CBR Predictive Models for Granular Bases Using Physical and Structural Properties

: The California bearing ratio (CBR) test evaluates the structure of the layers of pavements. Such a test is laborious, time-consuming, and its results are generally a ﬀ ected by sample disturbance and tests conditions. The main objective of this research was to build a numerical model for the prediction of CBR tests that might substitute laboratory tests. The model was based on structural and physical parameters of granular bases. Four di ﬀ erent materials from the central region (Quer é taro) and north (Mexicali) of Mexico were used for the experimental work. Using the above-mentioned materials, 36 samples were fabricated, and six of them were used for the evaluation of the model presented in this research. Numerical and experimental comparisons demonstrated the adequacy of the model to predict the result of CBR tests from soil


Introduction
The pavement is a structure formed by several soil layers designed to provide and maintain a smooth surface for several applications. It requires supporting and distributing the stresses as well as minimizing permanent deformations on it. In general terms, the structure is formed by the main pavement layer, the base, and the sub-base, which are built on a prepared subgrade surface [1,2]. In addition, the pavement structure can be constructed using reinforced concrete or simply asphalt emulsion.
Among the main factors affecting the performance and quality of pavements are the mechanical and hydraulic characteristics of materials employed for each layer, climatic conditions, equipment and technology used in the site, and the skills of workmen involved in the construction. Due to these and other factors, it is complicated to provide quality control schemes in the field of engineering of pavements [3]. Hence, it is important to check and verify certain parameters used in pavements during their construction. Otherwise, the predictions made for the durability and serviceability of pavements will not be realistic, affecting the costs of maintenance and rehabilitation [4]. The design of pavements requires the knowledge of soil mechanics and specifically the behavior of compacted soils. This discipline establishes the laboratory and field tests required to evaluate the quality of compacted layers as well as the needed conditions in terms of durability and serviceability of a pavement subjected to certain loading conditions. In general, field and laboratory tests must meet the following requirements: (a) simple and standardized, (b) swift, (c) easy to interpret, and (d) use inexpensive tools easy to calibrate and use [5]. In the condition received from the quarry, Table 2 presents some of the main characteristics of the materials studied in this research. The CBR values corresponded to samples compacted at the optimum water content, resulting from the Modified Proctor compaction tests. Besides, for their classification, the following 6 tests were performed: (1) consistency limits according to standard ASTM D4318-05 [15], (2) dry loose volumetric mass according to standard M-MMP-1-08/03 [16], (3) relative density of solids according to standard ASTM C127-12 [17], (4) modified Proctor compaction test according to standard ASTM D1557-09 [18], (5) CBR test according to standard ASTM D1883-07 [19], and (6) water content according to standard ASTM D2216-10 [20]. For the thirty-six different samples, their grain size distribution was obtained. Also, the main volumetric and gravimetric parameters for these samples were obtained, as well as the results of the CBR test. Seven different samples were tested from quarry 1 (samples 1 to 7). These samples were prepared with different grain sizes distributions and water contents. In this way, sample 1 showed the original grain size distribution of the quarry. The grain size distribution for samples 2, 3, and 4 was modified to produce samples with 50% gravel and 50% sand. Samples 5, 6, and 7 only contained sand. Samples with different characteristics were prepared from quarries 2, 3, and 4:10 for quarry 2, 9 for quarry 3, and 10 for quarry 4 (samples 8 to 36). Samples from the same quarry presented similar grain size distributions. Figure 1 shows the grain size distribution for the different samples according to standard ASTM C136-06 [15]. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 14 Some samples were compacted according to the Modified Proctor compaction method and considering different water contents. Other samples were prepared using different compaction energies (CE) with the purpose of analyzing its effect on the CBR results. For such samples, the number of blows was modified. Thus, samples 1 to 4 from quarry 1, samples 8 to 11 from quarry 2, samples 18 to 21 from quarry 3, and samples 27 to 31 from quarry 4 were compacted according to the Modified Proctor compaction method, and the samples 5 to 7 from quarry 1, 12 to 17 from quarry 2, samples 22 to 26 from quarry 3, and samples 32 to 36 from quarry 4 were compacted with the same equipment and procedure but applying a different number of blows. The compaction energy and water content for each sample are summarized in Table 3.
In addition to the main characteristics of the different samples, their volumetric and gravimetric parameters after compaction were obtained according to the procedures established by [21]. Such parameters are shown in Table 3 and include the volumetric weight (γm), the volumetric weight of solids (γs), the specific density of solids (ss), the relative density (Cr), the void ratio (e), the porosity (n), the degree of saturation (Gw), the degree of concentration of air (GA), the volumetric water content (θ), the degree of compaction with respect to dry volumetric weight from a compaction test (GC).
The low CBR values of some samples from the same quarry were related to their low compaction energy. Therefore, CBR values were influenced by both the grain size distribution and compaction energy. Some samples were compacted according to the Modified Proctor compaction method and considering different water contents. Other samples were prepared using different compaction energies (CE) with the purpose of analyzing its effect on the CBR results. For such samples, the number of blows was modified. Thus, samples 1 to 4 from quarry 1, samples 8 to 11 from quarry 2, samples 18 to 21 from quarry 3, and samples 27 to 31 from quarry 4 were compacted according to the Modified Proctor compaction method, and the samples 5 to 7 from quarry 1, 12 to 17 from quarry 2, samples 22 to 26 from quarry 3, and samples 32 to 36 from quarry 4 were compacted with the same equipment and procedure but applying a different number of blows. The compaction energy and water content for each sample are summarized in Table 3.
In addition to the main characteristics of the different samples, their volumetric and gravimetric parameters after compaction were obtained according to the procedures established by [21]. Such parameters are shown in Table 3 and include the volumetric weight (γ m ), the volumetric weight of solids (γ s ), the specific density of solids (s s ), the relative density (C r ), the void ratio (e), the porosity (n), the degree of saturation (G w ), the degree of concentration of air (G A ), the volumetric water content (θ), the degree of compaction with respect to dry volumetric weight from a compaction test (G C ).
The low CBR values of some samples from the same quarry were related to their low compaction energy. Therefore, CBR values were influenced by both the grain size distribution and compaction energy.

