the R² value shown on each plot, also, puts models in the same order obtained above.

Table 2. Correlation Coefficients’ Data for Models Used

(*) Refers to equation number.

Model 8, however, is of a different form when compared with the other four models. In the original form of the model, M_R is multiplied by deviator stress, which makes the values predicted by the model a function of deviator stress. Thus, the model output is not directly comparable with those of the other four models. Therefore, in order to compare model 8 with other models the model output is divided by the corresponding deviator stress to obtain “predicted M_R”. Figure 5 shows a plot of measured vs. predicted M_R using model 8. The level of scatter is the highest among all models resulting in relatively low R² value. In addition, Figure 5 shows that negative prediction values were obtained for M_R at low values of measured M_R. As mentioned earlier, this is a property of hyperbolic models. That is why a limiting minimum value is usually set in order to preclude such unrealistic values from appearing.

In order to get a clear picture of the relative differences between various models, Figure 6 shows a bar chart for R², standard error and average R² for each model. R² and standard error were obtained by conducting one-way analysis of variance (one way ANOVA) for measured vs. predicted M_R values for each model. Standard error for model 8 is very high compared with those of other models. Therefore, it was not possible to plot it in Figure 6 with other models. R², average R² and standard error show the superiority of model 6 followed by model 3 then model 2 then model 7. Model 8 shows a relatively high R² value and average R² is higher than those of models 2 and 7. However, the unrealistic predicted values obtained using model 8 and discussed above and its high standard error value make this model inferior to other models regardless of high R² value.

Various models were, also, compared in terms of trend line proximity to equality line of measured vs. predicted M_R values. Comparison of trend lines of various models is shown in Figure 7. It is clearly seen that all models under estimate measured M_R values at the upper range of values while they over estimate them at the lower range. Model 8 has the most bias one followed by model 7 then model 2 and finally by model 3 then model 6, which is very close to the equality line.

CONCLUSIONS

In conclusion, the ranking of the selected models starts with model 6 then model 3 then model 2 then model 7 and finally model 8 comes last. Although model 8 gives a relatively high R² value (higher than model 7), however, the unrealistic predicted values at the lower range of M_R values as well as the very high standard error rank it last among investigated models.

REFERENCES

AASHTO (1993) Guide for Design of Pavement Structures.

Al-Suhaibani, A., T. Al-Refeai, and A. S. Noureldin (1997) “Characterization of Subgrade Soils in Saudi Arabia; A Study of Resilient Behavior,” Final Report submitted to King Abdulaziz City for Science and Technology (KACST), Project No. AR-12-51, Riyadh, Saudi Arabia

Barksdale, R. (1972) “Repeated Load Testing Evaluation of Base Coarse Materials,” GHD Research Project No. 7002, Final Report, U.S. Department of Transportation, FHWA.

Brown, S. and J. Pappin (1981) “Analysis of Pavements with Granular Bases Layered Pavement Systems,” TRR 810, TRB, Washington, D.C.

Drumm, E., Y. Boateng-Poku, and T. Pierce (1990) “ Estimation of Subgrade Resilient Modulus from Standard Tests,” ASCE, Journal of Geotechnical Engineering, Vol. 116, No. 5, May.

Hicks, R. and C. Monismith (1971) “Factors Influencing the Resilient Response of Granular Materials,” Highway Research Record 345, Highway Research Board, Washington, D.C.

Houston, W., S. Houston, and T. Anderson (1992) “Stress State Considerations for Resilient Modulus Testing of Pavement Subgrade,” 71st Annual Meeting of Transportation Research Board, Jan. 12-16, Washington, D.C.

Johnson, T., R. Berg, and A. DiMillio (1986) “Frost Action Predictive Techniques: An Overview of Research Results,” TRR 1089, TRB, Washington, D.C.

Pezo, R. (1993) “A general Method of Reporting Resilient Modulus Tests of Soils, A Pavement Engineer’s Point of View,” 72nd Annual Meeting of Transportation Research Board, Jan. 12-14, Washington, D.C.

Santha, B. (1994) “Resilient Modulus of Subgrade Soils: Comparison of Two Constitutive Equations,” TRR 1462, TRB, Washington, D.C.

Shook, J., F. Finn, M. Witczak, and C. Monismith (1982) “Thickness Design of Asphalt Pavement – The Asphalt Institute Method.” 5th International Conference on the Structural Design of Asphalt Pavement, Delft, The Netherlands.

Uzan, J. (1985) “Characterization of Granular Materials,” TRR 1022, TRB, Washington, D.C.

Witczak, M. and J. Uzan, (1988) “The Universal Airport Design System, Report I of IV: Granular Material Characterization,” Department of Civil Engineering, University of Maryland, College Park, MD.

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