## 4 Numerical Examples
### Static Cohesive Crack Propagation Simulation of Koyna Dam  
After a severe earthquake in 1967, the neck of the Koyna Dam suffered serious damages. [PERSON] et al. [35] (1992) simulated the generated crack based on linear elastic fracture mechanics and [PERSON] et al. [36] applied the smeared model to analyze the crack propagation. The geometry and material parameters of the Koyna Dam are shown in Figure 16. The dam concrete was assumed to be homogeneous and the Koyna Dam was analyzed based on the planar strain assumption and the bilinear softening curve, where \(\omega_{c}=0.256\) mm, \(\omega_{0}=0.04\) mm, \(\omega_{1}=0.075\) mm, \(f_{t}=1\) MPa, \(G_{f}=100\) N/m, and \(f_{t}=0.25\) MPa. A crack was set at a horizontal orientation at an elevation of 66.5 m on the upstream face of the dam in advance. The initial length of the crack was 1.93 m, which equaled 1/10 of the width of the dam at an elevation of 66.5 m. The loads considered included the body load of the dam, hydrostatic pressure of the full reservoir, and the overloading head load applied at the ultimate fracture of the dam. The crack expansion step assumed that \(\Delta a=2\) m.  
Figure 16: The geometry and material parameters of the Koyna Dam. Figure 15: Variation of crack opening displacement and distribution of cohesive traction: (**a**) crack opening displacement and (**b**) the distribution of cohesive traction. During the numerical simulation, the crack initially expanded horizontally. When the overflow increased gradually, the crack trajectory gradually expanded downward owing to the increase of the compressive stress in the downstream area of the dam. Figure 17 is the schematic diagram of the ultimate crack propagation path and corresponding cohesive tractions when the overflow reached 10.35 m.  
Figure 18 compares the crack paths corresponding to an overflow of 10.35 m using the present method with the results of [PERSON] [34], [PERSON] and [PERSON] [36], and [PERSON] and [PERSON] [35]. In the results reported in these published studies, the overflow was approximately 10.2 m, 10 m, and 14 m, respectively. The crack path predicted by [PERSON] and [PERSON] [36] was initially nearly horizontal. It then turned downwards when the crack became equal to half of the width of the dam neck. The crack path based on X-SBFEM in this study is consistent with those predicted by [PERSON] [34] and [PERSON] and [PERSON] [35]. These studies reported the formation of very short horizontal extensions that initially curved downwards and towards the dam's heel. Figure 17: Ultimate crack propagation path and corresponding cohesive tractions for the Koyna Dam (step 11, overflow = 10.35 m). Figure 18: Comparison of crack paths with existing results. Figure 19 shows the relationships between the overflow and the crest displacement obtained herein and four previously published numerical simulations [34; 35; 36]. It can be seen from Figure 19 that the results of [PERSON] et al. [35] are consistent with those of [PERSON] et al. [34], [PERSON] and [PERSON] et al. [36], and [PERSON] et al. [23] and the method presented herein during the initial loading, before the obvious nonlinearity took place. Subsequently, the results obtained by the method presented in this study are comparatively different from those reported by [PERSON] et al. [34] and [PERSON] et al. [23] based on the linear elastic fracture mechanics method. Nevertheless, they match closely to the results of [PERSON] et al. [35] and [PERSON] and [PERSON] et al. [36]. In Figure 19, the resistance of the dam increases as a function of the crack length and there is no postpeak region owing to the stabilization effect of the self-weight of the dam. The results from these examples also reveal the minor difference encountered between the numerical simulation of the scaled-down dam model and the actual dam.