## 4 The Critical Moment for Lateral Torsional Buckling
In this study, _McrLT_elastic_warping_rotation_sym.cal.nb_ programme was formulated, in analogous terms, in the environment of the _Mathematica(r)_ package. The programme is geared towards symbolic \"computations\", and it accounts for both elastic restraint against warping (\(\kappa_{\omega}\)) and against rotation in the plane of lateral torsional buckling (\(\kappa_{u}\)). In this case, for the loading diagrams shown in Tables 2 and 3, the first term of the twist angle function (8) and the first or the second term \(\{b_{i}((1-\kappa_{u})W_{Pi}+\kappa_{u}W_{(1L)})\}_{i\ =\ 1\ {\rm or}\ 2}\) of the lateral deflection function (9) were employed. Figure 2: Exemplary diagram of beam loading. \(q_{z}(x)\)—function of transverse load. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Load Diagram** & **Coefficients** \\ \hline \(1\) & \(2\) & \(3\) \\ \hline \multirow{8}{*}{\(1\)} & & \(B_{1}\ =\ 11.52\cdot(1.2-\kappa_{u})\left(1.563-2.5\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(B_{2}\ =\ 30.719\cdot B_{4}(1.2-\kappa_{u})\left(1.457-2.4\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(B_{3}\ =\ 36.638\cdot B_{4}(1.2-\kappa_{u})\cdot(1.2-\kappa_{u})\); \(B_{4}\ =\ \alpha_{1}-\alpha_{2}\kappa_{u}+\alpha_{3}\kappa_{u}^{2}\) \\  & & \(D_{1}\ =\ \beta_{1}+\beta_{2}z\) \\  & & auxiliary coefficients: \\  & & \(a_{1}\ =\ 1.488\left(1.761-2.65\kappa_{u}+\kappa_{u}^{2}\right)\alpha_{2}\ =\ 2.44 \left(1.752-2.647\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(a_{3}\ =\ 1.742-2.64\kappa_{u}+\kappa_{u}^{2}\) \\  & & \(\beta_{1}\ =\ 0.043\left(2.1855-\kappa_{u}\right)+0.004\left(-1.399+\kappa_{u}^{ 5}\right)\kappa_{u}+0.081\left(-0.398-\kappa_{u}^{2}\right)\kappa_{u}^{5}\) \\  & & \(\beta_{2}\ =\ 0.001\left(0.42+\kappa_{u}\right)+0.0001\left(0.182-\kappa_{u}^{2} \right)\kappa_{u}+0.001\left(0.327+\kappa_{u}^{5}\right)\kappa_{u}^{5}\) \\ \hline \multirow{8}{*}{\(2\)} & & \(B_{1}\ =\ 7.5\cdot(1.2-\kappa_{u})\left(1.476-2.429\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(B_{2}\ =\ 18.749\cdot B_{4}(1.2-\kappa_{u})\cdot(1.457-2.4\kappa_{u}+\kappa_{u}^{2})\) \\  & & \(B_{3}\ =\ 22.500\cdot B_{4}(1.2-\kappa_{u})\cdot(1.2-\kappa_{u})\); \(B_{4}\ =\ \alpha_{1}-\alpha_{2}\kappa_{u}+\alpha_{3}\kappa_{u}^{2}\) \\  & & \(D_{1}\ =\ \beta_{1}+\beta_{2}z_{\rm g}\) \\  & & \(L\) & auxiliary coefficients: \\  & & \(a_{1}\ =\ 1.474\left(1.675-2.588\kappa_{u}+\kappa_{u}^{2}\right)\alpha_{2}\ =\ 2.429 \left(1.664-2.58\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(a_{3}\ =\ 1.653-2.571\kappa_{u}+\kappa_{u}^{2}\) \\  & & \(\beta_{1}\ =\ 0.029\left(3.337-\kappa_{u}\right)+0.006\left(-0.432-\kappa_{u} \right)\cdot\kappa_{u}+0.103\left(-0.293+\kappa_{u}^{5}\right)\kappa_{u}^{5}\) \\  & & \(\beta_{2}\ =\ 0.001\left(0.246+\kappa_{u}^{2}\right)+0.0002\left(0.015-\kappa_{u}^{5} \right)\kappa_{u}+0.001\left(-0.002-\kappa_{u}^{5}\right)\kappa_{u}^{5}\) \\ \hline \multirow{8}{*}{\(3\)} & & \(B_{1}\ =\ 7.68\cdot(1.2-\kappa_{u})\left(1.476-2.429\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(B_{2}\ =\ 19.66\cdot B_{4}(1.2-\kappa_{u})\cdot(1.457-2.4\kappa_{u}+\kappa_{u}^{2})\) \\  & & \(B_{3}\ =\ 235.929\cdot B_{4}(1.2-\kappa_{u})\cdot(1.2-\kappa_{u})\); \(B_{4}\ =\ \alpha_{1}-\alpha_{2}\kappa_{u}+\alpha_{3}\kappa_{u}^{2}\) \\  & & \(D_{1}\ =\ \beta_{1}+\beta_{2}z_{\rm g}\) \\  & & \(L\) & auxiliary coefficients: \\  & & \(a_{1}\ =\ 1.474\left(1.675-2.588\kappa_{u}+\kappa_{u}^{2}\right)\alpha_{2}\ =\ 2.429 \left(1.664-2.58\kappa_{u}+\kappa_{u}^{2}\right)\) \\  & & \(a_{3}\ =\ 1.653-2.571\kappa_{u}+\kappa_{u}^{2}\) \\  & & \(\beta_{1}\ =\ 0.033\left(-2.9373-\kappa_{u}\right)+0.005\left(-0.512-\kappa_{u} \right)\cdot\kappa_{u}+0.103\left(-0.279+\kappa_{u}^{5}\right)\kappa_{u}^{5}\) \\  & & \(\beta_{2}\ =\ 0.001\left(0.174+\kappa_{u}^{2}\right)+0.0002\left(-0.054-\kappa_{u} ^{5}\right)\kappa_{u}+0.001\left(0.024-\kappa_{u}^{5}\right)\kappa_{u}^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients \(B_{1}\), \(B_{2}\), \(B_{3}\), \(B_{4}\) and \(D_{1}\) for selected diagrams of transverse load.