# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
## 2 Materials and Methods  
### Principles of Fourth-Moment Method  
In reliability theory [38], the probability of failure \(P_{f}\) is usually expressed in terms of the state function \(g\) as  
\[P_{f}=P(g<0) \tag{1}\]\(g\) can be normalized to \(g_{s}=\frac{g-\mu_{g}}{\sigma_{g}}\), with \(\mathrm{E}(g_{s})=0\) and \(\mathrm{E}({g_{s}}^{2})=1\). \(\mu_{g}\) and \(\sigma_{g}\) are the mean and standard deviation of \(g\), respectively.  
\(g_{s}\) can be further converted to  
\[y=g_{s}+c{g_{s}}^{2} \tag{2}\]  
where \(y\) is assumed to be normally distributed and \(c\) is a constant, so the statistical moments of \(y\) are expressed as follows:  
\[\mu_{y}=\mathrm{E}(y)=c \tag{3}\]  
\[{\sigma_{y}}^{2}=\mathrm{E}\big{[}\big{(}y-\mu_{y}\big{)}^{2}\big{]} \tag{4}\]  
\[\alpha_{3y}=\frac{\mathrm{E}\big{[}\big{(}y-\mu_{y}\big{)}^{3}\big{]}}{{\sigma _{y}}^{3}} \tag{5}\]  
\(y\) is a normal random variable, so  
\[\alpha_{3y}=c^{3}\big{(}\alpha_{6g}-3\alpha_{4g}+2\big{)}+3c^{2}\big{(}\alpha_ {5g}-2\alpha_{3g}\big{)}+3c\big{(}\alpha_{4g}-1\big{)}+\alpha_{3g}=0 \tag{6}\]  
where \(\alpha_{6g}=\mathrm{E}\big{[}{g_{s}}^{6}\big{]}\), \(\alpha_{5g}=\mathrm{E}\big{[}{g_{s}}^{5}\big{]}\), \(\alpha_{4g}=\mathrm{E}\big{[}{g_{s}}^{4}\big{]}\), and \(\alpha_{3g}=\mathrm{E}\big{[}{g_{s}}^{3}\big{]}\) are the sixth-, fifth-, fourth-, and third-order dimensionless central moments of \(g\), respectively.  
In the first proposed fourth-moment method (FM) [15], \(c\) was approximated as  
\[c=\frac{-\alpha_{3g}}{3\big{(}\alpha_{4g}-1\big{)}} \tag{7}\]  
The standardized \(y\) can be expressed as  
\[v=\frac{y-\mu_{y}}{\sigma_{y}} \tag{8}\]  
According to FORM, the failure probability \(P_{f}\) in (1) can be expressed as  
\[P_{f}=P(g<0)=P\bigg{(}g_{s}<\frac{-\mu_{g}}{\sigma_{g}}\bigg{)}=\Phi\bigg{(} \frac{-\mu_{g}}{\sigma_{g}}\bigg{)} \tag{9}\]  
The corresponding reliability index is expressed as  
\[\beta=\frac{\mu_{g}}{\sigma_{g}} \tag{10}\]  
In the case where the first four moments of \(g\) are obtained, Equations (2)-(4) are brought into (8) to obtain  
\[v=\frac{g_{s}+c{g_{s}}^{2}-c}{\sqrt{\big{(}\alpha_{4g}-1\big{)}c^{2}+2\alpha _{3g}c+1}}=\frac{-{g_{s}}3\big{(}1-\alpha_{4g}\big{)}-\alpha_{3g}\big{(}{g_{s }}^{2}-1\big{)}}{\sqrt{5{a_{3g}}^{2}\big{(}1-\alpha_{4g}\big{)}+9\big{(}1- \alpha_{4g}\big{)}^{2}}} \tag{11}\]  
Based on the definition of failure probability, the first proposed FM reliability index can be derived as follows:  
\[\beta_{\text{FM}}=\frac{3\big{(}\alpha_{4g}-1\big{)}\beta+\alpha_{3g}\big{(} \beta^{2}-1\big{)}}{\sqrt{5{a_{3g}}^{2}\big{(}1-\alpha_{4g}\big{)}+9\big{(}1- \alpha_{4g}\big{)}^{2}}} \tag{12}\]  
On this basis, by considering the neglected higher-order infinitesimals, the expression for \(c\) is further derived [18] as  
\[c_{z}=\frac{-5\alpha_{3g}}{3\big{(}3\alpha_{4g}+1\big{)}} \tag{13}\]which leads to  
\[v_{z}=\frac{3\big{(}3\alpha_{4g}+1\big{)}g_{s}-5\alpha_{3g}\big{(}{g_{s}}^{2}-1 \big{)}}{\sqrt{9\big{(}1+3\alpha_{4g}\big{)}^{2}-5\alpha_{3g}\big{(}13\alpha_{4g }+11\big{)}}} \tag{14}\]  
Then, the improved fourth-moment reliability index can be derived as follows:  
\[\beta_{\text{F}\text{M}z}=\frac{3\big{(}3\alpha_{4g}+1\big{)}\beta+5\alpha_{3g }\big{(}\beta^{2}-1\big{)}}{\sqrt{9\big{(}1+3\alpha_{4g}\big{)}^{2}-5\alpha_{3g }\big{(}13\alpha_{4g}+11\big{)}}} \tag{15}\]  
This improved fourth-moment method has higher accuracy [18], and hence, it is adopted for the reliability analysis in this paper.