# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## 2 The Analysis of the Structural Reliability Using the Traditional SORM
### Polar Transformation For State Function with Randomness and Intervals  
The structural state function is defined as:  
\[G(\mathbf{x},\mathbf{z})=b-y(\mathbf{x},\mathbf{z}) \tag{7}\]  
where \(\mathbf{x}=(x_{1},x_{2},\cdots,x_{n}),\mathbf{z}=(z_{1},z_{2},\cdots,z_{m})\) is the basic uncertain vectors of \(G(\mathbf{x},\mathbf{z})\). \(x_{1},x_{2},\cdots,x_{n}\) are random variables and \(z_{1},z_{2},\cdots,z_{m}\) are interval variables. \(y(\mathbf{x},\mathbf{z})\) is the structural response and \(b\) is the threshold for the \(y(\mathbf{x},\mathbf{z})\), whose exceeding defines the failure state. The random variables \(\mathbf{x}=(x_{1},x_{2},\cdots,x_{n})\) are standard Gaussian transformations into \(\mathbf{u}=(u_{1},u_{2},\cdots,u_{n})\), where \(\forall u_{i}=T_{random}(x_{i})\sim N(0,1)\), \(i=1\), \(2\), \(\cdots\), \(n\), and \(T_{random}(\bullet)\) denotes the standard independent Gaussian transformation that converts \(\mathbf{x}\) into \(\mathbf{u}\). The interval variables \(\mathbf{z}=(z_{1},z_{2},\cdots,z_{m})\) are normalized into \(\boldsymbol{\delta}=(\delta_{1},\delta_{2},\cdots,\delta_{m})\), where \(\forall\delta_{j}=T_{interval}(z_{j})\in[-1,1]\), \(j=1\), \(2\), \(\cdots\), \(m\), and \(T_{interval}(\bullet)\) denotes the standard independent interval transformation that converts \(\mathbf{z}\) into \(\boldsymbol{\delta}\). Hence, we denote \(\boldsymbol{\omega}=(T_{random}(\mathbf{x}),T_{interval}(\mathbf{z}))=(u_{1},u_{2},\cdots,u_{n},\delta_{1},\delta_{2},\cdots,\delta_{m})\). (7) can be expressed as follows:  
\[G(\boldsymbol{\omega})=b-y(\boldsymbol{\omega}) \tag{8}\]  
In this paper, the polar coordinate is improved. The polar axis is set as the cosine of the angle between the mass center vector of the failure domain and the sample point in a rectangular coordinate system. The pole is the coordinate origin. It is denoted that \(v_{2}=\cos\theta\), \(\theta=\angle(\boldsymbol{\omega},\boldsymbol{\omega}^{+})\), where \(\boldsymbol{\omega}^{+}\) is the centroid vector of a failure domain in cartesian coordinate [45], as shown in Figure 1.  
Figure 1 suggests that \(\mathbf{\omega}^{+}\) is paralleled with the gradient direction of state function \(G(\mathbf{\omega})\). \(\mathbf{\omega}^{+}\) denotes the unit vector of \(G(\mathbf{\omega})\) at the design point along the gradient direction. The distance \(\parallel\mathbf{\omega}^{+}\parallel=1\):  
\[\mathbf{\omega}^{+}=\frac{\
abla G(\mathbf{\omega}^{*})}{\left\|\
abla G(\mathbf{\omega})^ {*}\right\|}=-\frac{\
abla y(\mathbf{\omega}^{*})}{\left\|y(\mathbf{\omega})^{*}\right\|} \tag{9}\]  
\[\overline{v}_{1}=\sqrt{\sum_{i=1}^{n}{u_{i}}^{2}+\sum_{j=1}^{m}{\delta_{i}}^{ 2}} \tag{10}\]  
\[\overline{v}_{2}=\cos\theta=\frac{(\mathbf{\omega}\mathbf{,}\mathbf{\omega}^{+})}{\left\| \mathbf{\omega}\right\|\left\|\mathbf{\omega}^{+}\right\|}=\frac{(\mathbf{\omega}\mathbf{,} \mathbf{\omega}^{+})}{\left\|\mathbf{\omega}\right\|\left\|\mathbf{\omega}\right\|\left\| \mathbf{\omega}\right\|}=\frac{(\mathbf{\omega}\mathbf{,}\mathbf{\omega}^{+})}{\overline{v}_{ 1}} \tag{11}\]  
(8) is rewritten in the polar space as:  
\[G(\overline{v}_{1},\overline{v}_{2})=b-y(\overline{v}_{1},\overline{v}_{2}) \tag{12}\]  
for \(\forall n\in N\), it is suggested that:  
\[\sqrt{\sum_{i=1}^{n}{u_{i}}^{2}}\leq\overline{v}_{1}=\sqrt{\sum_{i=1}^{n}{u_{ i}}^{2}+\sum_{j=1}^{m}{\delta_{i}}^{2}}\leq\sqrt{\sum_{i=1}^{n}{u_{i}}^{2}+m} \tag{13}\]  
We denote \(\Delta=\sum_{j=1}^{m}{\delta_{i}}^{2}\), and for \(\forall u_{i},i=1,2,\cdots,n\), \(u_{i}\) obeys standard Gaussian distribution. It is reduced so that \(\overline{v}_{1}{}^{2}-\Delta=\sum_{i=1}^{n}{u_{i}}^{2},\Delta\in[0,m]\), in which \(\sum_{i=1}^{n}{u_{i}}^{2}\sim\chi^{2}(n)\), where \(n\) is the number of random variables and \(m\) is one of the intervals.