# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 2 Materials and Methods
### Monte Carlo Method
The mean value and variance of the estimator are given, respectively, as follows:  
\[\mathrm{E}\bigg{[}\widetilde{\mathrm{P}}_{\mathrm{f}}\bigg{]}=\mathrm{E}[ \frac{1}{\mathrm{K}}\sum\limits_{\mathbf{k}=1}^{\mathrm{K}}\mathrm{I}_{\Omega _{t}}(\mathbf{X_{k}})]=\frac{1}{\mathrm{K}}\mathrm{KE}[\mathrm{I}_{\Omega_{t}} (\mathbf{X})]=\frac{1}{\mathrm{K}}\mathrm{KP}_{\mathrm{f}}=\mathrm{P}_{ \mathrm{f}} \tag{21}\]  
\[\mathrm{Var}\bigg{[}\widetilde{\mathrm{P}}_{\mathrm{f}}\bigg{]}=\mathrm{Var} \bigg{[}\frac{1}{\mathrm{K}}\sum\limits_{\mathbf{k}=1}^{\mathrm{K}}\mathrm{I}_ {\Omega_{t}}(\mathbf{X_{k}})\bigg{]}=\frac{1}{\mathrm{K}^{2}}\mathrm{Var}[ \sum\limits_{\mathbf{k}=1}^{\mathrm{K}}\mathrm{I}_{\Omega_{t}}(\mathbf{X_{k}} )]=\frac{1}{\mathrm{K}^{2}}\mathrm{KP}_{\mathrm{f}}(1\ -\ \mathrm{P}_{\mathrm{f}})=\frac{1}{\mathrm{K}}\mathrm{P}_{\mathrm{f}}(1-\ \mathrm{P}_{ \mathrm{f}}) \tag{22}\]  
The coefficient of variation of the estimator take the form of the following:  
\[\mathrm{v}_{\widetilde{\mathrm{P}}_{\mathrm{f}}}=\frac{\mathrm{\sigma}_{ \widetilde{\mathrm{P}}_{\mathrm{f}}}}{\widetilde{\mathrm{P}}_{\mathrm{f}}^{0}}= \sqrt{\frac{1\ -\ \mathrm{P}_{\mathrm{f}}}{\mathrm{K}\cdot\mathrm{P}_{\mathrm{f}}}} \tag{23}\]  
The above formula shows that obtaining a coefficient of variation of the estimator of 0.1 with a predicted probability of failure, which for real designs ranges from \(10^{-7}\) to \(10^{-4}\), requires \(K=10^{6}\)-\(10^{9}\) simulations, which, even with the capabilities of today's multiprocessor computers, is an overwhelming task, if at all feasible in an acceptable time. An alternative to determining the point estimator, together with its error, is to determine the so-called confidence interval in which, with the assumed confidence level, the probability of failure \(\mathrm{P}_{\mathrm{f}}\) is contained. We can determine the confidence interval of the estimator (20) using the information of the following random variable:  
\[\mathrm{Y}_{\mathbf{k}}=\frac{\frac{1}{\mathrm{K}}\sum\limits_{\mathbf{k}=1}^ {\mathrm{K}}\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X_{k}})-\mathrm{E}\big{[} \mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})\big{]}}{\sqrt{\mathrm{v}_{ \widetilde{\mathrm{P}}_{\mathrm{f}}}}} \tag{24}\]  
angles \(\alpha_{i}\), and then, using gradientless methods, search the line in the generated direction to find a point for which the limit state function takes the value zero. The distance of this point from the origin is denoted as \(\mathrm{r_{i}}\). Based on the total number of N samples, the probability of failure \(\mathrm{P_{f}}\) can be approximated as follows:  
\[\mathrm{\widetilde{P_{f}}}=\frac{1}{n}\sum_{i=1}^{n}[1-\chi_{n}^{2}\left( \beta_{i}^{2}\right)] \tag{29}\]  
where \(\chi^{2}\) is the CDF of the chi-square distribution with \(n\) degrees of freedom. This method can handle problems that have multiple (local) design points or if the failure is defined as a combination of different failure modes. However, directional sampling is not suitable for problems with multiple random variables. Interesting comments on this topic are given in [26; 27; 28; 44; 45].