## 6 Examples  
To make a comparative analysis, steel beams (\(E\) = 210 GPa, \(G\) = 81 GPa) made from IPE300, HEA300, HEB300 sections with a span of \(L\) = 5 and 7 m, and beams fabricated from IPE500, HEA500, HEB500 sections with a span of \(L\) = 8 and 10 m were assumed. In computations, the loads were as those in the diagrams shown in Tables 2 and 3. For the diagrams in Table 2, transverse loads were applied to the top flange (\(z_{g}\) = +/2), to the section weight axis (\(z_{g}\) = 0) and to the bottom flange (\(z_{g}\) = -/2). When the loads were moments concentrated at supports (Table 3), the whole range of variation of the ratio of the moments (\(-1\leq\psi\leq 1\)) was taken into account for the following parameter values \(\psi\) = {\(-1\); \(-0.75\); \(-0.5\); \(-0.25\); \(0.25\); \(0.5\); \(0.75\); \(1\)}. Analyses were carried out for the full range of variation of the index of fixity against warping \(\kappa_{\omega}\) (from 0 to 1) and against lateral rotation \(\kappa_{u}\) (from 0 to 1) for the following values of \(\kappa_{i}\) = {0; 0.25; 0.5; 0.75; 0.9; \(1\)}. Computations were run for various combinations of the values of indexes \(\kappa_{\omega}\) and \(\kappa_{u}\). For each of the beams analysed, the critical moment for lateral torsional buckling was determined acc. \(M_{LTB,EL,2}\) programme using 3 terms of both series (8) and (9). The critical moment was estimated with Formulas (14), (15) and (16), and then compared with the values obtained from FEM (_LTBeam_). As the number of received values of the critical moment of beams was large, the paper reports only selected results of detailed cases (Tables 4 and 5, Figure 4; Figure 5). Cumulative analyses of results (for all cases included in the paper) are presented in Table 6.  
Figure 3: Exemplary beam modeled using the _LTBeam_ software: (a) the critical moment for lateral torsional buckling of beam, (b) deformed shape of beam. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multirow{2}{*}{**Load Diagram**} & \multirow{2}{*}{\(\kappa_{\omega}\)} & \multirow{2}{*}{\(\kappa_{u}\)} & \multicolumn{6}{c}{\(M_{cr}\) [kNm]} \\ \cline{5-10}  & & & & \(LTBeam\) & \(M_{LTB,EL,2}\) & \% 6-5 & Formula (14) & \% 8-5 \\ \hline \(1\) & \multirow{5}{*}{0} & \multirow{5}{*}{0} & \multirow{5}{*}{111.2} & \multirow{5}{*}{111.9} & \multirow{5}{*}{0.6} & \multirow{5}{*}{110.5} & \multirow{5}{*}{-0.6} \\ \cline{5-1 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Item** & **Load Diagram** & \begin{tabular}{c} \(M_{\textit{LTEB,EL,2}}\) **vs.** \\ _LTBeam_ \\ \end{tabular} & \
\begin{tabular}{c} **Formulas (14), (15)** \\ **vs. LTBeam** \\ \end{tabular} &
\begin{tabular}{c} **Formula (16)** \\ **vs. LTBeam** \\ \end{tabular} \\ \hline
1 &
\begin{tabular}{c} \(P_{z}\) \\ \(L/2\) \\ \(q_{z}\) \\ \end{tabular} & from +0.5 to +2.6\% & from \(-\)3.8 to +4.1\% & \(-\) \\
2 &
\begin{tabular}{c} \(q_{z}\) \\ \(L\) \\ \end{tabular} & from 0 to +0.7\% & from \(-\)3.0 to +2.6\% & \(-\) \\
3 &
\begin{tabular}{c} \(q_{z}\) \\ \(L\) \\ \end{tabular} & from \(-\)0.5 to +0.7\% & from \(-\)2.3 to +3.3\% & \(-\) \\
4 &
\begin{tabular}{c} \(M_{\textit{L}}\) \\ \(\psi\in(-1\times 1)\) \\ \end{tabular} & from 0 to +4.9\% & \(-\) & from \(-\)3.2 to +4.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Percentage differences in the values of \(M_{\textit{cr}}\) for all analysed beams. Figure 5: The critical moment of lateral torsional buckling of beam as a function of the index of fixity \(\kappa_{u}\), for selected index of fixity \(\kappa_{u}\): (**a**) beam with concentrated force load, (**b**) beam with uniformly distributed load. Table 4 lists exemplary results of calculations obtained for IPE300 beam, with a span of \(L=5\) m, loaded with a concentrated force applied to the upper flange (\(z_{g}=+h/2\)) at the midspan. Percentage differences in the results obtained with \(M_{LTB,EL,2}\) programme (Column 6) relative to the _LTBeam_ programme (Column 5) are shown in Column 7.