## 3 Case Study  
### Model Information  
For a certain arch dam, the maximum dam height is 76.30 m, the crest elevation is 126.30 m, the arc length of the crest is 419.00 m, the thickness of the crest is 8.00 m, the thickness of the bottom is 53.25 m, and the normal storage level of the reservoir is 119.00 m [55]. After finite element model calculation, it is recommended to select the appropriate range of foundation modeling for this arch dam, which is three times the dam height along the depth direction of the foundation, the length direction of the left and right banks, and the upstream and downstream length direction. For the calculation and analysis in this paper, it is assumed that the dam concrete and the foundation are isotropic linear elastic materials, in which the modulus of elasticity of the dam concrete is 24.75 GPa, the Poisson's ratio is 0.2, and the density is 2450 kg/m\({}^{3}\). The elastic modulus of foundation material is 15 GPa, Poisson's ratio is taken as 0.2, and the density is taken as 1 \(\times\) 10\({}^{-6}\) kg/m\({}^{3}\). The two methods of the Westergaard added mass model and FSCM are adopted to simulate the reservoir dynamic hydraulic pressure, where the reservoir density is taken as 1000 kg/m\({}^{3}\), the bulk modulus of the reservoir water is taken as 2.07 \(\times\) 10\({}^{9}\) Pa when the compressibility of the reservoir water is considered, and the bulk modulus of the reservoir water is taken as 2.07 \(\times\) 10\({}^{20}\) Pa when the compressibility of the reservoir water is not considered. The construction of this dam spanned three phases. Phase I involved concrete pouring from 1959 to 1962. Phase II was completed in 1969-1972 and poured to the original design elevation of 125.00 m. The phase III of concrete pouring was completed in 1978 and poured to the current elevation of 126.30 m [56]. Therefore, it is necessary to consider the impact of dam material zoning on OSP. This paper divides the dam body into three regions based on the actual project, the specific zoning situation is shown in Figure 3. The initial elastic modulus of the concrete arch dam is 16.5 GPa, and the corresponding dynamic elastic modulus is taken as 1.5 times the static elastic modulus. By 2023, the dam has been in operation for 45 years. According to Equation (13), the degradation of elastic modulus in various regions of dam body is presented in Table 1.  
During the construction of this arch dam, insufficient attention was given to temperature control measures, resulting in the generation of numerous cracks in the dam body. Some of these cracks have posed a significant threat to the dam's safety. The main severe cracks include the following: (1) horizontal cracks near elevations of 105 m and 111.5 m in the downstream surface, penetrating through 24 and 16 dam sections, with depths reaching 6 m and 12 m, respectively; (2) longitudinal vertical cracks on the dam crest, with depths exceeding 8 m [57]. The schematic distribution of these horizontal and longitudinal vertical cracks is illustrated in Figure 3. Additionally, based on measured data, there are numerous \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Element**} & **Material** & **Years of** & **Phase I** & **Phase II** & **Phase III** \\  & **Parameters** & **Degradation** & **(GPa)** & **(GPa)** & **(GPa)** \\ \hline \multirow{2}{*}{Dam} & Elastic modulus & \multirow{2}{*}{0} & 24.27 & 24.84 & 24.75 \\  & & & & & \\  & & & & & \\ Dam & Elastic modulus & 10 & 22.92 & 24.27 & 24.72 \\  & & & & & \\ Dam & Elastic modulus & 20 & 20.97 & 22.92 & 23.81 \\  & & & & & \\ Dam & Elastic modulus & 30 & 18.56 & 20.97 & 22.20 \\  & & & & & \\ Dam & Elastic modulus & 40 & 15.84 & 18.56 & 20.04 \\  & & & & & \\ Dam & Elastic modulus & 45 & 14.43 & 17.22 & 18.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical table of elastic modulus degradation in various regions of the arch dam after 45 years of operation. Figure 3: Schematic diagram of arch dam material zoning. vertical cracks on the upstream surface of the dam.