1. IntroductionThe 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) [1]. This behavior led scientists to define the resilient modulus, which replaces henceforth Young’s modulus, in order to better characterize the real behavior of unbound materials. The resilient modulus is the primary input on materials properties in the NCHRP 1-37 (2004) guide. It is very important to calculate the stress, strains and deflections of pavement under wheel load. However, the determination of resilient modulus is time consuming, requires sophisticated equipment which is costly as well as skilled personnel for laboratory. It is so important to predict resilient modulus in order to reduce cost and time in road project. In addition, the resilient modulus is influenced by many factors, the most important of them are stress level and material properties (Lekarp et al., 2000 [2]). This article presents the estimation of resilient modulus of unbound materials from Senegal based on physical properties, since the physical properties are easy to obtain in laboratory and are not expensive.
2. BackgroundSince 1960, numerous research efforts have been developed to characterize the resilient behavior of granular materials (Lekarp et al., 2000 [2]). In Senegal, the investigation on mechanical behavior of unbound materials goes up since twenty years. Fall (1993) [3] has announced for the first time, the importance of adequacy input for pavement design. He underlines the first advanced mechanical properties of lateritic soil from Senegal. Ba (2012) [4] also studied the mechanical behavior under cyclic wheel load of granular materials (Diack Basalt, Quartzite from Bakel and limestones from Bargny and Bandia) and defined resilient modulus of these materials. This concept of resilient is introduced by Seed et al. in 1960 (Angelone and Martinez, 2000 [5]) and can be defined as the ratio of the repeated axial deviatory stress to the recoverable axial strain (Figure 1).
Mr: resilient modulus (kPa), σ1: major stress (kPa), σ3: minor stress (kPa), σd: deviatoric stress (kPa), εa: 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) [7] developed two correlation equations for A-7-6 subgrade soil in California, combining the water content and the degree of saturation. The results show a coefficient of 0.96 for compacted soil and 0.45 for undisturbed soil. Carmichael and Stuart (1985) [8] developed two relations result from tests conducted on 250 soils. Yau and Von Quintus (2002) [9] proposed an equation for predicting resilient modulus. They found that the equation did not fit all situations. George (2004) [10] found linear and nonlinear relationships for estimating resilient modulus for fine and coarse grained soils from physical properties. Kim et al. (2007) [11] established relationships to estimate models parameters of resilient modulus. Van Aswegen and Steyn (2013) [12] also study the statistical modeling of the resilient behavior of unbound granular material of South Africa soils. These studies are very important for estimating resilient modulus of unbound granular materials.
4. Statistical AnalysisThe resilient modulus test results and physical properties of materials collected from Ba (2012) [4] were analyzed using statistica software. The data base is not large, so it is not adequate to perform suitable statistical analysis. But it is important to define basic knowledge for correlating resilient modulus of unbound granular materials of Senegal.
4.1. Description of VariablesThe 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 (Wopt), dry density (γd), maximum dry density (γdmax), the percentage of fine particles (% fine), the percent passing on sieve 2 mm (P2), the maximum size of particle (Dmax), 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) [13]. They are determined from states of stresses usually found in base course defined by NCHRP (2004) [14]. There is summary resilient modulus which is used to perform correlations. The following equations show the models as well as the states of stresses used.
Mr: resilient modulus (MPa), θ: bulk stress = 208 kPa, τoct: octaedral shear stess = 48.55 kPa, Pa: atmospheric pressure = 101.3 kPa, σd: deviatoric stress 103 kPa, ki: model parameters (kPa).
4.2. Relationships between VariablesAfter 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 (Figure 4) that the resilient modulus is explained at 67.64%, and is posi-
tively dependent on γd, γdmax and Dmax and negatively dependent on Wopt, P2 W and % fines. For the Diack Basalt (Figure 5), the resilient modulus is positively dependent on γd and negatively dependent on W, it is explained at 80.88%. For limestones (Figure 6) the resilient modulus is explained at 81.21% and is positively dependent on γd 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.
4.3. Correlation AnalysisIn this analysis, regression method by stepwise is used. It is based on the coefficient of determination R2 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), β0: intercept, βi: model parameters, Xi: 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 R2 of 0.34 and 0.32 for Uzan and NCHRP (2004) models. For limestones, there are a R2 of 0.96 for Uzan model and a R2 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:
GB for UZAN model• 
GB for NCHRP model• 
wang#title3_4:spLimestone for UZAN model• 
wang#title3_4:spLimestone for NCHRP model
SRM: Summury Resilient Modulus (MPa), W: water content (%), γd: dry density (kN/m3).
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 R2 of 0.38 and
0.34 for Uzan and NCHRP (2004) models. For limestones, there are a R2 of 0.98 for Uzan model and a R2 of 0.98 for NCHRP model. The result also showed that the R2 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:
GB for UZAN model• 
GB for NCHRP model• 
wang#title3_4:spLimestone for UZAN model• 
wang#title3_4:spLimestone for NCHRP model
SRM: Summary Resilient Modulus (MPa), W: water content (%), γd: dry density (kN/m3).
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.
5. ConclusionThe 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.