## 2 Axis Orthogonal Importance Sampling Method  
As we know, the importance sampling function has a significant impact on accuracy of the probability of structural failure in the importance sampling method. The probability of structural failure is calculated as:  
\[p_{f}=\int_{-\infty}^{+\infty}\frac{I[\mathsf{g}_{X}(\mathbf{v})]f_{X}(\mathbf{v})}{p_{ V}(\mathbf{v})}p_{V}(\mathbf{v})d\mathbf{v}=E\bigg{\{}\frac{I[\mathsf{g}_{X}(\mathbf{v})]f_{X}(\mathbf{v})} {p_{V}(\mathbf{v})}\bigg{\}} \tag{1}\]  
where \(p_{V}(\mathbf{v})\) and \(g_{X}(\mathbf{v})\) is the sampling probability density function (PDF) and the performance function of structure at \(\mathbf{v}\) in the standard normal space \(V\), \(I(\cdot)\) is the indicator function, and \(f_{X}(\mathbf{v})\) is the joint PDF in the normal space \(V\) transformed from the origin space \(X\). The importance sampling PDF is commonly defined based on the most probable failure point (MPFP) that are obtained by the first-order reliability method. The unbiased estimate \(\hat{p}_{f}\) of the probability of structural failure \(p_{f}\) can be yielded as the following by \(N\) samples \(\mathbf{v}_{i}=\left(v_{i1},v_{i2},\cdots v_{in}\right)^{T}\), \(i=1,2,\cdots,N\) that are generated from the important PDF:  
\[\hat{p}_{f}=\frac{1}{N}\sum_{i=1}^{N}\frac{I[\mathsf{g}_{X}(\mathbf{v}_{i})]f_{X} (\mathbf{v}_{i})}{p_{V}(\mathbf{v}_{i})} \tag{2}\]  
where \(N\) is the sample size, \(n\) is the number of random variables. The variance of \(\hat{p}_{f}\) is calculated by:  
\[ \begin{split}\sigma_{\hat{p}_{f}}^{2}&=\frac{1}{N} E\Bigg{\{}\frac{I[\mathsf{g}_{X}(\mathbf{v})]f_{X}{}^{2}(\mathbf{v})}{p_{V}(\mathbf{v})} \bigg{\}}-\frac{1}{N}\hat{p}_{f}{}^{2}\\ &=\frac{1}{N}\int_{-\infty}^{+\infty}\frac{I[\mathsf{g}_{X}(\mathbf{v })]f_{X}{}^{2}(\mathbf{v})}{p_{V}(\mathbf{v})}\mathrm{d}\upsilon-\frac{1}{N}\ \hat{p}_{f}{}^{2}\end{split}  \tag{3}\]  
A sample plan of the axis orthogonal importance sampling in the transformed normal space is established in the direction of the approximate tangent hyperplane, as shown in Figure 1, where \(\mathsf{g}_{X}(\mathbf{v})=0\) is the limit state surface in standard normal space, \(\beta\) is the reliability index. The axial orthogonal importance sampling method takes the MPFP as the center and samples in the tangent plane which at the MPFP. \(N\) samples in the local coordinate system of the tangent plane form an \(N\times(n-1)\) matrix \(S\), which just samples in the \(n-1\) dimensions with more efficiency in the axis orthogonal importance sampling method. These samples projects onto the limit state surface along the normal direction \(\mathbf{m}\) of the MPFP. The calculation of reliability sensitivity is equivalent to summing the reliability sensitivity of a group of performance functions defining those hyperplanes (parallel line \((\mathrm{PL}),PL_{i},i=1,\cdots,N\)) parallel to the tangent plane of the limit state surface through the abovementioned projection points on the limit state surface. Suppose \(S_{i}\) is the \(i\) th realization of \(S\) in the local coordinate system, the coordinates of the projection point in the local coordinate system \(\mathbf{v}_{1}^{\prime}O\mathbf{v}_{2}^{\prime}\) is \([b_{i}\ S_{i}]\). In order to determine \(b_{i}\), the projection point \([b_{i}\ S_{i}]\) coordinate is firstly transformed to \(S_{i}^{*}\) in the global coordinate system \(\mathbf{v}_{1}O\mathbf{v}_{2}\) by a coordinate transformation matrix \(A\), and then substituted into \(\mathsf{g}(\mathbf{v})=0\) to solve the equation. The MPFP is \(\mathbf{m}\), and the unit vector of \(\mathbf{m}\), \(\mathbf{\mathrm{a}}\) reads as [17]:  
\[\mathbf{\mathrm{a}}=\frac{\mathbf{m}}{\|\mathbf{m}\|} \tag{4}\]  
The transformation matrix \(A\) of the local and global coordinate system can be defined as:  
\[A=(\mathbf{\mathrm{a}},\cdots) \tag{5}\]  
Except for the first column \(\mathbf{\mathrm{a}}\), the other columns of \(A\) are all orthogonal to \(\mathbf{\mathrm{a}}\) with the property \(A^{T}A=I\), where \(I\) is the identity matrix.