# A Comprehensive Safety Analysis Study for Concrete Core Dams
## 1 Introduction
### Proposed Calculation Model  
#### 1.1.1 Theoretical Model  
A nonlinear elastic model is used based on an incremental generalized Hooke's law. The model embodies a nonlinear relationship between strain and stress. It includes nine material parameters, all of which have specific physical significance and can be easily determined and used in numerical calculations. [PERSON] and [PERSON] proposed a model for the relationship to obtain the tangent Poisson ratio. The tangent elastic modulus is expressed as:  
\[E_{t}=\bigg{[}1-R_{f}\frac{(1-\sin\varphi)(\sigma_{1}-\sigma_{3})}{2c\cos\varphi +2\sigma_{3}\sin\varphi}\bigg{]}^{2}E_{t}=\bigg{[}1-R_{f}\frac{(1-\sin\varphi) (\sigma_{1}-\sigma_{3})}{2c\cos\varphi+2\sigma_{3}\sin\varphi}\bigg{]}^{2}kp_{ a}\bigg{(}\frac{\sigma_{3}}{p_{a}}\bigg{)}^{n} \tag{1}\]  
where \(c\) is the cohesion, \(\varphi\) is the friction angle, \(R_{f}\) is the break ratio, \(K\) is the magnitude of the initial elastic modulus, and n is the index of elastic modulus, \(P_{a}\) is the atmospheric pressure, \(K\) is the principal stress ratio, \(E_{i}\) is the initial elastic modulus, \(E_{t}\) is the tangent elastic modulus, \(\sigma_{1}\) is the large principal stress, and \(\sigma_{3}\) is the small principal stress. For a resilient modulus,  
\[E_{ur}=K_{ur}p_{a}\bigg{(}\frac{\sigma_{3}}{p_{a}}\bigg{)}^{n} \tag{2}\]  
where \(K_{ur}\) is the unloading modulus. The final expression for the tangent Poisson ratio is:  
\[\mu_{t}=\frac{G-F\lg\left(\frac{\sigma_{3}}{P_{a}}\right)}{\left(1-A\right)^{ 2}} \tag{3}\]  
and  
\[A=\frac{(\sigma_{1}-\sigma_{3})d}{KP_{a}\left(\frac{\sigma_{3}}{P_{a}}\right) ^{n}\cdot\left(1-\frac{R_{f}(1-\sin\varphi)(\sigma_{1}-\sigma_{3})}{2c\cos \varphi+2\sigma_{3}\sin\varphi}\right)} \tag{4}\]  
In Equations (3) and (4), \(G\) is the initial Poisson ratio and \(F\) is the decrease rate of the initial Poisson ratio with the increase of \(\sigma_{3}\). The parameters \(G\) and \(F\) can be obtained from triaxial tests, and \(d\) is the reciprocal of the incremental value for the assumed axial strain. lg is the logarithm with base 10 (the common logarithm), such as lg 10 = 1. Thus, there are nine parameters for the model, \(K\), \(n\), \(R\)f, \(c\), \(\varphi\), \(F\), \(G\), \(d\) and \(Kur\), which can be obtained from a conventional triaxial specimen. The shear strength mode of dam shell filling materials shall be linear. The expression is:  
\[\left(\sigma_{1}-\sigma_{3}\right)_{f}=2(c\bullet\cos\phi+\sigma_{3}\bullet \sin\phi)/(1-\sin\phi) \tag{5}\]  
A linear elastic model is adopted for the rigid concrete cutoff walls and bedrock, and its shear strength can be approximately estimated by the following formula:  
\[\left(\sigma_{1}-\sigma_{3}\right)_{f}=(N_{f}-1)\sigma_{3}+\sigma_{a} \tag{6}\]  
where \(\sigma_{a}\) is the compressive strength of unconfined pressure, \(N_{f}\) is the flow value and generally taken as 2-3, which is defined as:  
\[N_{f}=tg^{2}(45^{0}+\phi/2) \tag{7}\]  
\(tg\) is the abbreviation of tangent, mathematical trigonometric function. The stress level of a linear material is defined as:  
\[S\updownarrow= \begin{cases}\frac{(\sigma 1-\sigma 3)}{(\sigma 1-\sigma 3)f}&(\sigma 3\geq 0)\\ \max(\frac{\sigma 1}{\sigma 4},\frac{\sigma 3}{\sigma 7})&(\sigma 3<0)\end{cases}  \tag{8}\]  
where: \(\sigma t\) is the uniaxial tensile strength of linear materials. The contact surface between the cutoff wall and dam material is simulated by [PERSON]'s hyperbolic contact element model. The tangent shear stiffness reflecting the relationship between shear stress and relative displacement on the contact surface is:  
\[Kst=Ks\gamma_{\omega}\big{(}\frac{\sigma_{n}}{pa}\big{)}^{n_{s}}(1-\frac{\tau.R_ {fs}}{\sigma_{n}tg\delta_{s}+Cs})^{2} \tag{9}\]  
where: \(\tau\), \(\sigma_{n}\), \(R_{fs}\), \(\delta_{s}\), \(Cs\) are shear stress, normal stress, failure ratio, shear strength parameters on the contact surface, and controllable stress respectively. \(\gamma_{\omega}\) are water unit weight, while \(Ks\) and \(n_{s}\) are dimensionless coefficients.