## 2 Materials and Methods
The relationship between the dependent and independent variables can be expressed as follows:  
\[Y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\ldots+a_{n}x_{n}+e_{i}\]  
where \(Y\) is the dependent variable; \(a_{0}\) is the constant term of the equation; \(a_{0}\), \(a_{1}\), \(\ldots\), \(a_{n}\) are the coefficients of the independent variables \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{n}\); \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{n}\) are the independent variables; and \(e_{i}\) is the error term, which represents the difference between the observed value and the predicted value (\(Y\)-\(\hat{Y}\)). This model allows the dependent variable to be estimated by considering the effect of each independent variable on the bearing capacity and consolidation of the soil [4,63]. This research is a study in which only the test results of gravel soils belonging to 27 drilling data are examined; these test results are statistically analysed, the relationships of bearing capacity and settlement with other parameters of the soil are examined, and mathematical models are created. Estimation of the allowable bearing capacity of granular soil requires an intensive field research program. This research proposes empirical correlations to estimate the allowable bearing capacity and elastic settlement of shallow foundations on granular soils. The existing correlation and estimation of bearing capacity can be obtained by using only standard penetration blow number and soil unit weight. Such correlations can be used in the preliminary stage of estimating the allowable bearing capacity and elastic settlement of shallow foundations on granular soils, and can help field engineers make immediate decisions in case of field changes given in soil reports. Some researchers have studied this issue and adopted various approaches [64; 65; 66; 67; 68; 69]. For the foundation design of any structure, two important criteria need to be met: one is bearing capacity and other is settlements. Many authors have conducted research on the bearing capacity of soil considering various parameters like cohesion, Maximum Dry Density (MDD), Optimum Moisture Content (OMC), depth and width of foundation, and shear resistance angle by conducting various tests like the Direct Shear Test and standard proctor test. Safe bearing capacity of soil has been calculated using various methods like Terzaghi and IS Code. The effect of foundation shape and size and water table location \begin{table}
\begin{tabular}{c c} \hline \hline
**Correlation Range** & **Level of Relationship** \\ **(Can Be + and – Values)** & **Very weak relationship** \\ \hline
0.00–0.25 & Very weak relationship \\ \hline
0.26–0.49 & Weak relationship \\ \hline
0.50–0.69 & Moderate relationship \\ \hline
0.70–0.89 & High relationship \\ \hline
0.90–1.00 & Very high relationship \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation ranges and relationship levels. on ultimate bearing capacity in the case of local shear failure has been studied [70]. Some studies were conducted on various aspects related to displacements. Poisson-final ratio, foundation stiffness, and foundation geometry, etc. have been developed graphically in the case of shallow foundations [71; 72; 73; 74]. Moreover, some studies were made on finite and infinite soil modulus, which increase with soil depth [75; 76; 77]. Allowable bearing capacity and settlement are the two main criteria that control the design practices of shallow foundations to meet safety and usability requirements. Geotechnical engineers face significant uncertainties in terms of soil variability, time effects, construction effects, human error, and sampling and laboratory test inaccuracies while evaluating the soil. All these factors make the use of experimental correlations preferable to traditional methods because they are practical and less costly than traditional methods.