Development of Empirical Models for the Estimation of CBR Value of Soil from Their Index Properties: A Case Study of the Ogbia-Nembe Road in Niger Delta Region of Nigeria ()

Jonathan O. Irokwe^{}, Ify L. Nwaogazie^{*}, Samuel Sule^{}

Department of Civil and Environmental Engineering, University of Port Harcourt, Port Harcourt, Rivers State, Nigeria.

**DOI: **10.4236/ojce.2022.124036
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Department of Civil and Environmental Engineering, University of Port Harcourt, Port Harcourt, Rivers State, Nigeria.

This study developed empirical-mathematical models to predict the California
Bearing Ratio (CBR) using soil index properties in Ogbia-Nembe road in the
Niger Delta region of Nigeria. The determination of CBR of soil is a laborious
operation that requires a longer time and materials leading to increased cost and schedule; this can be reduced
by adopting an empirical-mathematical model that can predict the CBR
using other simpler soil index properties such as Plastic Limit (PL), the
Liquid Limit (LL), the Plasticity Index (PI) and the Moisture Content (MC), which
are less laborious and take lesser time to obtain. Thirteen models were
developed to understand the relationship between these soil index properties:
the independent variable and the California Bearing Ratio (CBR): the dependent
variable; Six linear, Six quadratic and One multiple linear regression models
were developed for this relationship. Analysis of variance (ANOVA) on the
thirteen models showed that the Optimum Moisture Content (OMC) and the Maximum Dry Density (MDD) are better
independent variables for the prediction of the CBR value of Ogbia-Nembe soil
generating a quadratic model and a multiple linear regression model having a
better coefficient of determination R^{2} = 0.96 and 0.94 respectively, mean square error
(MSE) of 0.74 and 1.152 respectively with Root mean square errors of 0.861 and
1.073 accordingly. These models were used to predict the CBR of the soil. The
CBR values predicted by the model were further compared with those of the actual experimental test and found to be relatively consistent with minimal variance. This establishes that CBR of
any soil can be predicted from the Index Property of the soil and this is more
economical and takes lesser time and can be universally adopted for soil
investigation.

Keywords

Multiple Regression Model, Soil Index Properties, Analysis of Variance, California Bearing Ratio, Coefficient of Determination

Share and Cite:

Irokwe, J. , Nwaogazie, I. and Sule, S. (2022) Development of Empirical Models for the Estimation of CBR Value of Soil from Their Index Properties: A Case Study of the Ogbia-Nembe Road in Niger Delta Region of Nigeria. *Open Journal of Civil Engineering*, **12**, 648-664. doi: 10.4236/ojce.2022.124036.

1. Introduction

Every geotechnical engineering work, both horizontal and vertical construction requires the investigation of the soil carrying capacity which is aimed at determining the sustainability of such soil for the purpose of taking the resultant load resulting from such construction and the service load at use [1]. This is important as the functionality of such structure has a direct relationship with the soil upon which it is situated [2]. The weaker the soil, the more tendency for structures erected on it to become weak and collapse, while the stable the soil is, the lesser the tendency for the structure erected on it to get weak and collapse [3]. More so, construction cost depends on the stability of the soil as more work may need to be done on improving the quality of weak soil for sustainability for any meaningful Engineering works [4]. Surrounded by these uncertainties and the fact that soil conditions vary from one location to another and from one point to another even within a given vast expanse becomes an issue of concern [5]. To solve this concern, an investigation must be carried out on the condition of the soil on every site at intervals to determine the soil condition for proper engineering design and construction [6].

