## 2 Methods
### Modeling Dam Deformation
#### 2.1.1 Hydrostatic Pressure Component  
A simplified two-dimensional model of a homogeneous gravity dam is taken as an analysis example, as shown in Figure 1, we obtain the following explanation through the mechanical relationship between water level and deformation. Under hydraulic load, the horizontal displacement \(\delta_{H}\) generated at any measuring point of the dam is composed of four parts, as shown in Figure 1: the displacement \(\delta_{1H}\) caused by the deformation of foundation due to internal forces on foundation surface and the displacement \(\delta_{2H}\) caused by rotation of foundation surface due to the gravity of the reservoir water, the displacement \(\delta_{3H}\) caused by the rotation of reservoir water pressure acting on dam body, and the shear horizontal displacement \(\delta_{4H}\) caused by the internal force of reservoir water pressure acting on dam body. Therefore, the dam deformation is expressed by:  
\[\delta=\delta_{1H}+\delta_{2H}+\delta_{3H}+\delta_{4H} \tag{2}\]  
Figure 1: _Cont._  
According to [PERSON] theory [26], the water pressure to the bottom of a dam causes the horizontal displacement \(\delta_{1H}\) and rotation angel \(\theta_{1}\) of dam foundation as:  
\[\delta_{1H}{=(\gamma K_{1}/6E_{f}a)H^{3}+(\gamma K_{2}/2E_{f})H^{2}}. \tag{3}\]  
\[\theta_{1}{=(\gamma K_{3}/6E_{f}a^{2})H^{3}+(\gamma K_{1}/2E_{f}a)H^{2}}. \tag{4}\]  
where \(E_{f}\) denotes the elastic modulus of bedrock; \(\gamma\) is unit weight of water; the width of dam bottom is \(a\); \(H\) presents the upstream water depth; and \(K_{1}\), \(K_{2}\), and \(K_{3}\) are the coefficients that depend on the Poisson's ratio of the bedrock and the length-to-width ratio of the equivalent rectangle at the bottom of the dam [27]. If the length of the reservoir \(L\) is very large, the weight of water acting on reservoir in front of dam will deform reservoir bank and cause rotation \(\theta_{2}\) at the bottom of dam as:  
\[\theta_{2}{=-H(1+\mu_{f})\ln(n+\sqrt{n^{2}+1})/\pi E_{f}} \tag{5}\]  
where \(\mu_{f}\) is the Poisson ratio of bedrock and \(n\) is half the ratio of the width of water in front of dam to the distance \(a_{0}\) from the center of gravity of the dam to the heel of the dam. Therefore, the horizontal displacement \(\delta_{2H}\) of crest caused by rotation of dam foundation \(\theta\) as:  
\[\delta_{2H}=c\theta{=(\theta_{1}+\theta_{2})}c \tag{6}\]  
The horizontal displacement \(\delta_{3H}\) of dam crest caused by the rotation \(\theta_{3}\) of dam body as:  
\[\delta_{3H}=\int_{0}^{c}\int_{0}^{c}Mdydy/E_{d}I \tag{7}\]  
where \(M=\gamma H^{3}/6\) is the bending moment caused by water pressure, which is proportional to the third power of water depth; \(E_{d}\) is the elastic modulus of dam concrete material; \(I\) is the moment of inertia of the horizontal section of dam and \(c\) is dam height. The dam shear horizontal displacement \(\delta_{4H}\) as:  
\[\delta_{4H}=\int_{0}^{c}2k(1+\mu_{d})Qdy/AE_{d} \tag{8}\]  
where \(Q=\gamma h^{2}/2\) is shear force caused by water pressure, which is proportional to the third power of the water depth; \(A\) is the horizontal area section of dam; \(k\) is the shear force  
Figure 1: Schematic of the influence of hydraulic load on horizontal dam displacement: (**a**) horizontal displacement of dam foundation; (**b**) horizontal displacement caused by rotation of the dam foundation; and (**c**) horizontal displacement caused by dam rotation and shear horizontal movement. (unit: m). distribution coefficient on the section, about 1.2; and \(\mu_{d}\) is the Poisson's ratio of the concrete material of dam. From the above analysis, it can be found that the horizontal displacement of dam crest caused by the water depth \(H\) in front of the dam is a function of \(H\), \(H^{2}\), and \(H^{3}\).