# 3D In Situ Stress Estimation by Inverse Analysis of Tectonic Strains
## 3 Application in the In Situ Stress Determination of Three Gorges Dam
### Inverse Analysis  
In order to obtain a reasonable engineering-scale in situ stress field, the stress-based approach and the proposed approach were used to simulate the engineering-scale in situ stress field. The bottom of the calculating model was fixed, and the lateral boundary conditions differed with a different approach. For the stress-based approach, the linear distribution of stress at the boundary was selected and applied on lateral boundaries during FEM calculation. For the proposed approach, the displacement boundary conditions were applied (Equations (26)-(28)) during FEM calculation. For the purpose of this study, predicting the deformation induced by excavation is not a focus, therefore, the elastic model was used during FEM computing in this paper. For the objective function in this project, it was established based on the measured values and corresponding calculated values (Equation (12)). The stress relief method through overcoming and hydraulic fracturing was used to test the in situ stress field (Changjiang River Scientific Research Institute, 1996). Here, the results of seven representative test points are selected as the measured values in this inverse analysis of an engineering-scale in situ stress field, which are shown in Figure 8 and Table 2. From Figure 8, it can be seen that seven test points are sorted in order from D2 to D8, located at the same altitude, and the Z axis coordinate is 95 m (Figure 8b). The test points D4 and D7 are located between the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Rock Mass**} & **Weight (10\({}^{4}\) N/m\({}^{3}\))** & **Young (Modulus (GPa)** & **Poisson’ Ratio** & **Cohesion (MPa)** & **Friction Angle (\({}^{\circ}\))** \\ \hline \hline
\begin{tabular}{c} Slightly weathered \\ rock mass \\ \end{tabular} & 2.68 & 30 & 0.22 & 2.0 & 59 \\ \hline \hline F20, F22, F32 & 2.6 & 15 & 0.25 & 0.20 & 35 \\ \hline F84 & 2.6 & 5 & 0.30 & 0.10 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of rock mass. Figure 8: The spatial distribution of seven test points in 3D calculating model. structure planes F20 and F22 (Figure 8a), moreover, D6, D5 and D4 gradually become closer to the structure plane F84 (Figure 8c), and thus, D7, D3 and D8 are becoming gradually closely located to the structure plane F32 (Figure 8a,d), and D6, D7 and D4 are gradually close to the structure plane F20 (Figure 8a). In the proposed approach, only six strain variables need to be optimized, and therefore, the simplex optimization algorithm was adopted in this paper. This simplex optimization algorithm in the MATLAB software can be used and the optimizing process was implemented using the MATLAB code combining the simplex optimization algorithm and the in situ stress calculation with FEM. The optimization processes with both approaches are shown in Figure 9. It can be seen that the proposed approach has faster convergence and a better convergence process than the stress-based approach. It also shows that the simplex optimization algorithm is sufficient.