## 2 Review of the NRBM  
Consider a series structural system consisting of \(m\) failure modes, each of which can be defined by a performance function:  
\[g_{j}(\mathbf{X}),j=1,2,\cdots,m \tag{1}\]  
where \(\mathbf{X}=\left(X_{1},X_{2},\ldots,X_{n}\right)^{\mathrm{T}}\) is an n-dimensional vector; \(X_{i}\) is a variable associated with geometric dimensions, material properties, loads, etc., and all of its possible values form a marginal interval \(X_{i}^{\mathrm{I}}\); and \(g_{j}(\mathbf{X})=0\) denotes the failure surface. For the sake of simplicity, the performance functions involved are assumed to be linear, since a weak nonlinear performance function can be replaced by its linear approximation at the design point. The interval model and the ellipsoid model have been widely used to deal with uncertain variables. In the following text, the ellipsoid model is only considered to describe the uncertainty domain.  
\[E_{\mathbf{X}}=\left\{\mathbf{X}|\left(\mathbf{X}-\mathbf{X}^{\mathrm{c}} \right)^{\mathrm{T}}\mathbf{\Omega}(\mathbf{X}-\mathbf{X}^{\mathrm{c}})\leq 1\right\} \tag{2}\]  
where \(E_{\mathbf{X}}\) and \(\mathbf{X}^{\mathrm{c}}\) denote the ellipsoid and its midpoint, respectively; and \(\mathbf{\Omega}\) is a characteristic matrix determining the size and orientation of the ellipsoid.  
The uncertain vector \(\mathbf{X}\) is first mapped into the normalized vector \(\boldsymbol{\delta}=\left[\delta_{1},\delta_{2},\cdots,\delta_{n}\right]^{ \mathrm{T}}\) by using a linear transformation. This linear transformation generally consists of a parallel shift and a rotation of the coordinate system, and the reader can refer to Refs. [19; 26] for more information. Through such a treatment, the subsequent system reliability analysis becomes easy to perform. Correspondingly, the ellipsoid and the performance function become the unit sphere \(\boldsymbol{\delta}^{\mathrm{T}}\boldsymbol{\delta}\leq 1\) and the normalized performance function \(G_{j}(\boldsymbol{\delta})\), respectively.  
For a series system, the structural system will fail when any one of the failure modes is in a failure state. The failure surface of the system can be mathematically expressed as follows:  
\[G_{\text{series}}(\delta)=0 \tag{3}\]  
In Figure 1, a system with two failure modes for a two-dimensional problem is taken as an example. Here, the failure surface \(G_{\text{series}}(\delta)=0\) represents a geometrically continuous but unsmooth surface. The entire ellipsoid is divided into two different parts by the failure surface, one of which is in the safe domain (\(G_{\text{series}}(\delta)\geq 0\)), while the other is in the failure domain (\(G_{\text{series}}(\delta)<0\)).  
Intuitively, the non-probabilistic failure degree \(f_{\text{s}}\) and non-probabilistic reliability \(R_{\text{s}}\) of a series system can be given as follows [26]:  
\[\begin{cases}f_{\text{s}}=\frac{A_{\text{s}}}{A_{\text{w}}}\\ R_{\text{s}}=\frac{A_{\text{w}}-A_{\text{s}}}{A_{\text{w}}}\end{cases} \tag{4}\]  
where \(A_{\text{w}}=\frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{k_{\text{s}}2\pi}{2} \right)}\) denotes the volume of the whole ellipsoid, with \(\Gamma(\bullet)\) denoting a gamma function; \(A_{\text{s}}\) denotes the volume of the ellipsoid falling into the failure domain, which can be called the failure volume of the system.  
The authors proposed the NRBM to estimate the system's non-probabilistic failure degree when \(A_{\text{s}}\) is not always available, i.e.,  
\[\frac{A_{1}+\sum\limits_{j=2}^{m}\text{max}\left[\left(A_{j}-\sum\limits_{k=1 }^{j-1}A_{jk}\right),0\right]}{A_{\text{w}}}\leq\frac{A_{\text{s}}}{A_{\text{ w}}}\leq\frac{\sum\limits_{j=1}^{m}A_{j}-\sum\limits_{j=2}^{m}\text{max} \begin{pmatrix}A_{jk}\\ k<j\end{pmatrix}}{A_{\text{w}}} \tag{5}\]  
where \(A_{j}\) is the failure volume of the \(j_{\text{th}}\) failure mode and \(A_{jk}\) is the joint failure volume of the \(j_{\text{th}}\) and \(k_{\text{th}}\) failure modes. The reader can refer to Refs. [26; 30] for more information on \(A_{j}\) and \(A_{jk}\).