# The Elastic Critical Moment of Lateral Torsional Buckling of Steel Beams with Spatially Elastically Restrained at the Support Nodes

## TL;DR

This paper introduces a new formula to calculate the critical buckling moment of steel beams with elastic restraints at supports, improving engineering accuracy.

## Contribution

A general approximation formula for lateral torsional buckling of beams with spatially elastically restrained supports is introduced.

## Key findings

- The approximation formula shows good agreement with FEM results (mean 1.006, standard deviation 0.028).
- The formula accounts for three elastic fixity indexes for practical loading schemes.
- Detailed calculations were performed for different beam sections and restraint combinations.

## Abstract

This paper presents the results of a further stage of the authors’ research into the lateral torsional buckling of hot-rolled bisymmetric I-beams, spatially elastically restrained at the support nodes, i.e., against: (1) warping, (2) rotation in the lateral torsional buckling plane and (3) rotation in the main bending plane My. The analysis considered the entire range of variation in node stiffness, from free support in bending My and full freedom of warping and rotation in the lateral torsional buckling plane, to full restraint of the beam at the nodes. The authors introduced a general approximation formula (AF) for the critical moment of lateral torsional buckling Mcr, simultaneously considering the three elastic fixity indexes for basic and frequently occurring loading schemes in engineering practice. In order to facilitate the calculations, the authors have included the full sequence of formulas for the successive components of the critical moment, derived in the authors’ previous papers. The ability to more accurately consider the actual conditions of the spatial elastic restraint of the beam at the nodes leads to a more accurate calculation of Mcr. The results obtained were compared with FEM (LTBeamN software, v. 1.0.3) by performing a large number of calculations and numerical simulations. The agreement of the AF/FEM results was achieved at a level sufficient from the engineering point of view (mean value 1.006, standard deviation 0.028, coefficient of variation 2.8%). Detailed calculations were carried out for different section types (I, H) and different combinations of fixity index values. The application of approximation formulas in practical calculations is demonstrated on an example. The formulas derived in the paper can be used, among other things, to verify the correctness of FEM calculations, including the correct modelling of elastic support restraints, which is important in design practice.