# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 2 Materials and Methods
### First-Order Reliability Method--FORM, Hasofer-Lind Reliability Index
\begin{array}{c}\mathrm{Z}_{\mathrm{i}}^{*}\\ \mathrm{Z}_{\mathrm{n}}^{*}\\ \cdot\\ \cdot\\ \cdot\\ \mathrm{Z}_{\mathrm{n}}^{*}\end{array} \right\}\] (10)
6. Calculate the column vector of sensitivity coefficients: \[\{\alpha\}=\frac{\{\mathbf{K}\}}{\sqrt{\{\mathbf{K}\}^{\mathrm{T}}\{\mathbf{K }\}}}\] (11)
7. Determine a new design point in the space of reduced random variables for \(\mathrm{n}-1\) variables: \[\mathrm{Z}_{\mathrm{i}}^{*}=\beta\alpha_{\mathrm{i}}\] (12)
8. Determine the corresponding design point in origin coordinates for \(\mathrm{n}-1\) variables from step 7: \[\mathrm{X}_{\mathrm{i}}^{*}=\mu_{\mathrm{X}_{\mathrm{i}}}+\mathrm{Z}_{\mathrm{ i}}^{*}\sigma_{\mathrm{X}_{\mathrm{i}}}\] (13)
9. Calculate the values of the remaining random variable (i.e., undefined in steps 7 and 8) by solving the equation \(\mathrm{g}(\mathbf{X})=0\). Steps 3-9 are repeated until \(\beta\) and the coordinates of the design point \(\{\mathrm{X}_{\mathrm{i}}^{*}\}\) converge. The Hasofer-Lind reliability index is originally defined for the case when the distributions of random variables are unknown (we assume only normal distributions). In the case when the distributions are known, then \(\beta\) can be calculated using the Rackwitz-Fiessler method [29]. For this case, it is required to know the cumulative distribution function \(\mathrm{F}_{\mathrm{X}}(\mathbf{X})\), and the probability density function \(\mathrm{f}_{\mathrm{X}}(\mathbf{X})\) of the random variables present in the limit state function. Nonnormal distributions are replaced by normal distributions so that the cumulative distribution functions and probability density functions have the same value for the true and surrogate distributions at the design point. For each random variable \(\mathrm{X}\), two equations with two unknowns (\(\mu_{\mathrm{X}}^{\mathrm{e}}\) and \(\sigma_{\mathrm{X}}^{\mathrm{e}}\)) can be written:  
\[\mathrm{F}_{\mathrm{X}}(\mathrm{X}^{*})=\Phi\bigg{[}\frac{\mathrm{X}^{*}- \mu_{\mathrm{X}}^{\mathrm{e}}}{\sigma_{\mathrm{X}}^{\mathrm{e}}}\bigg{]} \mathrm{f}_{\mathrm{X}}(\mathrm{X}^{*})=\frac{1}{\sigma_{\mathrm{X}}^{\mathrm{ e}}}\varphi\bigg{[}\frac{\mathrm{X}^{*}-\mu_{\mathrm{X}}^{\mathrm{e}}}{\sigma_{ \mathrm{X}}^{\mathrm{e}}}\bigg{]} \tag{14}\]  
After solving the system of equations, the following standard deviation of the surrogate normal distribution:  
\[\sigma_{\mathrm{X}}^{\mathrm{e}}=\frac{1}{\mathrm{f}_{\mathrm{X}}(\mathrm{X}^ {*})}\varphi\Big{(}\Phi^{-1}(\mathrm{F}_{\mathrm{X}}(\mathrm{X}^{*}))\Big{)}\] (15a) and the following mean value of the surrogate normal distribution:\[\mu_{\chi}^{\mathrm{e}}=X^{*}-\sigma_{\chi}^{\mathrm{e}}\Big{(}\Phi^{-1}(\mathrm{F }_{\mathrm{X}}(X^{*}))\Big{)} \tag{15b}\]  
are obtained. In the next steps, we proceed according to the algorithm described above. Determining the correct design point is one of the most important issues for determining the correct value of the reliability index. There are many developed algorithms for calculating the design point. The articles by [PERSON] and [PERSON] [30] and [PERSON] et al. [31] are based on gradient methods, which, due to slow convergence, and in cases involving highly non-linear boundary surfaces, even no convergence, were identified as a limitation of the procedure. The incorporation of the step-length reduction procedure proposed by [PERSON] [34] into the Rackwitz-Fiessler algorithm partially solves this problem. An alternative to this approach is the sequential quadratic programming (SQP) algorithm developed by [PERSON] [35; 36; 37] and [PERSON] [38]. They use additional information about the second derivatives to determine the search direction. In practice, this information is obtained by approximating the Hessian matrix of the Lagrange function only with gradients. The most commonly used Hessian approximation method is the BFGS (Broyden-Fletcher-Goldfarb-Shannon [31]) method. Paper [39] presents the possibility of using the FORM to assess the reliability of the optimized structure.