# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 2 Materials and Methods
### Sensitivity Measures of Probability of Failure
The probability of failure formula is presented below:  
\[\mathrm{P}_{\mathrm{f}}=\int_{\Omega_{\mathrm{f}}}\mathrm{f}(\mathbf{x}) \mathrm{d}\mathbf{x}=\int_{\mathbb{R}^{n}}\mathrm{I}_{\Omega_{\mathrm{f}}}( \mathbf{x})\mathrm{d}\mathbf{x}=\mathrm{E}\big{[}\mathrm{I}_{\Omega_{ \mathrm{f}}}(\mathbf{X})\big{]} \tag{34}\]  
In order to simplify the notation of the following mathematical formulas, \(\mathrm{Y}=\mathrm{I}_{\Omega_{\mathrm{f}}}(\mathbf{X})\):  
\[\mathrm{Y}(\mathbf{X})=\mathrm{f}_{0}+\sum_{\mathrm{i}}^{\mathrm{M}}\mathrm{f }_{\mathrm{i}}(\mathrm{X}_{\mathrm{i}})+\sum_{\mathrm{i}<j}^{\mathrm{M}} \mathrm{f}_{\mathrm{ij}}\big{(}\mathrm{X}_{\mathrm{i}},\mathrm{X}_{\mathrm{j} }\big{)}+\cdots+\mathrm{f}_{1\ldots\mathrm{M}}(\mathrm{X}_{1},\ldots,\mathrm{ X}_{\mathrm{M}}) \tag{35}\]  
where \(\mathrm{f}_{0}=\mathrm{E}(\mathrm{Y})\) and \(\mathrm{E}\big{[}\mathrm{f}_{\mathrm{i}_{1}}\ldots\mathrm{x}_{\mathrm{i}_{8}} \big{)}=0\) for each subset of \(\big{[}\big{(}\mathrm{X}_{\mathrm{i}_{1}},\ldots,\mathrm{X}_{\mathrm{i}_{8}} \big{)}\big{]}\). The variance of the function \(\mathrm{Y}\) is presented as the sum of the variances \(\mathrm{V}_{\mathrm{i}}\) and \(\mathrm{V}_{\mathrm{ij}}\):  
\[\mathrm{Var}(\mathrm{Y})=\sum_{\mathrm{i}=1}^{\mathrm{M}}\mathrm{V}_{\mathrm{ i}}+\sum_{\mathrm{i}=1}^{\mathrm{M}}\mathrm{V}_{\mathrm{ij}}+\ldots+\mathrm{V}_{1 \ldots\mathrm{M}} \tag{36}\]  
where \(\mathrm{V}_{\mathrm{i}}=\mathrm{Var}(\mathrm{E}[\mathrm{Y}|\mathrm{X}_{ \mathrm{i}}])\), and \(\mathrm{V}_{\mathrm{ij}}=\mathrm{Var}\big{(}\mathrm{E}\big{[}\mathrm{Y}| \mathrm{X}_{\mathrm{i}},\mathrm{X}_{\mathrm{j}}\big{]}\big{)}-\mathrm{V}_{ \mathrm{i}}-\mathrm{V}_{\mathrm{j}}\). First-order Sobol indices \(\mathrm{S}_{\mathrm{i}}\) and second-order indices \(\mathrm{S}_{\mathrm{ij}}\) are defined as follows:  
\[\mathrm{S}_{\mathrm{i}}=\frac{\mathrm{V}_{\mathrm{i}}}{\mathrm{Var}(\mathrm{Y} )}\ \ \mathrm{S}_{\mathrm{ij}}=\frac{\mathrm{V}_{\mathrm{ij}}}{\mathrm{Var}(\mathrm{Y})} \tag{37}\]For the Sobol indices \(\mathrm{S_{i}}\) and \(\mathrm{S_{ij}}\), the following relationship occurs:  
\[\sum_{\mathrm{i}}^{\mathrm{M}}\mathrm{S_{i}}+\sum_{\mathrm{i<j}}^{\mathrm{M}} \mathrm{S_{ij}}+\ldots+\mathrm{S_{1\ldots M}}=1 \tag{38}\]  
The total Sobol index \(\mathrm{S_{T}}\) is expressed by the following formula:  
Figure 4:0  
\(\mathrm{Var}(\mathrm{Y})\) is known when calculating Sobol indices using the Monte Carlo method:  
Figure 4:1  
The first-order Sobol index assesses the effect of changing \(\mathrm{X_{i}}\) to the Y function. Interactions with other model input variables are not taken into account during the calculation. These interactions are included in the total Sobol index, which will not be analyzed in this example. Sensitivity measures based on Sobol indices [13; 14] can be seen as an extension of variance-based sensitivity analysis. The first-order indices indicate the proportion of variance of the individual components of the \(\mathbf{X}\) vector and can be regarded as a modified version of the moment-independent measure of failure probability sensitivity. The higher order and total effect indices represent the combined contribution of the components of vector \(\mathbf{X}\).