# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## 2 The Analysis of the Structural Reliability Using the Traditional SORM
### Implementation Details  
(1) For the random variables vector \(\mathbf{x}=(x_{1},x_{2},\cdots,x_{n})\) and interval variables vector \(\mathbf{z}=(z_{1},z_{2},\cdots,z_{m})\), \(z_{j}\in[z_{j}^{L},z_{j}^{U}]\) in the state function of the structure \(G(\mathbf{x},\mathbf{z})=b-y(\mathbf{x},\mathbf{z})\), transfer \(\mathbf{x}\) to the independent Gaussian vector \(\mathbf{u}=(u_{1},u_{2},\cdots,u_{n})\) and convert \(\mathbf{z}\) to the independent normalized interval vector \(\boldsymbol{\delta}=(\delta_{1},\delta_{2},\cdots,\delta_{m})\in[-1,1]\). Then, \(G(\mathbf{x},\mathbf{z})\) is converted into \(G(\boldsymbol{\omega})\) with \(\boldsymbol{\omega}=(\mathbf{u},\boldsymbol{\delta})\).  
(2) Given an initial design point \(\boldsymbol{\omega}^{*(0)}=(u_{1}^{*(0)},u_{2}^{*(0)}\cdots,u_{n}^{*(0)},\delta _{1}^{*(0)},\cdots,\delta_{m}^{*(0)})\), compute the gradient \(\left.\dfrac{\partial G}{\partial\omega_{i}}\right|_{\boldsymbol{\omega}^{*(0)}}\) of \(G(\boldsymbol{\omega})\), \(d^{*(0)}\) and \(D^{*(0)}\) at \(\boldsymbol{\omega}^{*(0)}\), and with Equation (27) transfer \(G(\boldsymbol{\omega})\) into \(G^{(0)}(v_{1},v_{2})\) in polar space.  
(3) Using Equations (29) and (30), compute the initial failure probability interval \([P_{f}^{U(0)},P_{f}^{U(0)}]\).  
(4) Compute \(\omega_{k}^{+(l)}(k=1,2,\ldots n+m)(l=1,2,\ldots)\) by Equation (31)  
\[\omega_{k}^{+(l)}=\dfrac{-\dfrac{\partial y}{\partial\omega_{k}}\bigg{|}_{ \boldsymbol{\omega}_{k}^{+(l-1)}}}{\left[\sum_{k=1}^{n+m}\left(\left.\dfrac{ \partial y}{\partial\omega_{k}}\right|_{\boldsymbol{\omega}_{k}^{+(l-1)}} \right)^{2}\right]^{1/2}}(l=1,2,\ldots) \tag{31}\]  
(5) Calculate the new design point \(\omega_{k}^{*(l)}(k=1,2,\ldots n+m)(l=1,2,\ldots)\) with \(\omega_{k}^{+(l)}\) by function (32)\[\omega_{k}^{*(l)}=\omega_{k}^{*(l-1)}+\left\|\omega_{k}^{*(l-1)}\right\|\omega_{k} ^{+(l-1)}\left\|\omega_{k}^{+(l)}=\omega_{k}^{*(l-1)}+\left\|\omega_{k}^{*(l-1)} \right\|\right.\frac{-\frac{\partial y}{\partial\omega_{k}}\bigg{|}_{\omega_{ k}^{*(l-1)}}}{\sum_{k=1}^{n+m}\left(\frac{\partial y}{\partial\omega_{k}} \bigg{|}_{\omega_{k}^{*(l-1)}}\right)^{2}\right]^{1/2}}(l=1,2,\ldots) \tag{32}\]  
and parameters \(d^{(l)}\) and \(D^{(l)}\) with Equation (27).  
(6) Compute the failure probability interval \(P_{failure}^{L}\)\({}^{(l)}\) and \(P_{failure}^{U}\)\({}^{(l)}\) with Equations (29) and (30). Taking the robust convergence into account, assume that the iterative errors are denoted as \(\frac{\left|P_{failure}^{U}\right.^{(l)}-P_{failure}^{U}\left.{}^{(l-1)} \right|}{1-P_{failure}^{U}\left.{}^{(l)}\right|}\) and \(\frac{\left|P_{failure}^{L}\right.^{(l)}-P_{failure}^{L}\left.{}^{(l-1)} \right|}{1-P_{failure}^{U}\left.{}^{(l)}\right|}\). The thresholds of them is \(\varepsilon\). If \(\frac{\left|P_{failure}^{U}\right.^{(l)}-P_{failure}^{U}\left.{}^{(l-1)} \right|}{1-P_{failure}^{U}\left.{}^{(l)}\right|}\leq\varepsilon\) and \(\frac{\left|P_{failure}^{L}\right.^{(l)}-P_{failure}^{L}\left.{}^{(l-1)} \right|}{1-P_{failure}^{U}\left.{}^{(l)}\right|}\leq\varepsilon\), stop the calculating procedure. Otherwise, repeat steps 2-6. Assign the design point \(\omega^{*(l)}=(u_{1}^{*(l)},u_{2}^{*(l)}\cdots,u_{n}^{*(l)},\phi_{1}^{*(l)}, \cdots,\phi_{m}^{*(l)})\) by the initial design point until the accuracy requirement is satisfied. The main steps for implementation are shown in Figure 2. The pseudo-code that illustrates the process of using the method in more detail is shown in Appendix A.  
Figure 2: The flowchart of the implementation.