# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## Appendix A
(4) Rewrite the limited state function \(G(\mathbf{x},\mathbf{z})=b-y(\mathbf{x},\mathbf{z})\) into \(G(\mathbf{\omega})\) with \(\mathbf{\omega}=(\mathbf{u},\mathbf{\delta})\)(5) Assume an initial design point \(\omega^{*(0)}=(u_{1}^{*(0)},u_{2}^{*(0)}\cdots,u_{n}^{*(0)},\delta_{1}^{*(0)}, \cdots,\delta_{m}^{*(0)})\), and \(\omega_{1}\) compute: \(\frac{\partial G}{\partial\omega_{1}}\bigg{|}_{\omega^{*(0)}}\), \({v_{1}}^{*(0)}\), \({v_{2}}^{*(0)}\), \({D}^{*(0)}\), \({\lambda}^{*(0)}\), \(d^{*(0)}\) as follows:  
\[\frac{\partial G}{\partial\omega_{1}}\bigg{|}_{\omega^{*(0)}} =-\frac{\partial y}{\partial\omega_{1}}\bigg{|}_{\omega^{*(0)}}(i=1,2, \cdots,n+m)\] \[{v_{1}}^{*(0)}=\sqrt{\sum_{k=1}^{n+m}\left(\omega_{i}^{*(0)} \right)^{2}}\]  
\[{v_{2}}^{*(0)}=\frac{\sum_{i=1}^{n}\left(\left.\frac{\partial y}{\partial u_{ i}}\bigg{|}_{\omega^{*(0)}}u_{i}\right)+\sum_{j=1}^{m}\left(\left.\frac{ \partial y}{\partial\delta_{j}}\bigg{|}_{\omega^{*(0)}}\delta_{j}\right) \right.}{{v_{1}}^{*(0)}\left[\sum_{i=1}^{n}\left(\left.\frac{\partial y}{ \partial u_{i}}\bigg{|}_{\omega^{*(0)}}\right)^{2}\right]^{1/2}}\]  
\[D^{*(0)}=\left[\sum_{i=1}^{n}\left(\left.\frac{\partial y}{\partial u_{i}} \bigg{|}_{\omega^{*(0)}}\right)^{2}+\sum_{j=1}^{m}\left(\left.\frac{\partial y }{\partial\delta_{j}}\bigg{|}_{\omega^{*(0)}}\right)^{2}\right]^{1/2}\right.\]  
(6) compute the initial failure probability interval \([P_{f}^{U(0)},P_{f}^{L(0)}]\) as follows:  
\[P_{failure}^{L}{}^{*(0)}=\int_{\Omega_{5}=\{\overline{\psi}|G( \xi_{1},\xi_{2})\leq 0\}}^{1}f_{\overline{\tau}_{1}}(\xi_{1},m)f_{\overline{ \tau}_{2}}(\xi_{2})d\xi_{1}d\xi_{2}\] \[P_{failure}^{U}{}^{*(0)}=\int_{\Omega_{5}=\{\overline{\psi}|G( \xi_{1},\xi_{2})\leq 0\}}^{1}f_{\overline{\tau}_{1}}(\xi_{1})f_{\overline{ \tau}_{2}}(\xi_{2})d\xi_{1}d\xi_{2}\]  
(7) For \(l=1,2,\ldots,s\), compute \(\omega_{k}^{+(l)}(k=1,2,\ldots n+m)\) as follows:  
(8) For \(l=1,2,\ldots,s\), calculate the new design point \(\omega_{k}^{*(l)}(k=1,2,\ldots n+m)\) with \(\omega_{k}^{+(l)}\) as follows:  
\[\omega_{k}^{*(l)}=\omega_{k}^{*(l-1)}+\left|\left|\omega_{k}^{*(l-1)}\right| \left|\omega_{k}^{+(l)}=\omega_{k}^{*(l-1)}+\left|\left|\omega_{k}^{*(l-1)} \right|\right|\frac{-\frac{\partial y}{\partial\omega_{k}}\bigg{|}_{\omega_{ k}^{*(l-1)}}}{\left[\sum_{k=1}^{n+m}\left(\left.\frac{\partial y}{ \partial\omega_{k}}\bigg{|}_{\omega_{k}^{*(l-1)}}\right)^{2}\right]^{1/2}}(l=1,2,\ldots)\right.\](9) For \(l=1,2,\ldots,s\), calculate \(\left.\frac{\partial\mathcal{G}}{\partial\omega_{i}}\right|_{\mathbf{\omega}^{*(l)}}\), \({\mathbf{\upsilon}^{*(l)}}\), \({\mathbf{\upsilon}^{*(l)}}\), \({\mathbf{\upsilon}^{*(l)}}\), \({\mathbf{\upsilon}^{*(l)}}\), \({\mathbf{\lambda}^{*(l)}}\), \({\mathbf{\lambda}^{*(l)}}\) as follows:  
\[{\mathbf{\upsilon}_{1}}^{*(l)}=\sqrt{\sum\limits_{k=1}^{n+m}\Bigl{(}{\mathbf{\omega} _{i}^{*(l)}}\Bigr{)}^{2}}\]  
\[{\mathbf{\upsilon}_{2}}^{*(l)}=\frac{\sum\limits_{l=1}^{n}\biggl{(}\frac{\partial y }{\partial u_{i}}\biggr{|}_{\mathbf{\omega}^{*(l)}}\biggr{)}+\sum\limits_{l=1}^{m} \biggl{(}\frac{\partial y}{\partial\delta_{j}}\biggr{|}_{\mathbf{\omega}^{*(l)}} \biggr{)}}{{\mathbf{\upsilon}_{1}}^{*(l)}\biggl{|}\sum\limits_{i=1}^{n}\biggl{(} \frac{\partial y}{\partial u_{i}}\biggr{|}_{\mathbf{\omega}^{*(l)}}\biggr{)}^{2}+ \sum\limits_{l=1}^{m}\biggl{(}\frac{\partial y}{\partial\delta_{j}}\biggr{|}_ {\mathbf{\omega}^{*(l)}}\biggr{)}^{2}}\biggr{]}^{1/2}\]  
\[{\mathbf{\lambda}}^{*(l)}=\frac{b-y(\mathbf{\omega}^{*(l)})+\sum\limits_{k=1}^{n+m} \biggl{(}\frac{\partial y}{\partial\omega_{k}}\biggr{|}_{\mathbf{\omega}^{*(l)}} \omega_{k}{}^{*(l)}\biggr{)}-{\mathbf{\upsilon}_{1}}^{*(l)}{\mathbf{\upsilon}_{2}}^{*( l)}}{2\bigl{(}{\mathbf{\upsilon}_{1}}^{*(l)}\bigr{)}^{2}-2{\mathbf{\upsilon}_{2}}^{*(l)}}\]  
\[{v_{1}}^{*(0)}=\sqrt{\sum_{k=1}^{n+m}\left(\omega_{i}^{*(0)} \right)^{2}}\]0  
(10) Compute the failure probability interval \({P_{failure}^{l}}^{l}\) and \({P_{failure}^{l}}^{l}\) as follows:  
\[{v_{1}}^{*(0)}=\sqrt{\sum_{k=1}^{n+m}\left(\omega_{i}^{*(0)} \right)^{2}}\]1  
(11) Output: compute \(\dfrac{\left|{P_{failure}^{upper}(l)-P_{failure}^{upper}(l-1)}\right|}{1-P_{failure }^{upper}(l)}\) and \(\dfrac{\left|{P_{failure}^{lower}(l)-P_{failure}^{lower}(l-1)}\right|}{1-P_{failure }^{lower}(l)}\).