# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## 2 The Analysis of the Structural Reliability Using the Traditional SORM  
Computing the multiple probability integrals is the essential problem in structural reliability analysis, defined as:  
\[P_{f}=\Pr[G(\mathbf{x})\leq 0]=\int_{G(\mathbf{x})\leq 0}f(\mathbf{x})d\mathbf{x} \tag{1}\]  
where \(\mathbf{x}=(x_{1},x_{2},\cdots,x_{n})^{T}\) is an n-dimension random variables vector representing the basic uncertain variables in the structure. \(G(\mathbf{x})\) is the state function that determines the abnormality of the structure by using the condition of \(G(\mathbf{x})\leq 0\). \(P_{f}\) and \(f(\mathbf{x})\) are the failure probability and joint probability density function of \(\mathbf{x}\), respectively. FORM is one of the most widely used computational methods [14; 16]. The accuracy is decreased for the nonlinear performance function when using the linear function-based approximation of the state function. The SORM was proposed to enhance the accuracy of the FORM [16], where the limit-state surface is approximated by a second-order surface [18; 20], for the design point \(\mathbf{u}^{*}\) in the n u-space \(G(\mathbf{u})\). The state function can be expressed by second-order Taylor expansion as follows:  
\[G(\mathbf{u})=G(\mathbf{u}^{*})+\
abla G(\mathbf{u}^{*})(\mathbf{u}-\mathbf{ u}^{*})+\frac{1}{2}(\mathbf{u}-\mathbf{u}^{*})^{T}\mathbf{H}(\mathbf{u}- \mathbf{u}^{*}) \tag{2}\]  
where \(\
abla G(\mathbf{u}^{*})=\left.\frac{\partial G(\mathbf{u})}{ \partial\mathbf{u}}\right|_{\mathbf{u}^{*}}\), \(\mathbf{H}\) is the Hessian matrix, which denotes \(\mathbf{H}_{ij}=\left.\frac{\partial G}{\partial u_{i}\partial u_{j}}\right| _{u^{*}}\).  
However, the structural reliability estimation becomes challenging to solve when the dimensionality of the state function is high [45]. This is because many samples in high-dimensional spaces are needed for the failure event estimation with low probability. Furthermore, the reliability problem cannot be visualized, which is desirable in a general design process [23]. Thus, we propose to reduce the dimensionality of the problem. The random samples are represented in polar form in MC simulation using the design point and the mass center vector in the failure probability [23; 29]. Two nonlinear variables: (1)the distance to the origin and (2) the angle information, are employed. The two nonlinear variables \(v_{1}\) and \(v_{2}\) are defined as follows:  
\[v_{1}=r=\left\|\mathbf{u}\right\|_{2} \tag{3}\]  
\[v_{2}=\cos\theta=\frac{(\mathbf{u},\boldsymbol{\alpha})}{\left\|\mathbf{u} \right\|}=\frac{(\mathbf{u},\boldsymbol{\alpha})}{v_{1}} \tag{4}\]  
where the design point unit vector \(\boldsymbol{\alpha}=\dfrac{\
abla G(\mathbf{u}^{*})}{\left\|G(\mathbf{u})^{*} \right\|}\).  
The PDFs of \(v_{1}\) and \(v_{2}\) is written as:  
\[f_{v_{1}}(x)=\frac{2^{1-n/2}x^{(n-1)}}{\Gamma(n/2)}\exp(-\frac{x^{2}}{2})\qquad (x>0) \tag{5}\]  
\[f_{v_{2}}(x)=\frac{1}{\sqrt{1-x^{2}}\int_{0}^{n}\sin^{n+m-2}\alpha da}(\sin^{n +m-2}(\arccos x)+\sin^{n+m-2}(\pi-\arccos x))\quad\ -1\leq x\leq 1 \tag{6}\]  
According to [PERSON], the \(v_{1}\) and \(v_{2}\) are independent [23; 29]. The uncertainty properties are subjectively assumed: e.g., using the truncated Gaussian distribution representation for interval uncertainty and uniform distribution. As described in the next section, the new polar transformation is used to avoid the severe reliability analysis errors that are caused by subjective assumptions.  
**3. UPSORM: The New SORM in Polar for Random and Interval Uncertainty**