# Non-Probabilistic Reliability Analysis of Slopes Based on a Multidimensional Parallelepiped Model
## 2 Non-Probabilistic Reliability Analysis Method Based on a Multidimensional Parallelepiped Model
### Construction of the Multidimensional Parallelepiped Model  
\(\mathbf{X}=(X_{1},X_{2},\cdots,X_{n})^{T}\) is a vector consisting of \(n\) uncertainty parameters of the structure, where the uncertainty parameter \(X_{i}\) can be expressed in terms of an interval as follows:  
\[X_{i}=\left[X_{i}^{C}-X_{i}^{W},X_{i}^{C}+X_{i}^{W}\right],i=1,2,\cdots,n \tag{1}\]  
where \(X_{i}^{C}\) is the median of the marginal interval of the uncertainty parameter \(X_{i}\), and \(X_{i}^{W}\) is the deviation of the marginal interval of the uncertainty parameter \(X_{i}\). Consider any two uncertainty parameters \(X_{p}\) and \(X_{q}\)\((p,q=1,2,\cdots,n.\)_and_\(p\
eq q)\) in \(\mathbf{X}=(X_{1},X_{2},\cdots,X_{n})^{T}\). Based on a dataset with a limited number of sample points, the parallelogram uncertainty domain ABCD can be constructed that satisfies the following four conditions [15] (as shown in Figure 1):  
1. The parallelogram uncertainty domain \(\Omega\) is contained in the rectangular domain \(\Omega_{s}\). 2. The vertices A, B, C, and D of the parallelogram uncertainty domain \(\Omega\) are located on the diagonal lines of the rectangular domain \(\Omega_{s}\). 3. The parallelogram uncertainty domain \(\Omega\) is the minimum area parallelogram containing all sample points. 4. When the uncertainty parameters \(X_{p}\) and \(X_{q}\) are independent of each other, the parallelogram uncertainty domain \(\Omega\) degenerates into the rectangular domain \(\Omega_{s}\). In Figure 1, the white circles are the sample points, the rectangle domain \(\Omega_{s}\) formed by the red dashed lines represents the uncertainty domain in the traditional interval  
Figure 1: The MP interval model (two-dimensional case): (**a**) \(\rho_{X_{p}X_{q}}\leq 0\) and (**b**) \(0\leq\rho_{X_{p}X_{q}}\). model, and the purple dashed lines are the diagonal lines of the rectangle. \(a\) denotes the distance from the center point \(X_{C}\) to the vertex \(A\) of the parallelogram uncertainty domain, and \(b\) denotes the distance from the center point \(X_{C}\) to the vertex \(B\). For the MP-1 multidimensional parallelepiped model [20], the correlation coefficient \(\rho_{X_{p}X_{q}}\) between the uncertainty parameters \(X_{p}\) and \(X_{q}\) is defined as follows:  
\[\rho_{X_{p}X_{q}}=\frac{b-a}{b+a} \tag{2}\]  
when the value of \(a\) is equal to the value of \(b\), the value of the correlation coefficient \(\rho_{X_{p}X_{q}}\) is equal to 0. At the same time, the uncertainty parameters \(X_{p}\) and \(X_{q}\) are independent of each other and the uncertainty domain degenerates into a traditional interval model. The uncertainty domain \(\Omega\) coincides with the rectangular domain \(\Omega_{s}\) (the rectangle enclosed by the red dashed line in Figure 1). When the value of \(a\) is not equal to the value of \(b\) and neither the value of \(a\) nor the value of \(b\) is equal to 0, the value of the correlation coefficient \(\rho_{X_{p}X_{q}}\) is greater than \(-1\) and less than 1. In this case, there is a certain degree of correlation between the uncertainty parameters \(X_{p}\) and \(X_{q}\). The uncertainty domain \(\Omega\) is represented by the parallelogram ABCD surrounded by the solid lines in Figure 1. Specifically, when the value of \(a\) is equal to 0 (the value of the correlation coefficient \(\rho_{X_{p}X_{q}}\) is equal to 1) or the value of \(b\) is equal to 0 (the value of the correlation coefficient \(\rho_{X_{p}X_{q}}\) is equal to \(-1\)), the uncertainty domain degenerates into the straight line BD or the straight line AC, which is represented by the purple dashed line in Figure 1.  
For \(n\) uncertainty parameters \(X_{1}\), \(X_{2}\),..., \(X_{n}\), the multidimensional correlation coefficient matrix \(\rho\) is defined as follows:  
\[\rho= \begin{pmatrix}\rho_{X_{1}X_{1}}&\rho_{X_{1}X_{2}}&\cdots&\rho_{X_{1}X_{n }}\\ \rho_{X_{2}X_{1}}&\rho_{X_{2}X_{2}}&\cdots&\rho_{X_{2}X_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \rho_{X_{n}X_{1}}&\rho_{X_{n}X_{2}}&\cdots&\rho_{X_{n}X_{n}}\\ \end{pmatrix}