## 2 Probabilistic Non-Linear Computation
### Reliability Levels  
In order to assess the reliability of a structure or a structural element in general, the limit state function \(g(x)\) is formulated as indicated in Equation (1),  
\[g(x)=0 \tag{1}\]  
where the vector \(x\) consists of m basic variables \(X_{i}\) (\(i\) = 1,..., m). For the present structural reliability problem, \(g(x)\) is formulated as the difference between the resistance \(R\) and the load \(E\):  
\[g(x)=K_{R}R-K_{E}E \tag{2}\]  
In general, due to simplifications in the modeling, model uncertainties \(K_{R}\) for the resistance and \(K_{E}\) for the load are introduced [23], as they are meant to reduce the deviation of the numerical model from the realistic model. The limit state function is formulated in a way that negative values indicate failure and the failure probability is defined as the probability that the random combination of the input values results in an outcome in the failure domain. Mathematically the latter can be expressed through Equation (3)  
\[P_{f}=\int_{g(x)<0}f_{x}(x)dx \tag{3}\]  
where \(f_{x}(. )\) is the \(m\)-dimensional joint probability density function (PDF) of the \(m\) basic variables \(X_{i}\). The structural reliability is quantified through the reliability index \(\beta\), which can be generally expressed and calculated--assuming a normal distribution for \(g(x)\)--through Equation (4). \[\beta=\Phi^{-1}\Big{[}1-P_{f}\Big{]} \tag{4}\]  
where \(\Phi^{-1}[. ]\) is the inverse of the standard normal cumulative distribution function. Several methods have been developed historically and are now available for determining these failure probabilities or the corresponding reliability, each of which has its own level of sophistication. Often in literature [24; 25], as well as in modern design standards [26] the following categorization is used, starting with the level of highest complexity:  
* Level III: limit state functions and distribution functions for the random variables are introduced without any approximation; calculations are usually based on Monte Carlo simulation or straightforward numerical integration;
* Level II: the amount of calculation efforts is reduced by adopting well-chosen linearization techniques, usually the so called First Order Reliability Method; the degree of accuracy may strongly depend on the details of the problem at hand;
* Level I: the variables Xi are introduced by one single value only; this value is referred to as the design value. This method does not actually calculate a failure probability but only checks whether some defined target level is attained or not. It is the basis for partial safety factor format (PSFM) which is defined in Eurocodes as the basic design format for new structures and it is the design and assessment procedure in everyday practice and is referred to as the semi-probabilistic level. A particular challenge of many civil engineering problems lies in the fact that decisions are made based on values which are remote from the most expected (median or mean) values for a parameter. In these cases, the probability densities are very small, and the obtained results are very sensitive to the tails of the distributions. Besides, the probability of failure, although a strictly mathematical term, remains a subjective perception because it quantifies the expectation regarding structural failure. This subjective probability is not an inherent property; rather, it depends very much on the amount of information available to the calculation procedure. For the present study two further limitations apply:It is assumed that the variables in a limit state function are in a first approach independent from each other. Although correlations between variables can be taken into account in computational programs, they are difficult to determine and convolute the algorithms. 2. The analysis does not take into account human error. The failure probabilities \(p_{f}\) discussed herein are conditional on the assumption that there are no errors affecting the resistance and loading condition of the case study.