# The Impact of Node Location Imperfections on the Reliability of Single-Layer Steel Domes
## 3 Numerical Results and Discussion  
A mesh of the lattice dome, consisting of 25 nodes and 56 elements, is illustrated in Figure 4. The dome geometry is shown in Table 1. The elements of the structure were assumed to be made of steel tubes RO 101.6 \(\times\) 6 with yield point \(f_{y}\) = 235 MPa and [PERSON]'s modulus \(E\) = 210 GPa. The load of the \(\mu\)_P_ value was applied to the keystone (\(P\) = 1 kN). Figure 3: Illustration of the elasticity of the function \(\beta\)(x) of one variable x. In the first deterministic step, the Krata software [50; 51; 52] was used to determine the load-displacement path and the value of the limit load multiplier \(\mu_{\text{cr}}\) = 26.15 kN. Figure 5 shows the dependence between the value of the load multiplier \(\mu\) and the vertical displacement of node No. 1-\(q_{1}\). In this paper, the constant arc length method is used for determination of the equilibrium path. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**No. of** & \multirow{2}{*}{**\(\mathbf{x}\) [m]**} & \multirow{2}{*}{**\(\mathbf{y}\) [m]**} & \multirow{2}{*}{**\(\mathbf{z}\) [m]**} & \multicolumn{2}{c}{**No. of**} & \multirow{2}{*}{**\(\mathbf{x}\) [m  
The second step started with the creation of a stochastic model in the Numpress Explore program [53]. Based on the coordinates of the limit point, the form of the limit function was formulated as a condition of non-exceedance of the admissible multiplier of the vertical load on node No. 1:  
\[g_{\mu}(\mathbf{x})=1-\mu(\mathbf{x})/26.15 \tag{11}\]  
The alternative version of the limit state function could be the condition of non-exceedance of the vertical displacement \(q_{1}\) on node No.1:  
\[g_{q1}(\mathbf{x})=1-q_{1}(\mathbf{x})/103.96 \tag{12}\]  
The software allowed us to define two types of random variables, namely, basic and external. When the reliability analysis was conducted, the following basic random variables were taken into account: load \(P\), coordinates \(Z_{i}\) of nodes \(i=1,\,\ldots,\,17\). Basic random variables were not correlated. Their description is shown in Table 2. The external variables were implicit functions of basic random variables, the values of which were obtained with the Krata software. In this paper, the external variable is the load multiplier \(\mu\). Figure 6 shows the procedure of the analysis. The values of external variables were read from the text sets that contained the results of computations with the Krata software. The next step involved the introduction of the limit function formula, using the standard mathematical notation as a dependence on basic and external random variables. In the study, the condition of non-exceedance of the admissible load multiplier was satisfied. Numpress Explore software communicates with the Krata software. The latter was used to compute the values necessary to define the limit function for subsequent sets of random variables. Next, the reliability analysis method was chosen and computations began. The task ended with the generation of information that contained the values of the failure probability, reliability index, and elasticity index. Figure 5: The dependence between the value of the load multiplier \(\mu\) and the vertical displacement \(q_{1}\). In this paper, the impact of changes in the standard deviation of random variables describing coordinates of nodes \(Z_{i}\)\(i=1,\ldots,17\) on the reliability index \(\beta\) is analyzed. Two cases were considered, in which the variation coefficient of coordinates \(Z_{i}\) was modified. The variation coefficient of 10% was adopted for the first case, whereas for the other case it was 5%. Random variable notations correspond to node numbers in Table 1. Changing the standard deviation of random variables \(Z_{i}\)\(i=1,\ldots,17\) reflects the imperfections of the nodes position in the probabilistic description. In both cases, a constant variation coefficient of 10% was assumed for load \(P\). Figure 7 shows how the reliability index \(\beta\) depends on the load multiplier \(\mu\) for both cases.