## 3 Case Study
### Analysis and Results  
(1) Function relation between the residual displacement and the damage coefficient. Typical damage modes of Guandi dam under selected 28 earthquake waves are concluded in Figure 3. The upstream crest corner was called point A. As shown in Figure 3, the main damaged locations are as follows: (Location 1) neck area at the change of downstream face slope; (Location 2) along lift joints at various elevations, the change of upstream face slope and around the drainage gallery; (Location 3) along with dam-foundation interface at the toe and heel of the dam. Figure 4 shows the graph of the relation between the residual displacement in point A and the PGA. It can be seen that the residual displacement of point A increases with increasing PGA. Specifically, as the PGA of earthquake wave 10 increases from 0.1 g to 0.8 g, the residual displacement increases from 2.23 mm to 644.45 mm. The residual displacement has a fluctuation range under different earthquake with the same PGA. Figure 5 shows the scatter diagram of the dimensionless damage coefficient of dams (\(D\)) and the dimensionless residual displacement in the dam crest (\(R\)) under 28 earthquake waves. To study the correlation between \(D\) and \(R\) variables, [PERSON]'s correlation coefficient is adopted in the paper, as seen in Equation (21). Results show that the [PERSON]'s correlation coefficient between \(D\) and \(R\) in the dam under 28 seismic waves is 0.93, indicating that there is a strong correlation between \(D\) and \(R\). \[r(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var[X]Var[Y]}} \tag{21}\]  
where \(r(X,Y)\) is the correlation coefficient between \(X\) and \(Y\) variables; \(Cov(X,Y)\) is the covariance; \(Var[X]\) and \(Var[Y]\) are the variances of \(X\) and \(Y\) variables, respectively. The best-fit curves are shown in Figure 6 for the relationship between the \(D\) and \(R\) under earthquake waves 1-4. It can be seen that there is an approximately linear functional relationship between the \(D\) and \(R\) under these earthquake waves. For earthquake waves 1-4, the best-fit equations are linear functions as follows,  
\[R_{1}=-0.010+1.051D_{1}\quad(R^{2}=0.97) \tag{22}\]  
\[R_{2}=0.013+1.023D_{2}\quad(R^{2}=0.98) \tag{23}\]  
\[R_{3}=0.137+0.882D_{3}\quad(R^{2}=0.95) \tag{24}\]  
\[R_{4}=0.072+0.909D_{4}\quad(R^{2}=0.95) \tag{25}\]  
Taking the residual displacement obtained from the fitting functions between \(D\) and \(R\) under 28 earthquake waves as the prediction value, the average relative error under 28 earthquake waves is 13.52%, with the fluctuation of 4.95%-25.04%. The relative error shows that the relationship between \(D\) and \(R\) in the Guandi concrete gravity dam can be described by a linear function with high accuracy. Therefore, the linear functions of the \(D\) and \(R\) can be defined by  
\[R_{i}=a_{i}+b_{i}D_{i}\quad(i=1,2,\cdots,28) \tag{26}\]  
Therefore, the residual displacement-based level limits are as follows  
\[RD_{i}^{k}=(RD_{i})_{\max}\bigg{(}a_{i}+b_{i}\frac{Dl^{k}}{(Dl_{i})_{\max}} \quad\bigg{)}(i=1,2,\cdots,28;k=1,2,3) \tag{27}\]  
\[RD^{k}=\frac{\sum\limits_{i=1}^{28}RD_{i}^{k}}{28}(k=1,2,3) \tag{28}\]Figure 4: Graph of the relation between the residual displacement in point A and the peak ground acceleration (PGA). \[r(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var[X]Var[Y]}} \tag{21}\]0 Scatter diagram of R and D under 28 earthquake waves. (2) Classification of seismic damage  
Damage-based classification of seismic damage has been commonly used in the field of concrete dams. According to the related researches [8; 23; 32], the damage-based classification is shown in \[r(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var[X]Var[Y]}} \tag{21}\]1 According to Equations (27) and (28), the residual displacement-based level limits can be calculated with \(Dl^{1}=0.05\), \(Dl^{2}=0.15\) and \(Dl^{3}=0.40\) (as seen in Table 5). Table 6 shows the residual displacement-based classification of seismic damage of the Guandi gravity dam.