# The Application of Structural Reliability and Sensitivity Analysis in Engineering Practice
## 2 Materials and Methods  
When we define the problem of time-independent reliability analysis, we assume that the design parameters, whose values are characterized by uncertainty, are represented by an n-dimensional random vector X. In this research, realizations of random vector X in space R\({}^{n}\) are described by vector x. Random parameters can be load multipliers, material constants (such as yield strength or Young's modulus), or cross-section dimensions. Random vector X = \(\{\text{X}_{1},\text{X}_{2},\text{X}_{3},\ldots,\text{X}_{n}\}\) takes values in the n-dimensional space of real numbers R\({}^{n}\). Reliability analysis assumes that a structure can be in one of two permissible states: either a safe state or a failure state. Failure can be understood as a failure to meet a constraint that has been imposed by the designer on the operation of the structure. In reliability analysis, this constraint is defined by a limit state function g(x). The limit state function is a function of vector X. If there is a joint probability density function f(x) for the random vector X = \(\{\text{X}_{1},\text{X}_{2},\text{X}_{3},\ldots,\text{X}_{n}\}\), then the failure probability P\({}_{\text{f}}\) can be expressed by the following integral:  
\[\text{P}_{\text{f}}=\text{P}(\text{g}(\text{x})\leq 0)\,\rightarrow\,\text{P}_{ \text{f}}=\int_{\Omega_{\text{f}}}\text{f}(\text{x})\text{d}\text{x} \tag{1}\]  
where \(\Omega_{\text{f}}\) represents the failure area. Calculating such a simple-looking integral in Formula (1) is very difficult in practice. The use of analytical methods is possible only in special cases. Standard methods of numerical integration also do not work. It is believed that quadrature can be used if the space of random variables has at most five dimensions. A popular method of integrating functions defined in a large number of dimensions is Monte Carlo simulation [23; 24; 25], together with variance reduction methods, for examples, Importance Sampling, Line Sampling, and Directional Sampling [26; 27; 28]. The First-Order Reliability Method (FORM) [29; 30] and Second-Order Reliability Method (SORM) [31; 32; 33] approximation methods are also used to estimate the integral. In assessing the reliability of building structures, the calculation of sensitivity measures is often used, providing information on the influence of input random variables on the probability of failure. Classical measures are vectors, whose coordinates are the first partial derivatives of reliability indices evaluated in the design point. These values are obtained as a by-product of the FORM. When a linear approximation of the limit function is insufficient, it is worth considering the SORM. This method is more time-consuming, which may cause problems when analyzing large engineering problems. It should always be remembered that the FORM gives the best results when there is only one design point, and the limit state function is not strongly non-linear and is differentiable. In case of problems with calculating the gradients of the limit state function, we cannot use the FORM. Then, we have to use one of the simulation methods in which we do not use gradient algorithms. This, of course, solves the problem of calculating the probability of failure, but we still cannot determine the sensitivity measures that are important to us. In this article, we will try to find an alternative solution using simple limit state functions. We will determine the sensitivity of the failure probability to input random variables using the analysis of variance of the characteristic function of the failure area. We will show the possibilities of calculating various measures of sensitivity on the example of a cantilever beam. We will formulate the ultimate and serviceability limit state functions. For each function, we will calculate the corresponding failure probability using two different methods, i.e., the FORM (SORM) and Monte Carlo. Additionally, in each method we will calculate qualitatively different sensitivity measures.