## 3 Case Study and Results
### Finite Element Model of a Typical Concrete Gravity Dam  
According to the basic information, a two-dimensional plane strain finite element model of the typical concrete gravity dam was established. The foundation rock is homogeneous characterized by the Mohr-Coulomb model [30]. The linear elastic model was adopted in dam concrete which is in good condition. A thin layer with a height of 0.2 m was used to simulate the contact between the dam and the foundation [30], and its material properties were consistent with foundation rock. Foundation boundary conditions were set to apply fixed constraints at the bottom of the foundation and horizontal constraints at the upstream and downstream of the foundation. The foundation and the dam were considered to be impervious. Dam weight and hydrostatic pressure acting on the upstream and downstream surfaces of the dam body and the foundation were taken into account. The main material properties of dam concrete, foundation rock, and contact are shown in Table 2.  
When the finite element method is used to calculate the dam stress, the calculation domain needs to include the dam foundation. To reflect the influence of foundation stiffness on dam stress, the foundation was selected to a certain range, but in fact, the foundation was a semi-infinite space. To simplify the infinite space with limited space, the general way is to simulate by applying boundary constraints. According to the Saint-Venant principle, if the range of foundation considered is larger, the influence of boundary constraints on dam stress is smaller. When it reaches a certain range, the dam stress is almost no longer affected by the expansion of the foundation range. However, scholars do not have the same standard for the selection of foundation range.  
In this section, the foundation range of the finite element model for the stress of the dam toe and heel and anti-sliding stability analysis was selected by calculation. The calculation diagram of foundation range selection is shown in Figure 3, in which a = 0, 1,... 9 and 1 h is equal to 1 times the height of the dam.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Material** & **Elasticity Modulus (GPa)** & **Density (kg/m\({}^{3}\))** & **Poisson Ratio** & **Friction Angle (\({}^{\circ}\))** & **Cohesion (MPa)** \\ \hline Dam concrete & 28 & 2400 & 0.17 & - & - \\ Foundation rock & 26 & 2600 & 0.25 & 35 & 15 \\ Contact & 26 & 2600 & 0.25 & 35 & 15 \\ \hline \end{tabular}
\end{table}
Table 2: Main material properties. Calculation results.  
Figure 2: Typical profile of a gravity dam.  
The results show that the values of \(K_{\text{c}}\) and \(P_{\text{min}}\) first increase and then decrease with the increase in the foundation range. The influence of the foundation range on \(K_{\text{c}}\) and \(P_{\text{min}}\) is small. With the foundation range increasing from 1 h to 10 h, the values of \(K_{\text{c}}\) and \(P_{\text{min}}\) change to 0.00518 and 480 Pa, respectively. The values tend to become stable after the foundation range is greater than 3 h. The value of \(P_{\text{max}}\) first decreases and then increases with the increase in the foundation range. The influence of the foundation range on the \(P_{\text{max}}\) is also small. After the foundation range is greater than 3 h, the value tends to be stable. Therefore, in the finite element model, the foundation range is as follows: the length of the upstream foundation, length of the downstream foundation, and depth of the foundation were taken as 3 h.