According to [7], The Niger Delta Region is characterized by unstable soil conditions which vary in strength and other physical characteristics. Engineering works executed within this region must be properly planned and adequate information on the supporting soil done to avoid failures. According to [8], one of the determinant investigative studies required of every soil for the purpose of engineering works is the determination of the California Bearing Ratio (CBR). It is a measure of the strength of the soil material which is usually used as the foundation layer (subgrade) of every Horizontal Engineering construction such as Road construction or Vertical Engineering construction such as skyscrapers [9]. The general processes leading to the determination of the CBR value of soil require an empirical method of investigation which involves the collection of the soil sample and subjecting it to pressure to achieve penetration using a pluger of standard area which is compared with an equal pressure required to achieve equal penetration on a standard crusted rock [10]. The value of the CBR of soil determines various design parameters of the structure coming on the soil [11]. For a road pavement, a lower CBR will inform the choice of an increased subgrade thickness while for structures like building, soil with a low CBR will require increasing the foundation depth to balance the loading on the soil [12].

The determination of the CBR of soil is usually a traditional practice and a prerequisite before any engineering work can be done on any soil and this whole process is a laborious and expensive test that requires collection of soil sample at locations along the site of investigation at pre-defined intervals. The collected sample forms a representation of the soil along the area or extent of the construction. The collected samples are remodeled at predetermined optimum moisture content and maximum dry density with standard proctor compaction for the purpose of the test [13]. This test is usually conducted for both the soaked and unsoaked conditions, making the determination of CBR of soil a time dependent empirical analysis. The constraints associated with conducting CBR test on a large area of construction site like in construction of roads, will involve increasing the number of trial points which will result to cost overrun and an increase in project duration and if it is not properly conducted may not accurately reflect the variation and distribution pattern of the soil strength over the large construction area.

In trying to bridge this gap, many literatures have been written with great attempts to develop means of solving the puzzle of determining the compaction characteristics of the soil without going through the rigors of conduction the actual CBR test which is rather costly and time consuming. All the contributors who have made attempts in solving this limiting situation have focused on varying one index property over another and all their attempts are in predicting the compaction index properties of the soil without the actual estimation of the California Bearing Ration of the Soil. [14] developed a correlation model that correlates the compaction characteristics of natural soils which included the Optimum Moisture Content (OMC) and the Maximum Dry Density (MDD) with the plastic limit (PL) after an adjusted grading of the soil. Like all other contributors to this noble attempt, their works have centered on the correlation of the various index properties of the soil to another and to the compaction characteristic of the soil with the index properties without a definite correlation of the various index properties and the compaction properties in determining the California Bearing Ration (CBR) of the soil. This study attempts to correlate the CBR value of a soil with other physical properties of the soil include Plasticity Limit (PL), Liquidity Limit (LL), Plasticity Index (PI), Maximum Dry Density (MDD) and Optimum Moisture Content (OMC) that do not require laborious testing technique and is aimed at bridging the cost and time constrains involved in carring out CBR test.

2. Methodology

2.1. Study Area

The study area is the Ogbia-Nembe Road. This road is joined to the Nembe- Ogidiama Road in Nembe Town and to Oloibiri-Ibelebiri-Mbiama Road in Ogbia axis and is located within longitudes 4.58˚23'72"N and 4.71˚94'46"N and latitudes 6.36˚51'86"E and 6.35˚94'50"E. (Figure 1). According to [15], the road was constructed by Niger Delta Development Commission (NDDC) in partnership with Shell Petroleum Development Company (SPDC), covering 29 kilometers through swampy terrain with spurs. The road is surrounded by thick mangroves and swampy land, and is situated about 10 km east of Yenagoa, the capital city of Bayelsa State. The major economic activities in the area are those related to the oil and gas industry business and other economic activities include subsistence agriculture, artisanal fishing and basic commerce.

2.2. Data Collection and Analysis

The Ogbia-Nembe Road was delineated and disturbed Soil samples collected from twenty trial pits along the alignment at predetermined chainages using soil hand ulger at varying depth but not exceeding 1.3 m depth. The collected soil samples from each of the twenty pits were subjected to Laboratory test for CBR value and other physical properties which included Plastic Limit, Liquid limit, Plasticity Index, Optimum Moisture Content, Maximum Dry Density, Particle size distribution in accordance to [10]: Methods of Testing for soils for Civil

Figure 1. Combined map showing Ogbia-Nembe road on the Ogbia Local government map.

Engineering Purposes. XLSTAT was used to perform basic calculations, data analysis, create tables and figures.

