# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 2 Materials and Methods
### Monte Carlo Method  
The Monte Carlo method was developed by [PERSON]'s team during World War II, while working on nuclear weapons at Los Alamos. [PERSON] utilized the simulation approach to describe the random nature of neutron diffusion. The Monte Carlo method has long been recognized as the technique with the highest accuracy of all methods requiring knowledge of the probability distribution of random parameters [40]. The Monte Carlo method utilizes statistical analysis of a random sample obtained from the simulation to solve the problem. The method is commonly used for numerical integration of functions of multivariate random variables; estimation of failure probabilities also belongs to this class of problems. An approximation of the probability of failure is obtained from an estimate of the expected value of the characteristic function of the failure area, obtained using an estimator appropriate for the simulation method adopted. The classical Monte Carlo simulation involves generating a realization of \(\mathbf{x}\) of a random vector \(\mathbf{X}\), according to the joint probability distribution density \(\mathbf{f}_{\mathbf{X}}(\mathbf{x})\), and then checking whether the realization lies in the safe area or in the failure area. We define the failure probability estimator as the number of \"hits\" in the failure region relative to the total number of simulations (Figure 3). Figure 3: Concept of the Monte Carlo method. The above concept can be written down by defining the characteristic function of the set (failure area) as follows:  
\[\mathrm{I}_{\Omega_{t}}(\mathbf{X})= \begin{cases}1\text{ if }\mathbf{X}\in\Omega_{ \mathrm{f}}\\ 0\text{ if }\mathbf{X}\
otin\Omega_{\mathrm{f}}\end{cases}  \tag{16}\]  
It is easy to see that each simulation involving the determination of \(\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})\) values corresponds to a single realization of the Bernoulli test. The number of unreliable states in K attempts can be considered as a random variable with a binomial distribution:  
\[\mathrm{P}\big{[}\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})=1\big{]}= \mathrm{P}_{\mathrm{f}}\ \ \mathrm{P}\big{[}\mathrm{I}_{\Omega_{t}}(\mathbf{X})=0\big{]}=1\ -\mathrm{P}_{\mathrm{f}} \tag{17}\]  
where \(\mathrm{P}_{\mathrm{f}}=\mathrm{P}[\mathbf{X}\in\Omega_{\mathrm{f}}]\). The expected value and variance of \(\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})\) take the form the following, respectively:  
\[\mathrm{E}\big{[}\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})\big{]}=\mathrm{ P}_{\mathrm{f}}+0\cdot(1\ \ \mathrm{P}_{\mathrm{f}})=\mathrm{P}_{\mathrm{f}} \tag{18}\]  
\[\mathrm{Var}\big{[}\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})\big{]}= \mathrm{E}\Big{[}\big{(}\mathrm{I}_{\Omega_{t}}(\mathbf{X})\big{)}^{2}\Big{]} -\big{(}\mathrm{E}\big{[}\mathrm{I}_{\Omega_{t}}(\mathbf{X})\big{]}\big{)}^{2} =\mathrm{P}_{\mathrm{f}}-\mathrm{P}_{\mathrm{f}}^{2}=\mathrm{P}_{\mathrm{f}}(1 \ -\mathrm{P}_{\mathrm{f}}) \tag{19}\]  
The Monte Carlo method uses an estimator of the mean value of the failure area characteristic function of the form to calculate the probability of failure:  
\[\frac{1}{\mathrm{K}}\sum\limits_{\mathbf{k}=1}^{\mathrm{K}}\mathrm{I}_{\Omega _{\mathrm{f}}}(\mathbf{X_{k}})=\widetilde{\mathrm{P}}_{\mathrm{f}} \tag{20}\]  
where \(\mathbf{X_{k}}\) are independent random vectors with a probability distribution defined by the density function \(\mathrm{f}_{\mathrm{X}}(\mathbf{x})\), and \(\mathrm{K}\) is the number of simulations.