## 2 Elastic Restraint Against Warping and Against Lateral Rotation at the Support Nodes  
The static diagram of the beam elastically restrained against warping and against lateral rotation at the support nodes is shown in Figure 1. The springs in colour symbolically represent the elastic restraint in the support sections, i.e., in red--warping restraint (\(\alpha_{\omega}\)), and in blue--lateral rotation restraint (\(\alpha_{u}\)).  
Figure 1: Static diagram of the beam: (**a**) bimoment (\(B\)) and moment (\(M_{2}\)) at the support, (**b**) elastic restraint against warping (\(\alpha_{\omega}\)) and against lateral rotation (\(\alpha_{u}\)), (**c**) twist angle \(\phi(x)\) and lateral deflection \(u(x)\).  
The degree of elastic restraint against warping \(\varepsilon_{\omega}\)[33; 34; 35] can be determined from formula:  
\[\varepsilon_{\omega}\ =\ \frac{\alpha_{\omega}L}{EI_{\omega}} \tag{1}\]  
where: \(\alpha_{\omega}\)--stiffness of the elastic restraint against warping [33; 34; 35] acc. formula:  
\[\alpha_{\omega}\ =\ \frac{-B}{\frac{d\phi}{dx}} \tag{2}\]  
where: \(B\)--bimoment at the site of beam support, \(\phi\)--twist angle, \(\frac{d\phi}{dx}\)--warping of the section.  
The degree of elastic restraint \(\varepsilon_{\omega}\) acc. formula (1), ranges from \(\varepsilon_{\omega}\) = 0 for complete warping freedom to \(\varepsilon_{\omega}\) = \(\infty\) for full prevention of warping.  
In this study, the degree of elastic restraint against rotation in the plane of lateral torsional buckling \(\varepsilon_{u}\), was derived. It was written in the following form:  
\[\varepsilon_{u}\ =\ \frac{\alpha_{u}L}{EI_{z}} \tag{3}\]  
where: \(\alpha_{u}\)--stiffness of the elastic restraint against lateral rotation acc. formula:  
\[\alpha_{u}\ =\ \frac{M_{z}}{\frac{du}{dx}} \tag{4}\]  
where: \(M_{z}\)--bending moment with respect to the minor axis of the section at the support, \(u\)--lateral deflection, \(\frac{du}{dx}\)--rotation about axis \(z\).  
The degree of elastic restraint \(\varepsilon_{u}\), acc. formula (3), ranges from \(\varepsilon_{u}\) = 0 for complete rotational freedom to \(\varepsilon_{u}\) = \(\infty\) for full prevention of rotation in the plane of lateral torsional buckling.  
In order to account, in an explicit manner, for the beam elastic restraint, in the twist angle function \(\phi(x)\) and the lateral deflection function \(u(x)\), the dimensionless indexes of fixity against warping \(\kappa_{\omega}\)[34] and against lateral rotation \(\kappa_{u}\) were introduced acc. formulas:  
\[\kappa_{\omega}\ =\ \frac{\alpha_{\omega}L}{2 EI_{\omega}+\alpha_{\omega}L} \kappa_{u}\ =\ \frac{\alpha_{u}L}{2 EI_{z}+\alpha_{u}L}\] (5 ab)  
Indexes of fixity range from \(\kappa_{\omega}\) = 0 (\(\kappa_{u}\) = 0) for complete freedom of warping or lateral rotation, respectively, to \(\kappa_{\omega}\) = 1 (\(\kappa_{u}\) = 1) for complete blockage of warping or lateral rotation.  
Inverse relations, i.e., \(\alpha_{\omega}(\kappa_{\omega})\) and \(\alpha_{u}(\kappa_{u})\), have the following form:  
\[\alpha_{\omega}\ =\ \frac{2\kappa_{\omega}EI_{\omega}}{(1-\kappa_{\omega})L} \alpha_{u}\ =\ \frac{2\kappa_{u}EI_{z}}{(1-\kappa_{u})L}\] (6 ab)  
The relations holding between the degree of elastic restraint \(\varepsilon_{i}\) (1) and (3) and the index of fixity \(\kappa_{i}\) (5 ab) (for \(i\) = \(\omega\), \(u\)) are as follows [34; 38; 41]:  
\[\kappa_{i}\ =\ \frac{\varepsilon_{i}}{2+\varepsilon_{i}}\hskip 28.452756 pt \varepsilon_{i}\ =\ \frac{2\kappa_{i}}{1-\kappa_{i}} \tag{7}\]