# 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
 \tag{3}\]  
where \(\rho\) is an \(n\times n\) symmetric matrix; \(\rho_{X_{i}X_{i}}\) is the autocorrelation coefficient of the uncertainty parameter \(X_{i}(i=1,2,\cdots,n)\), whose value is 1; and \(\rho_{X_{p}X_{q}}\) is the correlation coefficient between any uncertainty parameters \(X_{p}\) and \(X_{q}\), whose value can be obtained from Equation (2) by constructing the parallelogram uncertainty domain. The MP-1 model can be constructed using the following equation:  
\[\left|\rho^{-1}\mathbf{T}^{-1}\mathbf{R}^{-1}\mathbf{\left(X-X^{C}\right)}\right|\leq\mathbf{e} \tag{4}\]  
where  
\[\left. \begin{array}{c}\mathbf{T}=diag(w_{1},w_{2},\cdots,w_{n})\\ w_{p}=\frac{1}{\sum\limits_{i=1}^{n}\rho(\rho_{J})},p=1,2,\cdots,n\\ \mathbf{R}=diag\big{(}X_{1}^{W},X_{2}^{W},\cdots X_{n}^{W}\big{)}\\ \mathbf{X}^{C}=\big{(}X_{1}^{C},X_{2}^{C},\cdots,X_{n}^{C}\big{)}^{T}\\ \mathbf{e}=\left(1,1,\cdots,1\right)^{T}\\ \end{array} \right\} \tag{5}\]  
The most important difference between the MP-2 model and the MP-1 model is the different definitions of the correlation coefficient and the multidimensional correlation coefficient matrix for the uncertainty parameters. For the MP-2 model, the correlation coefficient \(h_{X_{p}X_{q}}\) between any uncertainty parameters \(X_{p}\) and \(X_{q}\) is defined as follows:  
\[h_{X_{p}X_{q}}=\frac{b^{2}-a^{2}}{b^{2}+a^{2}} \tag{6}\]The multidimensional correlation coefficient matrix \(\mathbf{h}\) is defined as follows:  
\[\mathbf{h}= \begin{pmatrix}h_{X_{1}X_{1}}&h_{X_{1}X_{2}}&\cdots&h_{X_{1}X_{n}}\\ h_{X_{2}X_{1}}&h_{X_{2}X_{2}}&\cdots&h_{X_{2}X_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ h_{X_{n}X_{1}}&h_{X_{n}X_{2}}&\cdots&h_{X_{n}X_{n}}\end{pmatrix} ^{\frac{1}{2}} \tag{7}\]  
where \(\mathbf{h}\) is an \(n\times n\) symmetric matrix; \(h_{X_{i}X_{i}}\) is the autocorrelation coefficient of the uncertainty parameter \(X_{i}(i=1,2,\cdots,n)\), whose value is 1; and \(h_{X_{p}X_{q}}\) is the correlation coefficient between any uncertainty parameters \(X_{p}\) and \(X_{q}\), whose value can be obtained from Equation (6) by constructing the parallelogram uncertainty domain. The MP-2 model can be constructed using the following equation:  
\[\left|\mathbf{h}^{-1}\mathbf{S}^{-1}\mathbf{R}^{-1}\Big{(}\mathbf{X}-\mathbf{X}^{\mathbf{C}}\Big{)} \right|\leq\mathbf{e} \tag{8}\]  
where  
\[\left. \begin{array}{c}\mathbf{S}=diag(v_{1},v_{2},\cdots,v_{n})\\ v_{p}=\frac{1}{\sum\limits_{q=1}^{n}|h(p_{(q)})|},p=1,2,\cdots,n\end{array} \right\} \tag{9}\]  
As shown in Figure 2, when the uncertainty domain \(\Omega\) is described by a multidimensional parallelepiped model, it can be normalized to the uncertainty domain \(\Omega^{*}\) by means of matrix transformation, regardless of whether the MP-1 model or the MP-2 model is used. The transformation relation equation from the original \(\mathbf{X}\)-space to the standard \(\mathbf{u}\)-space for the MP-1 model in the two-dimensional case is presented by [PERSON] et al. [21]:  
\[\left. \begin{array}{c}X_{1}=\frac{X_{1}{}^{W}}{\sum\limits_{q=1}^{2}|\rho(1 _{(}4)|)}u_{1}+\frac{X_{1}{}^{W}}{\sum\limits_{q=1}^{2}|\rho(1_{(}4)|)}\rho X _{1}X_{2}u_{2}+X_{1}{}^{C}\\ X_{2}=\frac{X_{2}{}^{W}}{\sum\limits_{q=1}^{2}|\rho(2_{(}4)|)}\rho X_{2}X_{1} u_{1}+\frac{X_{2}{}^{W}}{\sum\limits_{q=1}^{2}|\rho(2_{(}4)|)}u_{2}+X_{2}{}^{C} \end{array} \right\} \tag{10}\]  
Figure 2: Matrix transformation for the MP model. (**a**) The original \(\mathbf{X}\)-space; (**b**) The standard \(\mathbf{u}\)-space. Similarly, this paper derives a transformation relation equation from the original \(\mathbf{X}\)-space to the standard \(\mathbf{u}\)-space for the MP-2 model in the two-dimensional case:  
\[ \begin{array}{l}X_{1}=\frac{X_{1}{}^{m}\mathbf{n}(1,1)}{\sum\limits_{q=1}^{L}|\mathbf{ n}(1,q)|}u_{1}+\frac{X_{1}{}^{m}\mathbf{n}(1,2)}{\sum\limits_{q=1}^{L}|\mathbf{n}(1,q)|}u_{2 }+X_{1}{}^{C}\\ X_{2}=\frac{X_{2}{}^{m}\mathbf{n}(2,1)}{\sum\limits_{q=1}^{L}|\mathbf{n}(2,q)|}u_{1}+ \frac{X_{2}{}^{m}\mathbf{n}(2,2)}{\sum\limits_{j=1}^{L}|\mathbf{n}(2,q)|}u_{2}+X_{2}{}^ {C}\end{array}