## 2 Measure of Structure Reliability  
The basic structural reliability problem considers only one load effect, \(S\), resisted by one resistance, \(R\). It is important that \(R\) and \(S\) are expressed in the same units. Let the vector \(\mathbf{X}\) represent all the basic variables involved in the problem. The limit state function \(G(R,S)\) can be generalized when the resistance \(R\) is expressed as \(R=G_{R}(\mathbf{X})\) and load effect as \(S=G_{S}(\mathbf{X})\). In that case, the resulting limit state function can be written simply as \(G(\mathbf{X})\), where X is the vector of all significant basic variables and \(G\)is some function expressing the relationship between the limit state and the basic variables. The limit state equation \(G(\mathbf{X})=0\) now defines the boundary between the safe domain \(G>0\) and the failure domain \(G<0\) in \(n\)-dimensional basic variable space (Figure 2). The limit state equation is the \"safety margin\" \(Z=R-S\) and the probability of failure, \(p_{f}\), of failure of the structural element can be stated as follows [3]:  
\[p_{f}=P[G(X)\leq 0]=\int\limits_{G(X)\leq 0}f_{RS}(R,S)drds \tag{1}\]  
or, in general:  
\[p_{f}=\Phi(-\beta)\text{ where }\beta=\frac{\mu_{Z}}{\sigma_{Z}}, \tag{2}\]  
where \(\beta\) is the reliability index, \(\Phi\) is the standard normal distribution function, and the \(f_{RS}\) is the probability density function of a two-dimensional random variable describing the physical state of the structure. However, \(p_{f}\) defined in this equation is only the exact probability of failure when \(R\) and \(S\) have normal and log-normal distributions. \(\beta\) is a measure (in standard deviation, \(\sigma_{Z}\)) of the distance that the mean, \(\mu_{Z}\), is from the origin \(Z=0\). This point marks the boundary of the failure region. Hence, \(\beta\) is a direct measure of the safety of the structural element. This idea was first presented in the work of Reference [28]. More generally, there will be many basic random variables, \(\mathbf{X}=\{X_{i}\), \(i\)=1, 2,..., \(n\}\), describing the structural reliability problem:  
\[G(\mathbf{X})=Z(\mathbf{X})=a_{0}+a_{1}X_{1}+a_{2}X_{2}+\ldots+a_{n}X_{n} \tag{3}\]  
In this case, the function \(G(\mathbf{X})=Z(\mathbf{X})\) is normally distributed, and the mean (\(\mu_{Z}\)) and the standard deviation (\(\sigma_{Z}\)) can be evaluated as follows:  
\[\mu_{Z}=a_{0}+\sum_{i=1}^{n}a_{i}\mu_{Xi} \tag{4}\]  
\[\sigma_{Z}=\sqrt{\sum_{i=1}^{n}a_{i}^{2}\sigma_{Xi}^{2}} \tag{5}\]  
Figure 1: Algorithms of applied approaches to reliability analysis. \(FI\), abbreviation of FORM and implicit; \(FNE\), abbreviation of FORM, neuronal, and explicit; NN, neural network; FORM, First Order Reliability Method. In general, the limit state function \(G(\textbf{X})=0\) is not linear. In this case, the most suitable approach is to linearize \(G(\textbf{X})=0\) by expanding \(G(\textbf{X})=0\) as a first-order Taylor series expansion about some point \(x^{\star}\). Approximations which linearize \(G(\textbf{X})=0\) will be denoted First Order Reliability Method (FORM). The first-order Taylor series expansion which linearizes \(G(\textbf{X})=0\) at \(x^{\star}\) might be denoted \(G_{L}(\textbf{X})=0\). The useful step is to transform all variables to their standardized form N(0,1), using the following well-known transformation:  
\[Z_{i}=\frac{X_{i}-\mu_{Xi}}{\sigma_{Xi}} \tag{6}\]  
where \(Z_{i}\) has \(\mu_{Z_{i}}=0\) and \(\sigma_{Z_{i}}=1\). In this space, the joint probability density function, \(f_{Z}(\textbf{Z})\), is the standardized multivariate normal [29]. The limit state function also must be transformed and is given by \(G(\textbf{Z})=0\). In this case, the linearizing of the limit function by expanding it into a Taylor power series in the limit point, called design point (\(z^{\star}\)), was performed. This point is on the limit state surface perpendicular to \(\beta\) in \(n\)-dimensional space.