Results
In order to define which gravimetric and volumetric parameters have the largest influence on the values of CBR tests, dispersion graphics were used, and a tendency line was plotted for different parameters. This task was performed by plotting the coefficient R 2 , which indicated the reliability or accuracy of the correlation. In other words, the more R 2 coefficient approached unity, the more reliable or accurate was the correlation. Table 4 summarizes the values of coefficient R 2 for each one of the volumetric and gravimetric parameters described in Table 3 with respect to the thirty-four CBR tests.
It could be observed that the values of coefficient R 2 showed low values for all volumetric and gravimetric parameters of the soil. This means that not only a single parameter was influencing the CBR values but a combination of them. Also, different parameters affected CBR values, depending on the type of soil. For this reason, different equations were developed, depending on the type of material. Due to the nature of the soils tested, five groups of correlation analyses were performed for the different samples according to their classification: (1) samples with classification GW-GM and GP, (2) samples with classification SP, (3) samples with classification GW, (4) samples with classification GP, and (5) the combination of samples with classification GW or GP. Only these groups were created since the samples tested belong to such soil classifications. In order to develop other correlation analyses of materials with different soil classifications, it is necessary to perform tests on other materials with different graduation than those analyzed in this research. Table 5 shows the results of coefficient R 2 for the correlations, considering individually each one of the fourteen parameters for each group. Figure 2a-e shows these correlations.  In general, the parameters presenting the largest correlations with the CBR test are the dry volumetric weight (γ d ), the water content (w), and the void ratio (e). These correlations could be observed in Figure 2a. For materials GW-GM and SM, a linear relationship with γ d could be observed with R 2 = 0.83. For w, a polynomial correlation was observed with R 2 = 0.91. For e also, a linear relationship was observed with R 2 = 0.86.
In the case of the materials SP, CBR values correlated linearly with parameters γ d with R 2 = 0.73; w with R 2 = 0.7; e and n with R 2 = 0.73. Also, for materials GW, CBR values correlated linearly with the following four parameters: (1) γ d with R 2 = 0.97, (2) γ m with R 2 = 0.79, (3) e with R 2 = 0.96, and (4) n with R 2 = 0.96. Figure 2c illustrates these correlations. Materials GP showed also linear relationships with the following parameters γ d , e, and n with R 2 = 0.85 and γ m with R 2 = 0.71. These correlations are shown in Figure 2d. The combination of the materials with classification GW and GP showed correlations with parameters γ d with R 2 = 0.93; γ m with R 2 = 0.85; e and n with R 2 = 0.93. Such correlations are shown in Figure 2d.
From the results summarized in Table 5, it could be observed that water content influenced the results of sandy soils (materials SP). The largest correlations of parameters γ d , e, n, and γ m were obtained for gravels (materials GW and GP), while the lower for sandy materials. As the parameter w might show seasonal variations during the dry and wet season, also CBR values might be subjected to these seasonal variations.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 14 correlations with parameters γd with R 2 = 0.93; γm with R 2 = 0.85; e and n with R 2 = 0.93. Such correlations are shown in Figure 2d. From the results summarized in Table 5, it could be observed that water content influenced the results of sandy soils (materials SP). The largest correlations of parameters γd, e, n, and γm were obtained for gravels (materials GW and GP), while the lower for sandy materials. As the parameter w might show seasonal variations during the dry and wet season, also CBR values might be subjected to these seasonal variations. (a)

