## 3 Prediction of Excavation-Induced Ground Movement
### Determination and Validation of the Maximum Surface Settlement  
The maximum surface settlement (\(\delta_{\mathrm{vvm},0}\)) is influenced by the factors \(H_{\mathrm{e}}\), \(\lambda\), \(S_{\mathrm{u}}/\sigma_{\mathrm{v}}^{\prime}\), \(EI\), and \(\theta\). The results of the FEM numerical analyses of the excavations were collected, and the relationships between the maximum surface settlement and inclination angle were obtained by statistical comparison. Figure 8 shows that the exponential function is the most appropriate to simulate the relationship between \(\delta_{\mathrm{vvm},0}\) and \(\theta\).  
By applying the least-squares method to the computed results and all the parameters, the maximum surface settlement, \(\delta_{\mathrm{vvm},0}\), can be derived from the equations below:  
\[\delta_{\mathrm{vvm},0}=(k_{1}e^{v}+k_{2})e^{ux_{5}} \tag{1}\]  
\[\begin{array}{c}v=k_{3}x_{1}^{2}+k_{4}x_{1}+k_{5}x_{2}^{2}+k_{6}e_{2}+k_{7}x_ {3}^{2}+k_{8}x_{3}+k_{9}x_{4}+k_{10}x_{1}x_{2}\\ +k_{11}x_{1}x_{3}+k_{12}x_{1}x_{4}+k_{13}x_{2}x_{3}+k_{14}x_{2}x_{4}+k_{15}x_{ 3}x_{4}+k_{16}\end{array} \tag{2}\]  
\[u=k_{17}(k_{18}x_{2}+k_{19})(k_{20}x_{3}+k_{21})(k_{22}x_{4}+k_{23})+k_{24} \tag{3}\]  
where \(v\) is used to determine the maximum surface settlement, \(x_{1}=\lambda\), \(x_{2}=H_{\mathrm{e}}\), \(x_{3}=S_{\mathrm{u}}/\sigma^{\prime}\), \(x_{4}=\ln(\gamma/EI)\), \(x_{5}=\theta/100\), and \(=\gamma\) is the water density. \(k_{1}-k_{24}\) are the fitting coefficients: \(k_{1}=50.1\), \(k_{2}=0.1\), \(k_{3}=-1.2\), \(k_{4}=-0.3\), \(k_{5}=1.9\), \(k_{6}=-1.2\), \(k_{7}=-44.9\), \(k_{8}=-0.3\), \(k_{9}=0.0\)  
Figure 8: The relationships of the maximum surface settlement for different angles.  
Figure 7: Flowchart of the proposed simplified method.  
\(k_{10}=2.7\), \(k_{11}=0.1\), \(k_{12}=6.9\), \(k_{13}=-0.1\), \(k_{14}=4.1\), \(k_{15}=3.8\), \(k_{16}=5.5\), \(k_{17}=0.1\), \(k_{18}=-0.8\), \(k_{19}=11.1\), \(k_{20}=-0.3\), \(k_{21}=0.3\), \(k_{22}=-13.2\), \(k_{23}=1.1\), and \(k_{24}=-14.2\).  
Then, the subsurface settlements can be obtained [4]. The maximum settlement at a specific depth, \(z\), can be calculated by the following equation:  
\[\frac{\delta_{\mathrm{0}vm,z}}{\delta_{\mathrm{0}vm,0}}={k_{25}}{x_{6}}^{2}+{ k_{26}}{x_{6}}+{k_{27}} \tag{4}\]  
where \(x_{6}=z/H_{\mathrm{e}}\); \(\delta_{\mathrm{0}vm,z}\) is the maximum settlement at a specific depth, \(z\); and \(k_{25}=0.3\), \(k_{26}=-1.0\), and \(k_{27}=1.0\).  
By selecting the data points not used in the fitting process (2/3 of the data points from the database were used for fitting), the predicted results from the formula are compared with the FEM results, as shown in Figure 9. Almost all data points are within the error range of \(\pm 20\%\), and the determination coefficient, \(R^{2}\), is 0.96, which demonstrates the accuracy of the numerical model in predicting the maximum surface settlement.  
To ensure the applicability of the formula, a further error analysis was performed by calculating the base function, which is a ratio of the numerical simulation value to the formula calculation value. After fitting the histogram in Figure 10, it was found that the data distribution follows a normal distribution, with a mean of 1.0 and a standard deviation of 0.09, indicating that the formula is effective.