3. Results

Distribution of Soil Index Parameters

Results of the various physical properties of the soils collected from each of the twenty trial points is tabulated in Table 1. They include the Index properties: Plastic Limit, Liquid Limits, Plasticity Index and Specific Gravity; compaction characteristics—Maximum dry density, optimum moisture content and California bearing ratio obtained at optimum moisture content.

The result from Table 1 and Table 2, it can be seen that the plastic limit for the soil samples on Ogbia-Nembe road ranged from 12.37% to 37.00% with a mean value of 27.4%. The liquid limit for the soil samples ranged from 22.00% to 63.00% with a mean value of 49.80%. The plasticity index of the soil samples ranged from 9.00% to 30.00% with a mean value of 22.7%. The very high liquid limit and moderately high plasticity index give indication that the soil is highly plastic. For the moisture content of the soil samples, the moisture content ranged from 13.20% to 52.20% with a mean value of 21.40%. The specific gravity of the soil samples ranged from 2.53 to 2.63 with a mean value of 2.6 and the maximum dry density ranged from 1.60 to 1.84 g/cm^{3} with a mean value of 1.71 g/cm^{3}. The optimum moisture content of the soil samples ranged from 6.3% to 16.20% with a mean value of 10.42%.

4. Discussions

4.1. Classification Based on the Unified Soil Classification System (USCS)

The result of the classification of soil samples collected from Ogbia-Nembe roads as shown in Figure 2. The result from Figure 2 showed that eleven soil samples had a liquid limit greater than or equal to 50%. It also showed that three soil samples were inorganic clay of high plasticity (fat clay), eight soil samples were inorganic clay of very low to medium plasticity (lean clay), eight soil samples were either organic clay of high plasticity or inorganic silt of high compressibility, and two soil samples were inorganic silt of very low to medium compressibility or organic silt of low compressibility. Figure 3 showed that 40% of the total soil samples were inorganic clays with very low to medium plasticity and 40% of the total soil samples collected on Ogbia-Nembe road were either inorganic silt with high compressibility or organic clay with high plasticity. The result from the classification of the soil on Ogbia-Nembe road revealed that the subgrade soil existing is predominately either organic clay or lean clay [16].

4.2. Classification Based on American Association of State Highway and Transportation Officials AASHTO

Figure 4 presents the classification of soil samples collected on the Ogbia-Nembe

Table 1. Results of the various physical properties of the soil collected on east-west road.

Table 2. Soil Index properties for sample soil collected on Ogbia-Nembe road.

Figure 2. Soil classification of the flexible section of Ogbia-Nembe road.

road based on AASHTO method. The result from Figure 4 showed that two soil samples fell into the A-4 or A-2-4 class, one soil sample fell into A-2-6 or A-6 class, six soil samples fell into A-2-7 or A-7-5 class, and eleven soil samples fell into A-7-6 class. The result further revealed that the soil samples collected on Ogbia-Nembe road were predominately A-7-6 class soil. Based on AASHTO soil classification, subgrade soil that falls into the soil class A-7-6 would generally have fair to poor strength, from Table 3: Number of soil type based on AASHTO soil classification, therefore it is expected that the subgrade strength for Ogbia-Nembe to be weak.

Figure 3. Number of soil types on Ogbia-Nembe road.

Figure 4. AASHTO Soil classification of the flexible section of Ogbia-Nembe road.

Table 3. Number of soil type based on AASHTO soil classification.

No. of soil samples = 20.

4.3. California Bearing Ratio of the Subgrade Soils

The result after testing of the California Bearing Ratio for the subgrade for Ogbia-Nembe roads in the Niger Delta region is presented in Table 4. The CBR for the 20 sample points tested on Ogbia-Nembe road ranged from 15.1% to 30.4%, with the mean CBR for the 20 sample points to be 24.23% ± 4.10%.

