## 4 Stability Analysis of the Dam
### Analysis of Total Displacement of the Dam  
This study employs a unique approach by examining the relationship between total displacement and a reduction factor at the dam's crest and toe. This method offers a nuanced understanding of dam behavior, going beyond traditional stability metrics like factor of safety (FOS). Figure 13a presents data for the dam's crest, a critical point for assessing overall stability. Under high reservoir conditions, the total displacement increases substantially from 0 to 53.6 m as the reduction factor remains constant at 1.32. This paradoxical behavior--large displacement without a decrease in stability--challenges conventional wisdom. Typically, in geotechnical engineering, significant deformation is associated with reduced stability [11]. However, our result suggests that the dam's design allows for substantial crest movement without compromising overall stability. This finding aligns with recent research on flexible dam designs. [PERSON] et al. studied a rubber dam in Brazil, noting that its crest displaced up to 20 m under high loads without stability loss [54]. They attributed this to the dam's ability to redistribute stresses through deformation. Our study extends this concept to larger displacements (over 50 m), suggesting that modern dam designs can accommodate even greater flexibility. The crest's behavior is equally intriguing during the rapid drawdown, a critical condition often leading to upstream slope failures [45]. As displacement increases from 0 to 3.36 m, the reduction factor fluctuates minimally (0.972 to 1.006). This stability under varying displacements contradicts traditional slope stability theories, which predict a steady decrease in stability factors during drawdown [54]. Our findings resonate with those of [PERSON] et al., who used differential InSAR to monitor the Three Gorges Dam. They observed that during a 30 m drawdown, downstream displacement reached 2.8 m without significant stability changes [56]. Our study, with displacements up to 3.36 m and near-constant reduction factors, provides quantitative support to their qualitative observations. Figure 13b, focusing on the dam's toe, reveals even more counterintuitive behavior. Under high reservoir conditions, the displacement is minuscule (max 0.0144 m) despite the crest's large movements. This differential movement--high at the crest, low at the  
toe--suggests a \"rolling\" deformation mode, where the dam's body flexes without sliding at its base. This behavior resembles that of modern concrete-faced rockfill dams (CFRDs). [PERSON] and [PERSON] noted that in CFRDs, the concrete face can displace over 1 m while the toe moves less than 0.1 m [56]. Our study shows this principle extends to much larger displacements, with a 53.6 m crest and 0.0144 m toe movement. This extreme differential suggests advanced stress-transfer mechanisms in the dam's design. The toe's response during drawdown is equally novel. In slow drawdown, as displacement increases from 0 to 0.497 m, the reduction factor hovers around 0.97-1.00. This stability persists even as the crest moves over 4 m. Such behavior defies classical Bishop or Janbu methods for slope stability, which would predict toe destabilization during drawdown [54]. Instead, our findings support emerging \"adaptive stability\" theories in geotechnical structures. [PERSON] et al. proposed that some earthen dams can adapt their stress distribution during loading changes, maintaining stability despite deformations. They used fiber Bragg grating sensors to show stress redistribution in a 30 m dam [57]. With its unique reduction factor approach, our study provides macro-scale evidence for this theory across much larger displacements. Moreover, the consistent reduction factors across vast displacement ranges suggest a form of \"displacement-invariant stability.\" This concept, while radical, finds support in recent geophysical studies. [PERSON] et al.