Regression Analysis
Regression analysis is a statistical method used to identify the relationship between dependent and independent variables. It provided the coefficients of the best fitting relationship between dependent and independent variables. In this case, CBR values represented the dependent variable, while soil parameters γ d , γ m , w, and e represented the independent variables. 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, γ d was replaced by its equivalence (Equation (5)), where γ 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).

CBR Predictive Model
Soils GW-GM and SM (plastic): For soils SP with no traces of plastic soils: For clean GW soils with no traces of plastic soils: For soils GP with no traces of plastic soil: 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. This might suggest that Equation (1) was more accurate when it was applied to soils compacted at the maximum dry density. This was so because the model was built from samples compacted at the optimum level. In addition, the model applied to sample 38 showed a difference of 24, which was reasonable, considering the differences in materials coming from different quarries.
It is important to mention that the CBR predictive models could be applied to materials showing the same geologic conditions, mechanical parameters, and consistency limits. Thus, due to this important limitation, it is necessary to develop more models of prediction of CBR, particularly applicable to materials with different classifications than those analyzed in this research. In addition to the classification of soils, consideration should be given to the plasticity of the material.

Discussion
The adequacy in developing CBR predictive models comes from the fact that laboratory tests need to be quick, easy, with no interference of the operator. The use of an analytical model to predict the result of CBR tests from simpler and current laboratory tests may yield in time-saving while keeping the same precision. In addition, it must be considered that the CBR values for a similar soil may be very diverse; such a variation depends on the number of combinations of the factors that define soil resistance. However, once results are obtained, certain correlations can be established to estimate the CBR value for a particular type of soil. Finally, it is important to mention that the CBR models presented in this paper might be restricted, in a certain way, to the physical conditions of the selected soil samples.

Conclusions
Based on the results presented in this paper, the following conclusions could be stated.

•
The predictive models for CBR tests were applied according to the classification of the considered soil. Four different CBR predictive models were obtained: for gravel and sand with some plasticity (GW-GM and SM); for sands (SP); for clean gravel (GW); and clean gravel well or poorly graded (GW or GP).

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For the development of the regression models, 14 parameters of the soil were considered.

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The more influencing parameters on the results of CBR tests were: γ m , γ d , e, n, and w. The last parameter presented an important influence on plastic materials.

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The precision of the models presented in this research was tested using compacted samples from different quarries to those initially employed for the development of the models. In this sense, it was observed that Equation (1) was more precise for samples compacted at the optimum level.
On the other hand, Equation (4) presented important differences to experimental results, which might come from the origin of the parent rock.