4.4. Pearson R Correlation between the California Bearing Ratio and Soil Index Properties

The result of the Pearson correlations between soil index properties and the California Bearing Ratio for the Ogbia-Nembe road is presented in Table 5 and Figure 5. The result from Table 5 shows the Pearson correlations between soil index properties and CBR of the soil samples collected from Ogbia-Nembe road. Figure 5 showed that the plastic limit had a positive correlation with the liquid limit with Pearson correlation of 0.95, which implies that as the liquid limit of the soil increase positively linearly so does the plastic limit of the soil increase. The correlation between the plastic limit and liquid limit was significant at a level of significance of 5%. Also, both plastic and liquid limit were positively correlated with the plasticity index of the soil. The Pearson Correlation between the plastic limit and the plasticity index was 0.84 while the Pearson correlation between the liquid limit and plasticity index was 0.93, and the association between the three variables were significant (p < 0.05). The moisture content of the soil was positively correlated with the optimum moisture content with Pearson correlation of 0.54, indicating that as the moisture content of the soil increases so does the optimum moisture content. Investigating the linear association of the soil index properties and the CBR as shown in Figure 6, the results in Figure 6 showed that the optimum moisture content was strongly negatively correlated with the CBR with Pearson correlation of −0.97, indicating very strong negative association between the two variables. The result implies that as the optimum moisture content of the soil increases it result to a corresponding decrease in the CBR and vice versa. The result also showed that the maximum dry density had a positive association with the CBR with Pearson correlation of 0.91, indicating that an increase in the maximum dry density would result to an increase in the CBR and vice versa. The result also showed that the moisture content of the soil was negatively correlated with the CBR with Pearson correlation of −0.55

Table 4. CBR statistics of the selected section of the east-west road.

Table 5. Pearson R relationship between california bearing ratio and soil index properties in Ogbia-Nembe road.

Values in bold are different from 0 with a significance level alpha = 0.05.

Figure 5. Heatmap showing the pearson correlation between soil index properties and CBR for Ogbia-Nembe soil samples.

Figure 6. Linear model showing the relationship between the soil index properties and California bearing ratio for Ogbia-Nembe.

indicating that an increase in the moisture content will decrease the CBR and vice versa this is also in line with the literature of [17].

4.5. Modelling the Relationship between the California Bearing Ratio and the Index Properties of the Soil

The modeling was done in other to reduce the cost of conducting CBR experiment which is expensive to carry out and requires at least 4 days to conduct. The models developed will enable prediction of the CBR of any given soil with relatively similar soil index properties. Thirteen models were developed to understand the relationship between the soil index properties and CBR and also to investigate if the model would be a good predictor of the CBR value of the subgrade. Six linear models, six quadratic model and one multiple linear regression model were developed relating the variables.

Model 1: Sample Linear Regression Model (*y* = *a*_{0} + *a*_{1}*x*)

Model 1 is a simple linear regression model generating the linear model equations (Model No. 1-6) in Table 6. The relationship between the CBR and the Plastic Limit (PI) shows a linear relationship generating the linear Regression model: Model No 1. CBR =25.90 − 0.061PL and the coefficient of determination R^{2} was 0.009 which indicates that only 0.009 variability in the CBR can be explained by the plastic limit of the soil. The result of the Analysis of Variance (ANOVA) shows that the relationship between the CBR and plastic limit was not statistically significant. Same applies to the CBR relationship with the liquid limit Plasticity Index and the Moisture Content which have their coefficient of determinations R^{2} as 0.00, 0.01 and 0.30 respectively in model Nos. 2 - 4 in Table 6: indicating that the relationships are statistically insignificant. Model No. 5 & 6 relates the relationship between the CBR and the Maximum Dry Density (MDD) and Optimum Moisture Content (OMC) respectively and the corresponding linear model equations relating their relationship are: CBR = 69.89 + 54.944*MDD with R^{2} = 0.82 and CBR = 38.676 − 1.386OMC with R^{2} = 0.9374 respectively. The coefficient of differential is therefore statistically significant.

