# CBR Predictive Models for Granular Bases Using Physical and Structural Properties
## 3 Results
### CBR Predictive Model  
Table 6 shows the coefficients of the CBR predictive models for each type of soil obtained from the multi-linear regression analysis. For this technique, how the coefficient R\({}^{2}\) was obtained had no relevance, since this parameter was used to determine the level of influence on the CBR, so, it was valid to use multilinear regression analysis even though the coefficient R\({}^{2}\) of w was obtained by means of a polynomial function. Also, the linear multiple regression analysis yielded values for coefficients that made up equations whose variables are of degree 1 (linear). It could be observed that the standard deviation for material GP was larger when compared with the combination of models for GP and GW. For this reason, a model was proposed for both materials. In Table 6, it could be noticed that the predicted CBR values for gravels were closer to experimental results when the soil was clean with no traces of plastic soil. Therefore, four models had been established for the materials analyzed in this research. The four models (Equation (4)) could be used in materials GW or GP, but it was decided to apply only in GP materials since model 3 (Equation (3)) had greater reliability when applied to GW materials. Besides, \(\gamma_{d}\) was replaced by its equivalence (Equation (5)), where \(\gamma_{0}\) is the specific weight of distilled water (equal to 1 or an entire power of 10); n was eliminated; hence it is related to Equation (6). Soils GW-GM and SM (plastic):  
\[CBR=\ -0.6231\frac{S_{S}}{1+\omega S_{S}}\gamma_{O}-9.5447w-1319.1924e+1924.9925 \tag{1}\]  
For soils SP with no traces of plastic soils:  
\[CBR=1.6064\gamma_{d}-5.3303w+2462.2411e-3913.2472 \tag{2}\]  
For clean GW soils with no traces of plastic soils:  
\[CBR=0.5979\frac{S_{S}}{1+\omega S_{S}}\gamma_{O}+0.0024\gamma_{m}+469.4978e-13 07.6738 \tag{3}\]  
For soils GP with no traces of plastic soil:  
\[CBR=0.1856\frac{S_{S}}{1+\omega S_{S}}\gamma_{O}-0.0551\gamma_{m}-346.259e-11 3.4502 \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**USCS** & \(\gamma_{d}\) **(x\({}_{1}\))** & \(\gamma_{m}\) **(x\({}_{2}\))** & **w** & e & n & **Constant** & \(\mathbf{R^{2}}\) & **Error** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline GW-GM & \(-0.6231\) & — & \(-9.5447\) & \(-1319.1924\) & — & \(1924.9925\) & \(0.9052\) & \(26.3470\) & \(3.7380\) \\ \hline SP & 1.6064 & — & \(-5.3303\) & \(2462.2411\) & \(0\) & \(-3913.2472\) & \(0.9559\) & \(4.4894\) & \(1.7746\) \\ \hline GW & 0.5979 & 0.0024 & — & \(469.4978\) & \(0\) & \(-1307.6738\) & \(0.9686\) & \(3.4216\) & \(2.4894\) \\ \hline GP & 13.9330 & 0.0667 & — & \(4816.3003\) & \(292.0012\) & \(-36564.5609\) & \(0.8938\) & \(9.4288\) & \(2.4352\) \\ \hline GW and GP & 0.1856 & \(-0.0551\) & — & \(-346.259\) & \(0\) & \(-113.4502\) & \(0.9256\) & \(6.8143\) & \(6.0547\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Coefficients of the predictive models for each type of soil. \[\frac{S_{S}}{1+\omega S_{S}}\gamma^{\prime\mathcal{O}} \tag{5}\] \[\mathrm{n}=\frac{\mathrm{e}}{1+\mathrm{e}} \tag{6}\]  
The above-mentioned models were selected, depending on the soil classification, according to USCS. Hence, they required volumetric weight, void ratio, and water content of the compacted material according to the Modified Proctor test [18]. As previously mentioned, the precision of such models had been tested using six samples (37 to 42) of compacted material obtained from three different quarries. Samples 37 and 38 came from different quarries and were tested at the optimum water content, whereas samples 39 to 42 were compacted at a water content different from the optimum. For sample 41, the CBR test was performed at the maximum dry volumetric weight. Table 7 shows the CBR values obtained in the laboratory and those obtained with the corresponding predictive model according to the soil classification and the consistency limits. Samples 39 and 42 showed the largest deviation from the experimental CBR value.