## 4 The Critical Moment for Lateral Torsional Buckling  
The energy method was used to determine the critical moment (\(M_{cr}\)) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (\(\kappa_{\omega}\), \(\kappa_{u}\)) at the support nodes. The total potential energy of the beam--load system was determined from formula:  
\[\Pi\ =\ U_{s,1}+U_{s,2}+U_{s,3}-T \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Polynomials** & **Physical Interpretation** \\ \hline
1 & 2 & 3 \\ \hline
1 & \(W_{P1}\ =\ \rho-2\rho^{3}+\rho^{4}\) & \\
2 & \(W_{P2}\ =\ \rho-10\rho^{3}+15\rho^{4}-6\rho^{5}\) & \\
3 & \(W_{P3}\ =\ \rho-26\rho^{3}+73\rho^{4}-72\rho^{5}+24\rho^{6}\) & \\
4 & \(W_{U11}\ =\ \rho^{2}-2\rho^{3}+\rho^{4}\) & \\
5 & \(W_{U2}\ =\ \rho^{2}-4\rho^{3}+5\rho^{4}-2\rho^{5}\) & \\
6 & \(W_{U3}\ =\ 2\rho^{2}-13\rho^{3}+29\rho^{4}-27\rho^{5}+9\rho^{6}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Polynomials used and their physical interpretation (where \(\rho=x/L\)) [34; 43]. where: \(U_{s,1}\)--elastic strain energy of beam bending and torsion; \(U_{s,2}\)--energy of the elastic restraint against warping; \(U_{s,3}\)--energy of the elastic restraint against lateral rotation; \(T\)--work done by external forces. The elastic strain energy of the beam bending and torsion [10] was expressed with the equation:  
\[U_{s,1}\ =\ \frac{1}{2}\Bigg{(}EI_{2}\int_{0}^{L}\left(\frac{d^{2}u}{dx^{2}} \right)^{2}\!dx+GI_{t}\int_{0}^{L}\left(\frac{d\phi}{dx}\right)^{2}\!dx+EI_{ \omega}\int_{0}^{L}\left(\frac{d^{2}\phi}{dx^{2}}\right)^{2}\!dx\Bigg{)} \tag{11}\]  
The energy of the elastic restraint against warping (\(U_{s,2}\)) [34] and against lateral rotation (\(U_{s,3}\)) was determined from formulas:  
\[U_{s,2}\ =\ \frac{\alpha_{\omega}}{2}\Bigg{(}\left(\frac{d\phi}{dx}\right)^{2}_ {x\ =\ 0}+\left(\frac{d\phi}{dx}\right)^{2}_{x\ =\ L}\Bigg{)}\qquad U_{s,3}\ =\ \frac{\alpha_{u}}{2}\Bigg{(}\left(\frac{du}{dx}\right)^{2}_{x\ =\ 0}+\left(\frac{du}{dx}\right)^{2}_{x\ =\ L}\Bigg{)}\] (12 ab)  
The work done by external forces is a function of the loading diagram and the co-ordinate of the point of load application. For instance, for a simply supported beam (Figure 2), loaded with linearly varied distribution of transverse load, for an arbitrary co-ordinate (\(z_{g}\)) of the point of load application over the section height, the work done by external forces can be written as follows:  
\[T\ =\ \frac{q_{z}}{2}\Bigg{(}\!\int_{0}^{L}\phi\frac{d^{2}u}{dx^{2}}\! \left(\frac{L}{3}-\frac{x^{2}}{3L}\right)\!dx+\frac{z_{g}}{2}\int_{0}^{L}\phi^{ 2}dx\Bigg{)} \tag{13}\]  
To determine \(M_{\sigma}\), a programme for numerical computations, namely _McrLT_elastic_warping_rotation_2.nb_ (\(M_{LTB,EL,2}\) for short) was developed in the environment of the _Mathematica(r)_ package. The programme allows the determination of the critical load as a function of the indexes of fixity (\(\kappa_{\omega}\), \(\kappa_{u}\)), for arbitrary geometric parameters of the bisymmetric I-section, an arbitrary value of the co-ordinate (\(z_{g}\)) of the load application point (see Figure 2), for beam loading diagrams that are most commonly found in practice (Table 2; Table 3). In the programme, the first three terms (\(a_{1,2,3}\)) of the twist angle function acc. (8) and the first three terms (\(b_{1,2,3}\)) of the lateral deflection function acc. (9) were employed. In study [34], _McrLT_elastic_fix.on.warp._sym.cal.nb_ programme was developed to make symbolic \"computations\" for those cases, in which elastic restraint against warping (\(\kappa_{\omega}\)) occurs. In order to receive possibly simple approximation formulas, only the first term \(\{a_{1}((1-\kappa_{\omega})W_{P1}+\kappa_{\omega}W_{(1L)})\}\) of the beam twist angle function (8) was employed. Still, a very good congruence between the results thus obtained and those produced by FEM (_LTBeam_, _Abaqus_) was noted.