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The resilient modulus (M_{r}) is an important parameter which describes the mechanical behavior of unbound granular materials. However, this parameter can be determined from physical properties. This paper presents the relationship between resilient modulus and physical properties of Quartzite from Bakel (GB), Basalt from Diack and Bargny and Bandia limestones. Simple and multiple regression method by stepwise are used to establish linear and nonlinear relations to predict the resilient modulus. The results showed no significant correlation for Basalt, a weak estimation of the modulus for GB and good prediction of resilient modulus for limestone. These results also showed that the model of Uzan is more suitable to predict the resilient modulus than NCHRP model and the resilient modulus is better predicted in nonlinear relationship.

The stiffness of granular materials was characterized in Senegal as in lots of other countries in the world, by Young’s modulus (E). This parameter supposed that the unbound granular materials were elastic linear. Recent studies have shown that the mechanical behavior of unbound granular materials is nonlinear elastoplastic (Huang, 2004) [

Since 1960, numerous research efforts have been developed to characterize the resilient behavior of granular materials (Lekarp et al., 2000 [

M_{r}: resilient modulus (kPa), σ_{1}: major stress (kPa), σ_{3}: minor stress (kPa), σ_{d}: deviatoric stress (kPa), ε_{a}: recoverable strain.

However, since the laboratory determination of resilient modulus is complex, costly and time consuming, resilient modulus can be estimated based on correlation with physical properties. In Senegal no studies for estimating resilient modulus were performed. But in the world there are many studies for predicting resilient modulus from physical proprieties.

Jones and Witczak (1977) [

The materials used in this study are unbound aggregates coming from various geological formations of Senegal. They are amongst others like the Bakel Quartzite, the Diack Basalt and Bargny and Bandia Limestones (

The database is collected from Ba (2012) [

Specimens were subjected to the resilient modulus test procedure. A MTS closed-loop servo-electro-hydraulic testing system was used to apply the cyclic loading in a haversine waveform, with 0.1 second of loading duration and 0.9 second of rest period. Displacements were measured internally using “Linear Variable Displacement Transducer” (“LVDT”) mounted around the specimen inside the cell. The specimens have been tested using the NCHRP Protocol 1-28. Each specimen was conditioned with 103.5 kPa confining pressure, and 1000 cycles of 207 kPa deviator stress. The cycles are repeated 100 times for 30 loading sequences with different combinations of confining pressures and deviator stresses. The last five cycles of each sequence are used to calculate the resilient modulus (Ba, 2011) [

The resilient modulus test results and physical properties of materials collected from Ba (2012) [

The variables used in this analysis are resilient modulus

which is the dependent variable and physical properties of unbound granular materials represent the independent variables. There are water content (W), optimum water content (W_{opt}), dry density (γ_{d}), maximum dry density (γ_{dmax}), the percentage of fine particles (% fine), the percent passing on sieve 2 mm (P_{2}), the maximum size of particle (D_{max}), the coefficient of uniformity (Cu) and the coefficient of curvature (Cc). The resilient modulus was calculated using Uzan and NCHRP models, because they are more suitable to predict the resilient modulus (Ba, 2011) [

M_{r}: resilient modulus (MPa), θ: bulk stress = 208 kPa, τ_{oct}: octaedral shear stess = 48.55 kPa, Pa: atmospheric pressure = 101.3 kPa, σ_{d}: deviatoric stress 103 kPa, k_{i}: model parameters (kPa).

After describing the variables, we will carry out a Principal Component Analysis (PCA) to look for relationships between variables. This analysis allowed the detection of the relationships between variables and helped to choose the best type of regression, Figures 4-6 represent the PCA performed with GB, Diack Basalt and limestones. The result shows for GB (

tively dependent on γ_{d}, γ_{dmax} and D_{max} and negatively dependent on W_{opt}, P_{2} W and % fines. For the Diack Basalt (_{d} and negatively dependent on W, it is explained at 80.88%. For limestones (_{d} and negatively on W. These PCA do not justify a real connection with resilient modulus. They give just an overall vision on the touchiness of possible relations with the modulus.

There are also other methods such as the examination of the matrix of correlation and the stepwise method which guided the selection of the best variables in the models.

In this analysis, regression method by stepwise is used. It is based on the coefficient of determination R^{2} and the tests of Student and Fischer. Only the variables most correlated and satisfactory with tests are selected in the model. These tests are associated at the p-level, and all variables inferior at threshold (0.05) are significant. However, linear and nonlinear relationships are determined for all materials. Indeed, multicolinearity test is performed between independent variables, in order to prevent bias relations. For the majority of our relations, the following general equation is used.

Y: dependent variable (resilient modulus), β_{0}: intercept, β_{i}: model parameters, X_{i}: independent variables (physical properties), ε: error.

The results show no correlation for Diack Basalt because there is a lack of data. They also show that the resilient modulus is weakly explained for GB with R^{2} of 0.34 and 0.32 for Uzan and NCHRP (2004) models. For limestones, there are a R^{2} of 0.96 for Uzan model and a R^{2} of 0.94 for NCHRP model. The summary of the relations established in linear regression according to the models of Uzan and NCHRP (2004) gave the following equations:

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SRM: Summury Resilient Modulus (MPa), W: water content (%), γ_{d}: dry density (kN/m^{3}).

Figures 7-11 show the scatter between measured and predicted resilient modulus for several materials in linear models. They show a scatter plot widely dispersed for Basalt, moderately dispersed for GB and a scatter plot showing a good correlation for limestones.

There are also nonlinear relationships developed in this paper. They showed as in linear relations that the resilient is weakly explained for GB with R^{2} of 0.38 and

0.34 for Uzan and NCHRP (2004) models. For limestones, there are a R^{2} of 0.98 for Uzan model and a R^{2} of 0.98 for NCHRP model. The result also showed that the R^{2} is increasing in nonlinear models. Indeed; the used of nonlinear models are the best for predicting resilient modulus. The models in nonlinear relationships are represented by the following equations:

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•

SRM: Summary Resilient Modulus (MPa), W: water content (%), γ_{d}: dry density (kN/m^{3}).

In nonlinear model, Figures 12 and 13 show a bad distribution of points resulting from low correlation with GB. However, a good estimation was noted for limestones Figures 14 and 15 show a good distribution of points resulting from good estimation of resilient modulus of limestones from physical properties.

The estimation of the resilient modulus of GNT from Senegal, shows that the Uzan model is more suitable for predicting the materials. However, there is no correlation for the basalt due to the deficiency of significant variables, a weak correlation for GB and a good relationship for limestones which have a strong affinity with water. The results also show that the resilient is better predicted in nonlinear models. This study is important to get a way to estimate suitability of the resilient modulus, but the relations cannot be used for all situations, because of the small size of data base.

We would like to acknowledge Dr. Makhaly Ba to have placed at our disposal the unit of its experimental results, which were useful for this article. We also thank the French cooperation for the granting for a research grant in Paris VI.