## 6 Examples
Analogous comparison of the results obtained with Formula (14) (Column 8) and the _LTBeam_ programme (Column 5) can be seen in Column 9. When compared with _LTBeam_, the critical moments determined using \(M_{LTB,EL,2}\) programme showed the differences of \(+0.6\) to \(+1.9\%\) (Table 4). The application of the approximation Formula (14) produced the values that differed from \(-3.4\) to \(-0.5\%\) in comparison with FEM. Table 5 lists exemplary results of computations for IPE300 beam, \(L=5\) m, and selected values of indexes \(\kappa_{\omega}\) and \(\kappa_{u}\), and loading diagrams acc. Tables 2 and 3. The percentage differences in the results obtained with the \(M_{LTB,EL,2}\) programme (Column 7) relative to the _LTBeam_ programme (Column 6) are shown in Column 8. Analogous comparison of the results obtained from the Formulas (14) and (16) (Column 9) and the _LTBeam_ programme (Column 6) is given in Column 10. In addition to the comparison of the values of \(M_{cr}\), the results compiled in Tables 4 and 5 can be employed in the tests on the correctness of the design of Formulas (14) and (16) in spreadsheets. Table 6 lists the maximum percentage differences between the results obtained by the authors and those produced using _LTBeam_ for beams IPE300, HEA300, HEB300, IPE500, HEA500 and HEB500. In addition to the data shown in Table 6, it should be noted that for Scheme 1, the values received with \(M_{LTB,EL,2}\) programme differed from \(+0.5\) to \(+2.6\%\) (HEB300, \(L=5\) m), and the results obtained acc. Formula (14) showed differences of from \(-3.8\) (IPE500, \(L=10\) m) to \(+4.1\%\) (HEA300, \(L=5\) m) compared with _LTBeam_. As regards Scheme 2, \(M_{LTB,EL,2}\) program gave critical moments that differed from \(0\) to \(+0.7\%\) (HEA500, \(L=10\) m), and the differences for Formula (14) were from \(-3.0\) to \(+2.6\%\) (HEA300, \(L=5\) m). For Scheme 3, \(M_{LTB,EL,2}\) programme values differed from \(-0.5\) to \(+0.7\%\) (HEB500, \(L=10\) m), and Formula (14) values showed differences of \(-2.3\) to \(+3.3\%\) (HEA300, \(L=5\) m). Finally, for Scheme 4, \(M_{LTB,EL,2}\) programme generated critical moments that were \(0\) to \(+4.9\%\) (HEA300, \(L=5\) m) different, and acc. Formula (16), from \(-3.2\) (HEA300, \(L=5\) m) to \(+4.0\%\) (IPE300, \(L=7\) m). Figure 4 presents the courses of variation of \(M_{cr}\) for a beam made of an IPE300 profile with a span \(L=5\) m, depending on the value of the index of fixity against warping \(\kappa_{\omega}\) (from \(0\) to \(1\)) for selected values of the index of fixity against lateral rotation \(\kappa_{u}=\{0;0.25;0.5;0.75;0.9;1\}\). Concentrated force load (Figure 4a) or uniformly distributed load (Figure 4b) was applied to the upper flange of the beam (\(z_{g}=+h/2\)). Critical moments of the beam were determined with Formula (14). The comparison of the critical moments of lateral torsional buckling (Figure 4), obtained for the full warping restraint (\(\kappa_{\omega}=1\)) in relation to its full freedom (\(\kappa_{\omega}=0\)), shows \(+71\%\) (Figure 4a) and \(+81\%\) (Figure 4b) increase in \(M_{cr}\), basically regardless of the value of the \(\kappa_{u}\) index. The dependence \(M_{cr}(\kappa_{\omega})\) is strongly non-linear throughout the whole range of the \(\kappa_{\omega}\) restraint index (from \(0\) to \(1\)). Figure 5 shows the courses of variation of critical moments of lateral torsional buckling of the beam, for geometric parameters according to Figure 4, depending on the value of the index of fixity against lateral rotation \(\kappa_{u}\) (from \(0\) to \(1\)) for selected values of the index of fixity against warping \(\kappa_{\omega}=\{0;0.25;0.5;0.75;0.9;1\}\). The comparison of the critical moments of lateral torsional buckling (Figure 5), obtained for the full lateral rotation restraint (\(\kappa_{u}=1\)) in relation to its full freedom (\(\kappa_{u}=0\)), shows \(+29\%\) (Figure 5a) and \(+43\%\) (Figure 5b) increase in \(M_{cr}\), basically regardless of the value of the \(\kappa_{\omega}\) index.