## 3 Reliability Calculation of a Single Slip Pathway  
### Reliability Calculation of Elements  
If a slip pathway of the dam slope consists of m elements in a two-dimensional case, the limit state function for the \(i\)-th element can be formulated based on the Mohr-Coulomb strength criterion as follows:  
\[Z_{i}=\frac{\sigma_{1}-\sigma_{3}}{2}-\frac{\sigma_{1}+\sigma_{3}}{2}\sin\phi- c\cos\phi\left\{ \begin{array}{l}<0,\ Safe\ state\\ =0,\ Limit\ state\\ >0,\ Failure\ state\end{array} \right.\quad(i=1,2,\cdots,m) \tag{1}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) represent the major and minor principal stresses of the element, respectively, which are obtained as the average values of the stress components at the Gauss integration points of the element. \(c\) and \(\phi\) denote the cohesion and internal friction angle of the soil, respectively, which are treated as random variables. For the rubble materials, the cohesion \(c\) = 0, and the internal friction angle \(\varphi\) is a function of the minor principal stress [18; 19; 20], that is,  
\[\phi=\phi_{0}-\Delta\phi\log\left(\frac{\sigma_{3}}{p_{a}}\right) \tag{2}\]  
where \(\varphi_{0}\) and \(\Delta\varphi\) are the nonlinear shear strength parameters of the rubble materials, which are random variables; \(p_{a}\) is the standard atmospheric pressure. The element limit state function in Equation (1) contains stress terms, which need to be obtained through finite element calculations. Additionally, the material strength parameters of the element affect the element's stress. Therefore, the element limit state function is a random variable, which is an implicit function of the strength parameters \(c\), \(\varphi\), \(\varphi_{0}\), and \(\Delta\varphi\). The reliability calculation problem of a limit state function, being an implicit function of random variables, can be solved using the response surface method [21]. This method constructs an explicit function to replace the original implicit function using surface fitting, and the replacement function is referred to as the response surface function. In this study, a quadratic polynomial without interaction terms is used to construct the response surface function of the element's limit state function, i.e.,  
\[Z_{i}=a_{i}+\sum_{j=1}^{n}b_{ij}X_{j}+\sum_{j=1}^{n}c_{ij}X_{j}^{2}\ (i=1,2, \cdots,m) \tag{3}\]  
where \(X_{j}\) (\(j\) = 1, 2, \(\ldots\), \(n\)) are the random variables; \(a_{i}\), \(b_{ij}\), and \(c_{ij}\) are the undetermined coefficients. The undetermined coefficients in Equation (3) can be determined using the least squares method based on the designed numerical test points and their corresponding function values. The response surface, excluding cross-terms, has been widely applied to practical engineering problems. Excluding cross-terms may reduce fitting accuracy, but the corresponding computational demands for finite element analyses decrease. After establishing the response surface function of the element's limit state function, the reliability index of the element and the corresponding design verification point \(x_{i}\)* can be calculated using the improved first-order second-moment method [22]. Slope failure is generally progressive. However for conservative reasons, the dam slope is assumed to fail suddenly, so the elements within each slip pathway are assumed to fail simultaneously. When the elements constituting a slip pathway fail simultaneously, the overall instability of the dam slope increases. Therefore, the reliability calculation of a slip pathway composed of multiple elements essentially involves the reliability calculation of a parallel system. The reliability analysis of parallel systems has long been a challenging issue in structural reliability analysis. In this study, the stepwise equivalent linearization method is employed to calculate the reliability of the parallel system. This method is an approximation but offers high computational efficiency, and its accuracy is generally sufficient for engineering applications.