## 2 Structural Reliability
### Second-Order Reliability Method  
The SORM [6] transforms a reliability problem into the independent standard normal random variable space as the FORM, approximates a limit-state function through a second-order surface, and computes the probability of failure using the approximate surface. Consider a Taylor series expansion of the limit-state function in the standard normal space at \(\mathbf{u}^{*}\):  
\[\begin{split} G(\mathbf{u})&=\
abla G(\mathbf{u}*)( \mathbf{u}-\mathbf{u}*)+\frac{1}{2}(\mathbf{u}-\mathbf{u}*)^{\mathsf{T}} \mathbf{H}(\mathbf{u}-\mathbf{u}*)+O\\ &\cong\|\
abla G(\mathbf{u}*)\|\Big{[}(\beta-\alpha\mathbf{u})+ \frac{1}{2\|\
abla G(\mathbf{u}*)\|}(\mathbf{u}-\mathbf{u}*)^{\mathsf{T}} \mathbf{H}(\mathbf{u}-\mathbf{u}*)\Big{]}\end{split} \tag{6}\]  
where \(\mathbf{H}\) is the second-derivative matrix of \(G(\mathbf{u})\) at \(\mathbf{u}^{*}\), known as the Hessian matrix having elements \(H_{ij}=\partial^{2}G(\mathbf{u}*)/(\partial u_{i}\partial u_{j})\), \(i,j=1,\ldots,n\), and \(O\) is the higher-order error term. Consider an orthonormal matrix \(\mathbf{P}\) with \(\alpha\) that leads to a rotation of axes \(\mathbf{u}^{{}^{\prime}}=\mathbf{P}\mathbf{u}\) and positions the design point on the \(u^{\prime}{}_{n}\) axis (\(\equiv\mathbf{u}*\)). The limit-state function can be expressed as:  
\[G^{\prime}(\mathbf{u}^{{}^{\prime}})=\beta-u^{\prime}{}_{n}+\frac{1}{2}( \mathbf{u}^{{}^{\prime}}-\mathbf{u}^{{}^{\prime}}*)^{\mathsf{T}}\mathbf{D}( \mathbf{u}^{{}^{\prime}}-\mathbf{u}^{{}^{\prime}}*) \tag{7}\]  
where \(u^{\prime}{}_{n}=\alpha\mathbf{P}^{\mathsf{T}}\mathbf{u}^{{}^{\prime}}\) and \(\mathbf{D}=\mathbf{P}\mathbf{H}\mathbf{P}^{\mathsf{T}}/\|\
abla G(\mathbf{u} ^{*})\|\). Expanding the matrix product in Equation (7) and considering the tangent plane at the design point while neglecting the smaller order terms [6] yield:  
\[G^{\prime}(\mathbf{u}^{{}^{\prime}})=\beta-u^{\prime}{}_{n}+\frac{1}{2} \mathbf{u}^{{}^{\prime}}_{1}\mathbf{D}_{11}\mathbf{u}^{{}^{\prime}}_{1} \tag{8}\]  
where \(\mathbf{D}_{11}\) is the \((n-1)\times(n-1)\) matrix formed by the first \((n-1)\) rows and columns of \(\mathbf{D}\). After a set of linear transformations [6; 10; 28], such as rotation of the axes with \(\mathbf{u}^{{}^{\prime\prime}}_{1}=\mathbf{Q}\mathbf{u}^{\prime}_{1}\) (\(\mathbf{Q}\) is an \((n-1)\times(n-1)\) orthonormal matrix) and orthogonal diagonalization, the limit-state function in Equation (8) can be simplified as follows:  
\[G^{\prime}(\mathbf{u}^{{}^{\prime}})=\beta-u^{\prime}_{n}+\frac{1}{2}\sum_{i= 1}^{n-1}\kappa_{i}\mathbf{u}^{{}^{\prime\prime}2}_{i} \tag{9}\]  
where \(\kappa_{i}\) represents the eigenvalues of \(\mathbf{D}_{11}\) that define the principal curvatures of the limit-state function. Based on the asymptotic approximations derived by [PERSON] [8], the failure probability can be computed as:  
\[P_{\text{/SORM}}\approx\Phi(-\beta)\cdot\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+ \beta\kappa_{i}}} \tag{10}\]  
Note that the term \(1/\sqrt{1+\beta\kappa_{i}}\) acts as a correction for the FORM approximation. [PERSON] and [PERSON] [9] proposed an alternate correction to improve the calculation results, as:  
\[P_{\text{/SORM }-1}\cong\Phi(-\beta)\cdot\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+( \varphi(\beta)/\Phi(-\beta))\kappa_{i}}} \tag{11}\]  
The approximation of the limit-state function in the SORM is better than that of the FORM and results in a more accurate assessment of the failure probability. However, the SORM requires additional effort in terms of the function evaluation and construction of the Hessian matrix.