# The Impact of Node Location Imperfections on the Reliability of Single-Layer Steel Domes
## 1 Introduction  
The reliability theory is a well-established research area. The most well-known textbooks and monographs are: [PERSON], [PERSON], and [PERSON] [1], [PERSON] [2], [PERSON] and [PERSON] [3], [PERSON] and [PERSON] [4], and [PERSON], [PERSON], and [PERSON] [5]. Special attention should be paid to publications by [PERSON] [6] and [PERSON] and [PERSON] [7]. These publications present the basic concepts of reliability theory with a particular reference to their uses in civil engineering. The work of [PERSON] and [PERSON] [8], published in 1974, is recognized as the first important step towards the contemporary methods that make it possible to effectively and accurately estimate structural safety. [PERSON] and [PERSON] developed the concept of localization of the so-called 'design point', i.e., such realization of random variables from the failure surface, which corresponds to the greatest value of the probability density function. With the linearization of the limit state function at the design point, it is possible to obtain a measure of reliability that is invariant, due to the equivalent formulations of the boundary condition, i.e., the so-called Hasofer-Lind reliability index. The lack of invariance was thebasic shortcoming of the previously applied Cornell index [9]. It was difficult to use this measure of reliability for comparison of the degrees of safety of different structures. The concept of the Hasofer-Lind index was used in 1978 by [PERSON] and [PERSON] in [10]. Additionally, they applied transformations of independent random variables with arbitrary probability distributions to standard normal variables and they proposed an algorithm of the approximation of the design point. In [11], [PERSON] and [PERSON] proposed the utilization of [PERSON]'s transformation [12] to the transformation of dependent random variables to standard space. [PERSON] and [PERSON]'s transformations, applied for the first time by [PERSON] and [PERSON] in [13], are most frequently used at present. They make it possible to analyze reliability problems, both when the joint probability density function of random variables, as well as the marginal probability density function of variables, and correlation matrix are known. The approximation of the limit state function at the design point by the function of the first or second degree leads to the first-order reliability method (FORM) or the second-order reliability method (SORM). In the early applications of the FORM and SORM methods, it was most often assumed that the limit state function was an explicit function of random variables. Unfortunately, with the exception of a few simple examples, such functional dependency cannot be specified, and in most cases it is given through a definite numerical procedure, e.g., the finite element method [14; 15; 16]. However, with the first studies of [PERSON] and collaborators, the problems of the utilization of the finite element method in reliability analysis began to be very intensely developed. An overview and test of different discretization methods can be found in the works of [PERSON] and [PERSON] [17] and [PERSON] [18]. Popular reliability analysis codes, such as ANSYS PDS, DesignXplorer, CalREL/ERUM/OpenSees, COSSAN, NESSUS, PERMAS-RA/STRUREL, PHIMECA-SOFT, PROBAN, PROFES, and UNIPASS, are reviewed in [19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. The separate group of reliability analysis methods are simulation methods. The classical Monte Carlo method is best known. In the Monte Carlo simulation method, the estimation of failure probability consists in the n-th generation of basic random variables, in conformity with their joint probability density function and determination of the number of hits in the failure area. The lower the probability of failures that we want to estimate, the more simulations ought to be carried out, so that the error of this estimation could be situated in a fairly small trust range with a sufficiently high probability.