## 4 The Critical Moment for Lateral Torsional Buckling
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Load Diagram** & **Coefficients** \\ \hline \(1\) & \(2\) & \(3\) \\ \hline \multirow{8}{*}{\(1\)} & & \(C_{1}\ =\ 207.407\left(1.667-2.611\kappa_{u}+\kappa_{u}^{2}\right)\left(1.457-2.4\kappa_{u}+ \kappa_{u}^{2}\right)\) \\  & & \(C_{2}\ =\ 2488.889\left(1.667-2.611\kappa_{u}+\kappa_{u}^{2}\right)\left(1.2- \kappa_{u}\right)\) \\  & & \(C_{3}\ =\ \sqrt{\left(1-\psi\right)^{2}\left(4-\alpha_{2}\kappa_{u}+\alpha_{3} \kappa_{u}^{2}\right)}\) \\  & & \(D_{1}\ =\ \beta_{1}+\beta_{2}\kappa_{u}^{2}+\beta_{3}\kappa_{u}^{6}-\left(\beta_{4}+ \beta_{5}\kappa_{u}+\beta_{6}\kappa_{u}^{6}\right)\kappa_{u}^{3\psi}\) \\  & & \(\psi\in\left(-1\div-0.5\right)\) & \(a_{1}\ =\ 1.347\left(1.248-2.234\kappa_{u}+\kappa_{u}^{2}\right)\alpha_{2}\ =\ 2.321 \left(1.241-2.228\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(a_{3}\ =\ 1.235-2.22\kappa_{u}+\kappa_{u}^{2};\beta_{1}\ =\ 0.022\left(48.55-0.727\cdot \kappa_{u}+\kappa_{u}^{10}\right)\) \\  & & \(\beta_{2}\ =\ 0.206\left(-0.068-0.981\cdot\kappa_{u}^{2}+\kappa_{u}^{10}\right)\) \\  & & \(\beta_{3}\ =\ 0.615\kappa_{u}^{2};\ \beta_{4}\ =\ 0.132\left(12.909-5.114\kappa_{u}+ \kappa_{u}^{10}\right)\) \\  & & \(\beta_{5}\ =\ 0.167\left(0.593-1.06\kappa_{u}^{2}+\kappa_{u}^{10}\right)\) \\ \hline \multirow{8}{*}{\(M\)} & & \(C_{1}\ =\ 48. (1.2-\kappa_{u})\left(1.457-2.4\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(C_{2}\ =\ 576. (1.2-\kappa_{u})\cdot(1.2-\kappa_{u})\) \\  & & \(C_{3}\ =\ \sqrt{\left(1+\psi\right)^{2}\left(4-\alpha_{2}\kappa_{u}+ \alpha_{3}\kappa_{u}^{2}\right)}\) \\  & & \(D_{1}\ =\ \beta_{1}+\beta_{2}\kappa_{u}^{2}+\beta_{3}\kappa_{u}^{6}-\left(\beta_{4}+ \beta_{5}\kappa_{The formula for the critical moment for lateral torsional buckling, which addresses the indexes of fixity (\(\kappa_{\omega}\), \(\kappa_{u}\)) and an arbitrary ordinate (\(z_{g}\)) of the point of transverse load application with respect to centre of the section sheer (see Figure 2), has the following form:  
\[M_{cr}\ =\ D_{1}\frac{-B_{1}El_{2}z_{g}+\sqrt{EI_{2}\big{(}B_{2}Gl_{l}L^{2}+B_{3} EI_{\omega}+B_{1}^{2}El_{2}z_{8}^{2}\big{)}}}{B_{4}L^{2}} \tag{14}\]  
where: \(B_{1}\), \(B_{2}\), \(B_{3}\), \(B_{4}\) and \(D_{1}\)--coefficients acc. Table 2.  
For loads applied at the section shear centre (\(z_{g}=0\)), the formula for the critical moment is reduced to the following form:  
\[M_{cr}\ =\ D_{1}\frac{\sqrt{EI_{2}(B_{2}Gl_{l}L^{2}+B_{3}EI_{\omega})}}{B_{4}L^{2}} \tag{15}\]  
As regards the beam loaded with moments concentrated at the ends (for \(-1\leq\psi\leq 1\), Table 3), the formula for the critical moment for lateral torsional buckling, which accounts for arbitrary (\(0\div 1\)) values of the indexes of fixity (\(\kappa_{\omega}\), \(\kappa_{u}\)), has the following form:  
\[M_{cr}\ =\ D_{1}\frac{\sqrt{EI_{2}(C_{1}Gl_{l}L^{2}+C_{2}EI_{\omega})}}{C_{3}L^{2}} \tag{16}\]  
where: \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(D_{1}\)--coefficients acc. Table 3.  
Formula (16), derived for concentrated moments from the interval \(-0.5<\psi\leq 1\) (see Table 3), in which symmetric or slightly asymmetric (with respect to the beam midspan) lateral torsional buckling mode occurs, was approximated with the first term of the twist angle function (8) and the first term of the lateral deflection function (9). However, for the interval \(-1\leq\psi\leq-0.5\), where much more asymmetric mode of lateral torsional buckling is found, the best results were obtained for the first term of the twist angle function (8) and the second term of the lateral deflection function (9). The design of approximation formulas makes it possible to develop relatively simple spreadsheets.