## 3 Reliability of Structures and Probability of Failure  
Structural reliability is the ability of a structure to meet special requirements, including the design life for which the structure is designed. Reliability includes the safety, usability, and durability of a structure and it is expressed through probabilistic terms [10; 11; 12; 13; 14; 15]. The structure must be designed and constructed in such a way as to serve the intended purpose and withstand, during the expected service life, with an appropriate degree of reliability and economy, all the loads and influences which may occur during operation and use of the structure. The Eurocode also allows determination of individual coefficients, for example, the coefficients \(\psi\) based on the theory of statistical evaluation of experimental data according to the theory of reliability. For the ultimate limit states, the individual partial factors of the different materials and loads are calibrated so that the confidence levels for typical structures are as close as possible to the guide value of the reliability index \(\beta\). In the analysis, the accepted or assumed statistical variability of the effects of loads, resistances, and model uncertainties are considered. Reliability classes (RC) can be defined based on the reliability index \(\beta\). Classes RC1, RC2, and RC3 are related to consequence classes CC1--CC3. The reliability index on the numerical axis are shown in Figure 1.  
It is generally assumed that according to EN 1990-1999, the structure has a reliability index higher than 1.5 for a 50-year reference period. The probability of failure, \(P_{f}\), can be expressed by the limit state function \(g\), in which it is assumed that for \(g>0\) the structure will not fail and for \(g\leq 0\) the structure will fail:  
\[P_{f}=\mathrm{Prob}(g\ \leq\ 0). \tag{8}\]  
If \(R\) is the resistance and \(E\) is the load effect, then the limit state function or structural margin \(g\) is:  
\[g\ =R-E. \tag{9}\]  
where \(R\) is the strength of the structure as a function of random variables of input parameters and \(E\) is the load response as a function of the random variables of the input parameters. If the function \(g\) has a normal distribution, then, the \(\beta\) confidence index is equal to:  
\[\beta=\frac{\mu_{g}}{\sigma_{g}}=\frac{\mu_{R}-\mu_{E}}{\sqrt{\sigma_{R}^{2}+ \sigma_{E}^{2}}} \tag{10}\]  
where  
\(\mu_{g}\) is a mean value of \(g\)  
\(\sigma_{g}\) is the standard deviation (Figure 2). Then,  
\[\mu_{g}-\beta\sigma_{g}=0, \tag{11}\]  
\[P_{f}=\mathrm{Prob}(g\ \leq\ 0)\ =\ \mathrm{Prob}\Big{(}\mu_{g}-\beta\sigma_{g} \Big{)}. \tag{12}\]  
For other types of distribution \(g\), \(\beta\) is only a formal measure of confidence \(P_{s}=(1\)--\(P_{f})\). Figure 1: Reliability index \(\beta\) on the numerical axis for load-bearing elements of reliability class RC2. Probabilistic reliability assessment methods can be both analytical and simulation methods. Simulation methods include Monte Carlo (MC), Latin hypercube sampling (LHS), and stratified sampling (SC). Advanced simulation methods also exist such as adaptive sampling (AS), directional sampling (DS), importance sampling (IS), line Sampling (LS), and response surface method (RSM). The Monte Carlo method can be classified as a simulation method with direct simulation. The RSM method is based on the assumption that it is possible to define the dependence between input and output parameters using the following approximation function:  
\[Y=c_{0}+\sum_{i=1}^{N}c_{i}X_{i}+\sum_{i=1}^{N}c_{ii}X_{i}X_{j}+\sum_{i=1}^{N} \sum_{i=1}^{N}c_{ij}X_{i}X_{j} \tag{13}\]  
where \(c_{0}\) is the coefficient of the constant member of the polynomial, \(c_{i}\) is the coefficient of the linear member, and \(c_{ij}\) is the coefficient of the quadratic member. The polynomial coefficients are based on the minimum error condition, usually the Central Composite Design (CCD) sampling method or the Box Behnken Matrix (BBM) sampling method.