## 4 Sensitivity Analysis of Bearing Capacity Parameters of a Layered Foundation
### Grey Correlation Analysis of Influencing Parameters  
The previous section discussed and analyzed the effects of the width to depth ratio of the foundation, the ratio of the cohesive force, the ratio of the tangent value of the internal friction angle, and the thickness of the overlying soft clay on the bearing capacity of the foundation. To gain a more intuitive understanding of the impact of various factors on the ultimate bearing capacity of the foundation, in this section, grey correlation analysis is conducted based on MATLAB to understand the correlation level of each influencing factor. The specific formulas used are as follows:  
\[\Delta=\left|X_{i}-Y_{i}(j)\right| \tag{29}\]  
\[A_{ij}=\frac{\Delta\text{min}+\rho\Delta\text{max}}{\Delta ij+\rho\Delta\text{ max}} \tag{30}\]  
\[R_{j}=\left(\sum_{i=1}^{m}A_{ij}\right)/m \tag{31}\]where \(\Delta\) is the difference coefficient matrix; \(X_{i}\) is the column matrix, and is the \(i\)-th row element of the \(X\) matrix; \(Y_{i}(j)\) represents the \(i\)-th element of the influencing factor in column \(j\); \(A_{ij}\) is the correlation coefficient matrix; \(\Delta\)min is the minimum value of the coefficient of difference matrix, and \(\Delta\)min is the maximum value of the coefficient of difference matrix; \(\Delta_{ij}\) is the \(i\)-th row and \(j\)-th column element of the coefficient of difference matrix; and \(\rho\) is the resolution coefficient, with a value of 0-1. In this paper, a value of 0.5 is taken [24].  
In this paper, there are a total of 16 sets of operating conditions in the orthogonal experiment, including 3 sets of repeated operating conditions and 13 sets of non-repeated operating conditions. Therefore, \(Y_{i}(j)\) is a 13 \(\times\) 4 matrix composed of normalized input variables of various influencing factors. \(X_{i}\) is the corresponding sequence of the ultimate bearing capacity of the foundation, and the ultimate bearing capacity of each row in \(X_{i}\) is the normalized output value corresponding to each variable in \(Y_{i}(j)\). In this paper, each column of \(Y_{i}(j)\) from left to right represents the ratio of foundation width to depth, the ratio of cohesive force, the ratio of tangent value of internal friction angle, and the thickness of overlying soft clay. \(A_{ij}\) is the correlation coefficient matrix obtained from grey correlation analysis between various influencing factors and the ultimate bearing capacity of the foundation. By taking the mean of all values in column j of \(A_{ij}\), the grey correlation degree of the influencing factor can be obtained.  
The final grey correlation matrix is obtained as R = [0.7977 0.5613 0.6820 0.7243]. Figure 10 shows the correlation between the parameters selected for numerical simulation in this paper and the ultimate bearing capacity of the foundation. From Figure 10, it can be seen that the influencing factors considered in the numerical simulation in this paper are highly correlated with the ultimate bearing capacity of the foundation, so the selection of influencing factors is relatively reasonable. Among them, the width to depth ratio of the foundation has the greatest impact on the ultimate bearing capacity of the foundation, followed by the thickness of the overlying soft clay, the tangent ratio of the internal friction angle, and the ratio of the cohesive force. Therefore, in practical engineering, when designing foundations, it is necessary not only to consider the size design of the foundation but also to fully analyze the inherent properties of the foundation soil and the thickness of the soft clay interlayer.