## 3 Reliability Calculation of a Single Slip Pathway
### Reliability Calculation of Elements
#### 3.2.2 Stepwise Equivalent Linearization of the Joint Failure Boundary of Multiple Elements  
The joint failure boundary of elements \(i\) and \(j\) can be represented by the following system of simultaneous equations:  
\[ \begin{cases}f_{i}=\sum\limits_{k=1}^{n}\alpha_{ik}u_{k}+\beta_{i}=0\\ f_{j}=\sum\limits_{k=1}^{n}\alpha_{jk}u_{k}+\beta_{j}=0\end{cases}  \tag{13}\]  
Based on the condition of equal reliability indices, the failure boundary equivalent to the joint failure boundary can be constructed as follows:  
\[f_{E}=\sum\limits_{k=1}^{n}\alpha_{Ek}u_{k}+\beta_{E}=0 \tag{14}\]  
The coefficient \(\alpha_{Ek}\) and the reliability index \(\beta_{E}\) in the equation must satisfy the following relationship:  
\[\left\{ \begin{array}{l}\sum\limits_{k=1}^{n}\alpha_{Ek}^{2}=1\\ \beta_{E}=-\Phi^{-1}\left(\Phi\big{(}-\beta_{i},-\beta_{j},\rho_{ij}\big{)} \right)\end{array} \right. \tag{15}\]  
where \(\rho_{ij}\) is the correlation coefficient between elements \(i\) and \(j\), which is calculated as follows:  
\[\rho_{ij}=\sum\limits_{k=1}^{n}\alpha_{ik}\alpha_{jk} \tag{16}\]  
The coefficient \(\alpha_{Ek}\) can be determined by the sensitivity of the reliability index \(\beta_{E}\) to the mean of the standard normal random variable \(u_{k}\). If a small increment \(\varepsilon\) is applied to the mean of the \(k\)-th random variable \(u_{k}\) (\(k=1\), \(2\), \(\ldots\), \(n\)) in Equations (13) and (14), the increment \(\Delta\beta_{i}\) of \(\beta_{i}\) is \(\alpha_{ikx}\), the increment \(\Delta\beta_{j}\) of \(\beta j\) is \(\alpha_{jkx}\), and the increment \(\Delta\beta_{E}\) of \(\beta_{E}\) is \(\alpha_{Ekx}\). According to Equation (15), \(\Delta\beta_{E}\) can also be determined by the following formula:  
\[\Delta\beta_{E}=\Phi^{-1}\big{(}\Phi\big{(}-\beta_{i},-\beta_{j},\rho_{ij} \big{)}\big{)}-\Phi^{-1}\Big{(}\Phi\Big{(}-\beta_{i}-\alpha_{ik}\epsilon,- \beta_{j}-\alpha_{jk}\epsilon,\rho_{ij}\big{)}\Big{)} \tag{17}\]  
Therefore, the calculation formula for \(\alpha_{Ek}\) can be given as  
\[\alpha_{Ek}=\frac{\Phi^{-1}\big{(}\Phi\big{(}-\beta_{i},-\beta_{j},\rho_{ij} \big{)}\big{)}-\Phi^{-1}\Big{(}\Phi\Big{(}-\beta_{i}-\alpha_{ik}\epsilon,- \beta_{j}-\alpha_{jk}\epsilon,\rho_{ij}\Big{)}\Big{)}}{\epsilon} \tag{18}\]  
The \(\alpha_{Ek}\) calculated from the above formula generally does not satisfy the requirement of the first equation in Equation (15) and needs to be corrected, that is,  
\[\alpha^{\prime}{}_{Ek}=\alpha_{Ek}/\sqrt{\sum_{k=1}^{\frac{n}{2}}\alpha_{Ek} ^{2}} \tag{19}\]  
At the same time, \(\beta_{E}\) needs to be corrected as follows:  
\[\beta^{\prime}{}_{E}=\beta_{E}/\sqrt{\sum_{k=1}^{\frac{n}{2}}\alpha_{Ek}^{2}} \tag{20}\]  
The \(m\) elements of the sliding pathway are arranged in increasing order of reliability index. According to Equations (18) to (20), the equivalent failure boundary of the first and second elements is determined. Then, the equivalent failure boundary of this combined failure boundary is determined with the third element's failure boundary, and so on. Finally, the equivalent failure boundary of multiple elements and the corresponding parallel reliability can be obtained. Now, a numerical example is used to verify the computational accuracy of the stepwise equivalent linearization method. Assuming a parallel system consisting of four elements, the known element limit state functions are as follows:  
\[\left\{ \begin{array}{l}Z_{1}=X_{1}+2.5\\ Z_{2}=X_{2}+2.5\\ Z_{3}=0.2X_{1}+0.98X_{3}+3.25\\ Z_{4}=0.4X_{1}+0.9X_{2}-0.08X_{3}+0.15X_{4}+3.25\end{array} \right. \tag{21}\]  
where \(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{4}\) are standard normal random variables.