# The Impact of Node Location Imperfections on the Reliability of Single-Layer Steel Domes
## 1 Introduction
With a predicted failure, with the probability of building structures of the range from \(10^{-4}\) to \(10^{-7}\), the number of necessary simulations is estimated to be from \(10^{6}\) to \(10^{9}\), which is a huge task even for modern computers, and is hardly feasible in acceptable time. The computational outlay disqualifies, in the opinion of the author, the usefulness of the classical Monte Carlo method for the analysis of practical problems of the reliability of building structures. The 1980s marked the development of the so-called importance sampling method. Through a proper selection of probability density function, according to which random variables are generated, one can significantly decrease the sampling area and the number of simulations. Articles by [PERSON] and [PERSON] [29], [PERSON] and [PERSON] [30], and [PERSON] [31] stand out in works devoted to this method. Another variety of simulation methods is the directional sampling method. It is best applicable for the analysis of problems in which the limit area has a shape close to a hypersphere. An overview of various simulation methods can be found in the works of [PERSON] [32] and [PERSON] [2]. In the present paper, simulation methods are not used. The basic research tool is one of the approximation methods--FORM. The purpose of this article is an estimation of the influence of node location imperfections on the reliability of single-layer steel domes. They are subjected to big displacement gradients and are susceptible to stability loss from the condition of node snapping. When designing such structures, it is also possible to buckle individual bars locally. However, this is not a form of loss of stability that determines the load-bearing capacity of the structure. These truss structures are extremely sensitive, even to the slight changing of node locations [33; 34; 35]. Therefore, the imperfection description is very important for the correct design of such structures. In the analysis by means of the finite element method we used a space truss element, which is a simple two-node element with three degrees of freedom in each node. The shape functions of the displacement field are linear. The displacement field of an element contains three translational components: u and v and w. The strain tensor is reduced to one non-zero component of [PERSON]'s strain tensor \(\varepsilon_{11}\), which characterizes the elongation of the bar. Other components of the strain tensor are equal to 0. The stress tensor is represented by the component \(\sigma_{11}\) of [PERSON]'s second stress tensor. The stress-strain relationship is linear. In the paper, nonlinear geometrical relations are defined in the Lagrangian description. The stability analysis of structures by means of the finite element method involves the solution of large systems of nonlinear equations. The load-displacement history of a structure presents a curve in a (N + 1) dimensional space, called the equilibrium path. In this paper, the constant arc length method (Riks [36; 37]) is used for the determination of the equilibrium path. The problem of elastic stability is intimately connected with singularities that occur along the path. These singular points are referred to as critical points. The classification of critical points includes limit points and bifurcation points that are connected with the physical concepts of snapping and buckling, respectively [38; 39; 40; 41]. Let us now ask a question: What is the advantage of the inclusion of reliability analysis methods to the analysis of stability? Using methods of reliability analysis, we can determine the level of failure probability when we approach the critical point. This is a quantitative assessment. Reliability analysis is also employed to find the sensitivity of the reliability index to mean values or standard deviation of random variables by calculating the elasticity index.