# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 2 Materials and Methods
### First-Order Reliability Method--FORM, Hasofer-Lind Reliability Index  
Consider the limit state function g(X\({}_{1}\), \(\ldots\), X\({}_{n}\)) in which the random variables are uncorrelated and normally distributed (Figure 1). The limit state function can be represented in the standard form of random variables as follows:  
\[Z_{i}=\frac{X_{i}-\mu_{X_{i}}}{\sigma_{X_{i}}}\text{ and }X_{i}=\mu_{X_{i}}+Z_{i} \sigma_{X_{i}} \tag{2}\]  
Replacing X\({}_{1}\), \(\ldots\), X\({}_{n}\) with standard variables \(Z_{1}\), \(\ldots\), \(Z_{n}\), we obtain a new limit state function G(Z\({}_{1}\), \(\ldots\), \(Z_{n}\)). The Hasofer-Lind reliability index \(\beta\) is defined as the shortest distance from the origin in the space of standard random variables to the boundary delineated by the equation G(\(\mathbf{Z}\)) = 0. This definition means that in practice, \(\beta\) is the shortest distance from the point (0, \(\ldots\), 0) to the n-dimensional surface G(Z\({}_{1}\), \(\ldots\), Z\({}_{n}\)) = 0 (Figure 2). The point located on the G(Z\({}_{1}\), \(\ldots\), Z\({}_{n}\)) = 0--surface is called the design point, and its coordinates are de  
Figure 1: Illustration of limit state function g(x), safe area \(\Omega_{s}\), and failure area \(\Omega_{t}\) for two random variables. noted as (\(Z_{1}^{*},\ \ldots,Z_{n}^{*}\)). Therefore, the following formula can be used to determine the reliability index:  
\[\beta=\sqrt{\sum_{i=1}^{n}(Z_{i}^{*})^{2}} \tag{3}\]  
The value of the joint probability density function that corresponds to the failure at this point is the largest, due to the properties of the standard Gaussian space. This point reflects the most probable failure location among all points in this area. To perform the calculations, the iterative method can be used, which consists in solving the system of (\(2n+1\)) equations with (\(2n+1\)) unknowns: \(\beta\), \(\alpha_{1}\), \(\alpha_{2}\), \(\ldots\), \(\alpha_{n}\), \(Z_{1}^{*}\), \(Z_{2}^{*}\), \(\ldots\), \(Z_{n}^{*}\). The set of equations encompasses the relationships described by Formulas (4)-(7):  
\[G(Z_{1}^{*},\ \ldots,\ Z_{n}^{*})=0 \tag{4}\]  
\[Z_{i}^{*}=\beta\alpha_{i} \tag{5}\]  
\[\sum_{i=1}^{n}\left(\alpha_{i}\right)^{2}=1 \tag{6}\]  
\[\alpha_{i}=\frac{-\frac{\alpha_{G}}{\beta Z_{i}}\left| \begin{array}{c} \text{design}\\ \text{point}\end{array} \right. }{\sqrt{\sum_{i=1}^{n}\left[\begin{array}{c} \frac{\alpha_{G}}{\beta Z_{i}}\left| \begin{array}{c}\text{design}\\ \text{point}\end{array} \right.\right]^{2}}} \tag{7}\]  
The iterative procedure algorithm can be divided into the following steps:  
1. Define the limit state function and determine the corresponding parameters of the distributions of the random variables \(X\)i. 2. Find the initial design point \(\{X_{i}^{*}\}\) by assigning initial values to \(n-1\) variables \(X_{i}\) (mean value consideration is often used). Solve the equation \(g(X)=0\) with respect to the remaining random variable. This ensures that the initial point belongs to the boundary of the failure area. 3. Determine the values of the standard variables \(\{Z_{i}^{*}\}\) that correspond to the design point: \[Z_{i}^{*}=\frac{X_{i}^{*}-\mu_{X_{i}}}{\sigma_{X_{i}}}\] (8)  
Figure 2: Transformation of a limit state function to a standard Gaussian space. Determine the partial derivatives of the limit state function with respect to standard random variables. The column vector \(\{\mathbf{K}\}\) should be defined as follows:  
\[\{\mathbf{K}\}=\left\{ \begin{array}{c}\mathrm{K}_{1}\\ \mathrm{K}_{2}\\ \cdot\\ \cdot\\ \cdot\\ \mathrm{K}_{\mathrm{n}}\end{array} \right\}\text{where}\;\mathrm{K}_{\mathrm{i} }=-\frac{\partial\mathrm{G}}{\partial\mathrm{Z}_{\mathrm{i}}}\bigg{|}\text{ design} \tag{9}\]
5. Calculate the approximation \(\beta\) using the formula below: \[\beta=\frac{\{\mathbf{K}\}^{\mathrm{T}}\{\mathbf{Z}^{*}\}}{\sqrt{\{\mathbf{K }\}^{\mathrm{T}}\{\mathbf{K}\}}},\;\text{where}\;\{\mathbf{Z}^{*}\}=\left\{