## 3 Numerical Modeling of the Xulong High Arch Dam  
### Numerical Method  
Using a 3D nonlinear finite element analysis, the convergence of elastoplastic analysis solution shows the stability of the structure. The yielding condition of the ideal elastoplastic model adopts Drucker-Prager (D-P) yielding criterion [5].  
(1) Safety factors of the arch dam  
Both in the geomechanical model experiment and 3D nonlinear numerical simulation, overloading, strength reduction, and comprehensive method are main methods to analyze the ultimate state and safety factors of arch dams [2; 32], as illustrated in Figure 3. According to different overloading ways, overloading method includes increasing the bulk density of the upstream water and upstream water level (Figure 3a,b). The comprehensive method combines the overloading and strength reduction method.  
Figure 3: Overloading and strength reduction method. (**a**) Increasing the bulk density of the upstream water; (**b**) increasing the upstream water level; (**c**) strength reduction method.  
The arch dam is a high-order statically indeterminate structure. In the process of overloading or strength reduction, the deformation of arch dams can be divided into three stages, namely the elastic deformation stage, plastic deformation stage, and total failure stage. Corresponding to three deformation stages, three safety factors, \(K_{1},\ K_{2},\ K_{3}\), are employed to evaluate the overall stability of the dam [2].  
\(K_{1}\) represents the safety factor of crack initiation of the arch dam. A crack is generally initiated at the dam heel. As shown in Figure 3a, \(K_{1}=\frac{71}{70}\); in Figure 3b, \(K_{1}=\frac{h_{1}}{h_{0}}\).  
\(K_{2}\) represents the safety factor of structural nonlinear behavior initiation. The dam has a large displacement due to nonlinear deformation. Dam cracks rapidly propagate and coalesce with each other. As shown in Figure 3a, \(K_{2}=\frac{72}{70}\); in Figure 3b, \(K_{2}=\frac{h_{2}}{h_{0}}\).  
\(K_{3}\) represents the safety factor of the maximum loading of the dam-foundation system. The yielding regions connect. The dam foundation is totally destroyed and the undertaking capacity is lost. As shown in Figure 3a, \(K_{3}=\frac{72}{70}\); in Figure 3b, \(K_{3}=\frac{h_{2}}{h_{0}}\).  
(2) Evaluation of connection of yielding region  
Adopting to increase the bulk density of the upstream water (Figure 3a), the yielding region of dam and foundation is simulated under overloading condition. The connection of the yielding region means that it connects piece by piece and forms movement mechanism so that the integrity stiffness of arch dam-foundation is weakened. The cracking or yielding caused by excessive local stress reduces the constraint to adjust the internal stress. If the yielding region is not fully connected, the structure can still provide support. This method can fully reflect the adjustment process of nonlinear stresses and exert the bearing capacity of the high-order statically indeterminate arch dam.  
Failure criterion can be defined as Equation (1), that is, iteration does not converge.  
\[\varnothing(A_{i})\geq 0. \tag{1}\]  
where \(\varnothing\) is yielding surface. \(A_{i}\) is a mechanism composed of local yielding regions, that is, formed by the connection of the local yielding regions. \(A_{i}\) can be expressed as: \(A_{i}=\varphi_{i}\cup\varphi_{m}\cup\ldots\cup\varphi_{n}\). When any \(A_{i}\) is formed, the structure loses its integral stability. In addition, \(\det[\text{K}]\leq 0\), where [K] is the integral stiffness matrix in FEM analysis.