## 2 Methodology
### Relationship Determination of Residual Displacement and Damage Coefficient  
The effect of various earthquakes on the nonlinear behavior of concrete gravity dams can be directly reflected in the residual displacement. Under the action of a seismic wave with a big PGA, the state (e.g., damage) of concrete gravity dams will be worse than that with a small PGA [23]. [PERSON] et al. [24] concluded that the maximum displacement in the crest of the Pine Flat gravity dam is about between 30 mm to 480 mm with the PGA from 0.1 g to 0.8 g, the others are about 35 mm to 160 mm with the PGA from 0.1 g to 0.6 g in the Jin'anqiao concrete dam [25], about 23 mm to 329 mm with the PGA from 0.1 g to 1.0 g in the Koyna concrete gravity dam, and about 15 mm to 80 mm with the PGA from 0.1 g to 1.0 g [26]. Based on the research above, the displacement in concrete gravity dam crest and the damage degree of dams may have a certain correlation with the earthquake intensity, respectively. Therefore, the function to describe the potential and possible relationship between them under the \(i\)th earthquake wave can be expressed by  
\[R_{i}=f(D_{i}) \tag{5}\]  
where \(D_{i}\) and \(R_{i}\) are the dimensionless damage coefficient of dams and residual displacement in the dam crest under the \(i\)th earthquake, respectively; \(f(\cdot)\) is the function form.  
Peak ground acceleration (PGA) is the commonly used ground motion parameter when studying the seismic fragility of dams. Each earthquake wave is standardized at different levels, and eight PGAs, ranging from 1 m/s\({}^{2}\) to 8 m/s\({}^{2}\) are obtained. So the dimensionless damage coefficient (\(D_{ij}\)) and residual displacement (\(R_{ij}\)) of dams under the \(i\)th earthquake can be calculated by  
\[D_{ij}=\frac{DI_{ij}}{\left(DI_{ij}\right)_{\text{max}}},\;D_{i}=\left\{D_{i1},D_{i2},\cdots,D_{i8}\right\} \tag{6}\]  
\[R_{ij}=\frac{RD_{ij}}{\left(RD_{ij}\right)_{\text{max}}},\;R_{i}=\left\{R_{i1},R_{i2},\cdots,R_{i8}\right\} \tag{7}\]  
where \(DI_{ij}\) and \(RD_{ij}\) are the damage coefficient of dams and residual displacement in the dam crest under \(i\)th earthquake with a PGA of \(j\) m/s\({}^{2}\), with \(j\) = 1, 2, \(\ldots\), 8; \(\left(DI_{ij}\right)_{\text{max}}\) and \(\left(RD_{ij}\right)_{\text{max}}\) are the maximum values of \(DI_{ij}\) and \(RD_{ij}\); \(D_{i}\) and \(R_{i}\) are the sets of dimensionless damage coefficients and residual displacement under the \(i\)th earthquake; The damage coefficient of dams (\(DI\)) can be calculated by Equation (8) [21], where \(l_{Dp}\) is the length of the damage path in crack path \(p\); \(L_{p}\) is the total length to which crack path \(p\) is expected to grow; \(E_{p}\) is the damage dissipation energy at the crack path \(p\); and \(q\) is the number of crack paths.  
\[DI=\sum_{p=1}^{q}{(\frac{l_{Dp}}{L_{p}})E_{p}}\middle|\sum_{p=1}^{q}{E_{p}} \tag{8}\]  
Then, the parameters need to be solved in Equation (5) can be calculated based on the sets of \(D_{i}\) and \(R_{i}\) in Equations (6) and (7).  
In the field of the damage coefficient-based seismic damage classification analysis in gravity dams, the level limits of different damage degrees have been studied and lots of results have been made. Therefore, the residual displacement-based level limits under the \(i\)th earthquake can be determined based on the Equations (5)-(8) and the known level limit \(DI^{k}\) obtained by the damage coefficient-based seismic damage classification, that is  
\[RD_{i}^{k}=\left(RD_{ij}\right)_{\text{max}}\cdot f(\frac{DI^{k}}{\left(DI_{ ij}\right)_{\text{max}}}) \tag{9}\]where \(k\) is the classification level of seismic damage, \(k=1,2,3,4\) stands for the damage state of \"intact\", \"slight\", \"moderate,\" and \"severe\" in concrete dams.  
At last, the final level limits of the proposed classification method can be expressed by  
\[RD^{k}=(\sum_{i=1}^{n}RD_{i}^{k})/n \tag{10}\]  
where \(n\) is the number of earthquake waves.