## 3 Case Study
### Analysis and Results
As shown in Table 5, the residual displacement level limits of the intact, slight, moderate, and severe state are 13.29 mm, 39.60 mm, and 105.38 mm, respectively, and the level limits account for 0.009%, 0.027% and 0.073% of dam height. In addition, the coefficient of variations is between 34.17% and 55.47% due to the randomness of earthquake waves. The dam structure is a highly nonlinear system, it is reasonable that the response characteristics of the dam under different earthquake actions are quite different, so the average value of displacements under various earthquake waves is adopted for the further analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Damage State** & **Damage Index Range** \\ \hline DS1 & Intact & 0.00 \(<\) DI \(<\) 0.05 \\ DS2 & Slight & 0.05 \(<\) DI \(<\) 0.15 \\ DS3 & Moderate & 0.15 \(<\) DI \(<\) 0.40 \\ DS4 & Severe & DI \(\geq\) 0.40 \\ \hline \hline \end{tabular} Note: DS, damage state; DI, damage coefficient.  
\end{table} 
Table 4: Damage-based classification of damage states for gravity dams. Figure 6: Correlation of R and D under various earthquake waves. (**a**) Correlation of R and D under wave 1; (**b**) Correlation of R and D under wave 2; (**c**) Correlation of R and D under wave 3; (**d**) Correlation of R and D under wave 4. (3) Comparative fragility analysis  
Based on the seismic fragility analysis method of dams, a series of functions between the probability and the logarithm of PGA are as follows  
\[\text{Slight}:\ P_{f}{}^{DS2}=\Phi(1.609\ln(PGA)-1.912) \tag{29}\]  
\[\text{Moderate}:\ P_{f}{}^{DS3}=\Phi(3.004\ln(PGA)-2.268) \tag{30}\]  
\[\text{Severe}:\ P_{f}{}^{DS4}=\Phi(3.564\ln(PGA)-1.734) \tag{31}\]  
Figure 7 shows the seismic fragility curves of the Guandi dam. Generally speaking, Concrete gravity dams are designed based on two earthquake levels, the operating basis earthquake (OBE) and the maximum creditable earthquake (MCE), to ensure the seismic safety. The PGAs of OBE and MCE for the Guandi dam are 0.352 g and 0.415 g.  
Figure 7a shows the seismic fragility curves defined by the residual displacement-based classification of damage for the Guandi dam, and Figure 7b shows the seismic fragility curves defined by the damage-based classification of damage. Figure 7a shows that the probabilities of slight and moderate damage under the OBE are 59.2% and 19.3%, while the severe damage is only 2.3%. It indicates that the Guandi dam under the OBE is mainly the slight damage. The corresponding probabilities under the MCE are 69.0%, 35.4%, and 8.1%, respectively. The proportion of moderate damage probability has increased. In contrast to Figure 7a, the probabilities of different damage states under the OBE (MCE) in Figure 7b are 32.3% (51.9%), 22.7% (40.4%) and 2.2% (7.8%), respectively. Obviously, the difference between probabilities based on the residual displacement, and the damage is very small. It indicates that the residual displacement-based performance parameter and the corresponding classification method of seismic damage are reasonable and feasible. Displacements of dams are the must-be-monitored physical quantity, thereby making it easier to acquire the monitoring \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Damage State** & **Residual Displacement Range (mm)** \\ \hline DS1 & Intact & 0.00 \(<\) RD \(<\) 13.29 \\ DS2 & Slight & 13.29 \(<\) RD \(<\) 39.60 \\ DS3 & Moderate & 39.60 \(<\) RD \(<\) 105.38 \\ DS4 & Severe & RD \(\geq\) 105.38 \\ \hline \hline \end{tabular} Note: DS, damage state; RD, Residual displacement.  
\end{table} 
Table 6: Residual displacement-based classification of seismic damage.