# The Impact of Node Location Imperfections on the Reliability of Single-Layer Steel Domes
## 2 Reliability and Elasticity Indexes  
To conduct the reliability analysis is to find out how the structure state is affected by the random character of structure-describing parameters \(X_{1}\), \(X_{2}\),..., \(X_{n}\). The parameters can include material constants, loads, and geometric ratios. Those parameters can be grouped into the column vector (the random vector) \(\mathbf{X}=\{X_{1},\,X_{2},\,\ldots,\,X_{n}\}\), where: \(X_{1}\), \(X_{2}\),..., \(X_{n}\) are successive single-dimensional random variables (the basic random variables). In this defined n-dimensional real space \(R^{n}\), the random vector takes on an indefinite number of single sets of values of individual basic random variables that are equal to: \(x_{1}\), \(x_{2}\),..., \(x_{n}\), respectively. A single set, \(\mathbf{x}=\{x_{1},\,x_{2},\,\ldots,\,x_{n}\}\), is called the realization of the random vector \(\mathbf{X}\). The realization \(\mathbf{x}\) of the vector \(\mathbf{X}\) belongs to Euclidean space. The assessment of the structure state is possible, due to the failure criterion imposed by the designer (random limit function g(\(\mathbf{X}\))). It is usually a stress or displacement criterion. The values of the limit function divide the space as follows:  
\[ \begin{array}{l}g(\mathbf{x})\leq 0-\text{failure area }\Omega_{\text{f}}\\ g(\mathbf{x})>0-\text{safe area }\Omega_{\text{S}}\\ g(\mathbf{x})=0-\text{limit area}\end{array}  \tag{1}\]  
Using the definition of boundary conditions, failure probability is the probability of a random event specified by the failure area:  
\[P_{\text{f}}=P(g(\mathbf{x})\leq 0) \tag{2}\]Alternatively, failure probability can be expressed by an integral:  
\[P_{\mathrm{f}}=\int\Omega_{\mathrm{f}}f(\mathbf{x})\mathrm{d}\mathbf{x} \tag{3}\]  
where \(f(\mathbf{x})=f(\mathrm{x}_{1},\mathrm{x}_{2},\,\ldots,\,\mathrm{x}_{n})\) denotes the joint probability density function of random parameters. The reliability of the structure is the probability of the occurrence of the realizations of basic variables \(X\) in the area, which ensures safe performance of the structure. The nature of the concept above is illustrated in Figure 1. In the two-dimensional space of random variables, \(\mathbf{X}=\{X_{1},X_{2}\}\), the sets and contours of the joint probability density function \(f_{X}(x_{1},\,x_{2})\) were marked. The values of failure probability of a building are equal from 1.0-10 \({}^{-7}\) to 6.9-10 \({}^{-4}\)[46]. Relying on such values could be problematic. Consequently, a reliability measure (the reliability index \(\beta=-\,\phi\,^{-1}(P_{\mathrm{f}})\)) was introduced, where \(\phi\,^{-1}\) is the inverse function of the standard normal distribution function. From the technical standpoint, the most relevant problem concerns the means of the integral value determination. A large number of variables and the implicit form of the function \(g(\mathbf{X})\) make it impossible to numerically integrate the expression. The reliability analysis of real  
The reliability analysis problem was formulated using the limit function \(G(\mathbf{Z})\). The transformation of the basic random variables to the standard Gaussian space must ensure that the reliability problem is formulated in an equivalent way. In the FORM, after random variables had been transformed to the standard space \(\mathbf{Z}\), the limit surface \(G(\mathbf{z})=0\) was approximated with a hyperplane that was tangent to the surface at the design point (Figure 2).