# Non-Probabilistic Reliability Analysis of Slopes Based on a Multidimensional Parallelepiped Model
## 2 Non-Probabilistic Reliability Analysis Method Based on a Multidimensional Parallelepiped Model
### Non-Probabilistic Reliability Analysis  
In a multidimensional parallelepiped model, the non-probabilistic reliability index \(\eta\) denotes the shortest distance from the coordinate origin to the limit state surface in the extended space of the standardized interval variables measured via the minimum infinity norm \(\|\bullet\|_{\infty}\) (Figure 3 shows the two-dimensional case). It is consistent with the meaning of the non-probabilistic reliability index \(\eta\) in a traditional interval model. The non-probabilistic reliability index is defined as follows [22]:  
\[ \begin{array}{l}\eta=\min(\|\mathbf{u}\|_{\infty})\\ \text{s.t.}\ \mathrm{G}(u_{1},u_{2},\cdots,u_{n})=0\end{array}  \tag{12}\]  
The solution methods for calculating the non-probabilistic reliability index are mainly divided into three categories [23]: the definition method, the transformation method, and the optimization method. Theoretically, the definition method and the transformation method are accurate methods, while the optimization method is an approximation method. The optimization method is more adaptable in the case of complexity of the performance function, but there is the problem of interval expansion in the calculation. Commonly used in the definition method is the sequential quadratic programming method [24], which is backed up by good mathematical theory. However, it is limited to the fact that the performance function is a standard quadratic formula. What is w  
Figure 3: Definition of non-probabilistic reliability index. this iterative algorithm, there may be cases where the solution falls into a local optimum or the iterations do not converge. The transformation method is an analytic method, in which the global optimal solution method [25] is simple to operate and obtains a stable global optimal solution. The steps to compute non-probabilistic reliability indexes using the global optimal solution method are as follows. First, we solve all the minimum points of the limit state equation \(G(u_{1},u_{2},\cdots,u_{n})=0\) with respect to \(u_{1},u_{2},\cdots,u_{n}\). Second, we reduce the dimension of the limit state equation as follows:  
\[\left\{ \begin{array}{l}G(u_{1}=\pm u_{2},u_{2},\cdots,u_{n})=0,G(u_{1}=\pm u_ {3},u_{2},\cdots,u_{n})=0,\cdots,G(u_{1}=\pm u_{n},u_{2},\cdots,u_{n})=0\\ G(u_{1},u_{2}=\pm u_{3},\cdots,u_{n})=0,G(u_{1},u_{2}=\pm u_{4},\cdots,u_{n})= 0,\cdots,G(u_{1},u_{2}=\pm u_{n},\cdots,u_{n})=0\\ \cdots\\ G(u_{1},\cdots,u_{n-1}=\pm u_{n},u_{n})=0\end{array} \right. \tag{13}\]  
Then, we find all the minimum points of the above \(n\) (\(n-1\)) equations (calculated according to the \(n-1\) dimensional problem). The process of reducing the dimension of the limit state equation and searching for the minimum points is repeated until the limit state equation is reduced to two dimensions. Finally, we solve the equation \(G(u_{1}=\pm u_{2}=\cdots=\pm u_{n})=0\) (a total of \(2n-1\) equations) and obtain all the root value points. Then the non-probabilistic reliability index \(\eta\) is the minimum value of the distance between all the minimum points of the equation and the root value points to the coordinate origin measured via the infinity norm. In this paper, the global optimal solution method is used to solve the non-probabilistic reliability index. As shown in Figure 3, the solid squares represent the normalized parameters' uncertainty domain. When the non-probabilistic reliability index \(\eta=1\), the limit state surface of the standardized structure performance function is tangent to the parameter uncertainty domain, and the structure is in a critical state between being reliable and unreliable. When the non-probabilistic reliability index \(\eta>1\), the standardized parameter uncertainty domain is within the safety domain and the structure is reliable. The larger the value of \(\eta\) is, the more reliable the structure is. When the non-probabilistic reliability index \(\eta<1\), a portion of the area of the standardized parameter uncertainty domain is in the failure domain.