Model 2: Quadratic Model (*y* = *a*_{0} + *a*_{1}*x* + *a*_{1}*x*^{2})

Model 2 represents the quadratic model which also has six (6) quadratic models: Model No. 7 - 12 in Table 6. With the various soil index properties as independent variables and the CBR as the dependent variable, six quadratic models were generated showing the relationships of the CBR with each of the six index properties in Figure 7. Model Nos. 7 - 10 in Table 6 shows the quadratic relationship of the CBR with the Plasticity Limit (PL), Liquidity Limit (LL), Plasticity Index (PI) and the Moisture Content (MC). The coefficient of determination (R^{2}) of the four quadratic models are 0.11, 0.13, 0.08 and 0.35 respectively which again shows the relationship between them and the CBR of the soils are not statistically significant for the prediction of CBR outcome. CBR = −0.820.11 + 926.06MDD − 252.49MDD^{2} & CBR = 30.948 + 0.157OMC − 0.071OMC^{2} with coefficient of differentials R^{2} being 0.0897 and 0.96 respectively which are statistically significant for the prediction of the CBR.

Model 3: Multiple Linear Regression Model

Model 3 is a combined multiple linear regression model of the various soil index properties that have significant relationship with the CBR in the modeling equations in Model 1 and 2. It considers the two soil index properties that are statistically significant for the modeling of the CBR of the soil using linear regression model, *i.e*., Maximum Dry Density (MDD) and the Optimum Moisture Content and this generated one single equation.

Table 6. Regression analysis result of soil index properties with the CBR (Ogbia-Nembe).

Figure 7. Quadratic model showing the relationship between the soil index properties and California bearing ratio for Ogbia-Nembe.

CBR = 52.16 − 6.927*MDD − 1.5420*OMC

With a coefficient of determination (R^{2}) of 0.9387, which is very statistically significant having Mean Square Error (MSE) of 1.152 and Root Mean Square Error (RMSE) of 1.073. Result of the ANOVA provides sufficient evidence in stating that the Optimum Moisture Content (OMC) and the Maximum Dry Density are good predictors for evaluating the CBR of the subgrade soil of Ogbia-Nembe road.

Figure 8 shows a comparison of the actual CBR values obtained by laboratory investigation of the Obia-Nembe road and the predicted CBR values based on the regression models generated. Table 7 shows the variance in the CBR of

Figure 8. Comparison of the CBR between the predicted and actual value.

Table 7. Percentage variation in the predicted CBR.

Ogbia-Nembe soil when the actual and predicted are compared, indicating a reasonable fit to the soil type in the study area [18].

5. Conclusions

It was established that CBR value of soil can be estimated from the values of some of the index properties of the soil which include the Liquidity Limit (LL), Plastic Limit (PL), and Plasticity Index (PI) and also from the Compaction characteristics which includes Optimum Moisture Content (OMC), Maximum Dry Density (MDD) and Moisture Content (MC). Linear, Quadratic and multiple regression models were developed to predict the CBR value of the Ogbia-Nembe road in Niger Delta area of Nigeria from these characteristics of the soil. Thirteen models comprising Six linear, Six quadratic and One multiple linear regression models were developed from the correlation between these soil properties: the independent variable and the California Bearing Ratio (CBR): the dependent variable.

However, result from the Analysis of variance (ANOVA) showed that the Optimum Moisture Content (OMC) and the Maximum Dry Density (MDD) are better independent variables for the prediction of the CBR value of Ogbia-Nembe soil. Hence, the empirical models;

CBR = 30.948 + 0.1570MC − 0.0718*OMC^{2} and CBR = 52.16 − 6.927*MDD − 1.542*OMC

A quadratic model and a multiple linear regression model having a coefficient of determination R^{2 }= 0.96 and 0.94 respectively are most suitable with the best fitness. The CBR values predicted by the model were further compared with those of the actual experimental test and found to be relatively consistent with minimal variance.

These developed empirical-mathematical models can be used to predict the CBR values of any given soil having similar properties as those of Obia–Nembe Road and this will be a convenient and less laborious means for construction practitioners to determine the CBR of soil for the purpose of suitability for engineering works and will reduce the general time and cost required for the experimental process.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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