# Moment factor C1 for Lateral-Torsional Buckling

## Abstract

Within this article the determination of the moment factor C1 for lateral-torsional buckling according to EN 1993-1-1 is described using three distinct methods.

For each method a brief description is given and all methods are applied to the same example in order to compare the results.

## Introduction

The lateral-torsional buckling resistance as described in EN 1993-1-1 article 6.3.2 Ref.[1] is based on the so-called elastic critical moment for lateral-torsional buckling Mcr. The EN 1993-1-1 code however does not give any expression for calculating this variable but indicates to the user that the determination of this value should be based on the following principles:

- Using gross cross-sectional properties
- Taking into account the loading conditions
- Taking into account the real moment distribution
- Taking into account the lateral restraints

Using the general buckling theory an expression for Mcr can be derived in the following format in case of a doubly-symmetric section Ref.[2,3]:

Within this expression, the coefficient C1 takes into account the moment distribution while C2 takes into account the loading type and boundary conditions.

Within this article, the determination of the coefficient C1 is examined using three different methods available in SCIA Engineer:

- ENV 1993-1-1 Annex F

- ECCS 119/Galea

- Lopez, Yong, Serna

Each method is illustrated on the same example; a beam on two supports loaded by end moments and a point load in the middle. The moment diagram for this beam has the following shape:

## Method 1: ENV 1993-1-1 Annex F

The first method concerns the Annex F of ENV 1993-1-1 Ref.[4] This reference gives the values of the moment factor C1 for specific shapes of the moment diagram (uniform moment, point load or line load). The main disadvantage of this method is that only limited coefficients are given in case of combined loading, like the combined point load and end moments in the above example.

Within SCIA Engineer, the moment factors given in Annex F have been extended according to the rules given in Ref.[5] in order to account for combined loading.

For the above example, this method results in a moment factor C1 of 1,89. This is quite a significant value which indicates that the elastic critical moment is 89% higher than the Mcr for a uniform moment diagram. It is however questionable if this high value can be justified.

## Method 2: ECCS 119/Galea

The second method concerns ECCS 119 Annex B Ref.[3].

In the same way as the previous method, this reference gives the values of the moment factor C1 for specific shapes of the moment diagram. However, in addition, this reference also gives extensive coefficients in case of combined loading.

Based on the values of the loading and end moments the moment factor C1 can be derived from figures:

The figures given in this reference have been determined by numerical analysis. A thorough background of these figures including tabulated values can be found in Ref.[6].

For the above example, this method results in a moment factor C1 of 1,26.

In comparison with the previous method it can be seen that this value is much lower i.e. 26% compared to 89%. Since the ECCS 119 method was specifically derived for the case of combined loading it can thus be concluded that the old ENV 1993-1-1 Annex F is on the unsafe side.

## Method 3: Lopez, Yong, Serna

The final method is described in Lopez, Yong, Serna Ref.[7].

In contrast to the previous methods, this reference gives a closed form expression for the moment factor C1. The main advantage is thus that this method can be applied to any moment diagram without the need to identify the shape of the diagram.

The values of the actual moment diagram are taken on different sections of the beam and inserted in the closed form expression.

For the above example, this method results in a moment factor C1 of 1,20.

It is clear that this method is in close approximation to the ECCS 119 method.

## Conclusion

Within this article the determination of the moment factor C1 for lateral-torsional buckling according to EN 1993-1-1 has been described using three distinct methods.

Each method has been applied on a beam on two supports loaded by end moments and a point load in the middle.

The older method according to ENV 1993-1-1 Annex F produces a quite high value for the moment factor C1 due to the fact that this method gives only limited information in case of combined loading.

The method according to ECCS 119 contains specific figures for combined loading and thus leads to a much more accurate value of the moment factor C1.

The final method according to Lopez, Yong, Serna provides a closed form expression which is independent on the type of moment diagram and gives results which are in close approximation to the ECCS 119 method.

When comparing the different methods it seems the ENV 1993-1-1 Annex F method overestimates the moment factor.

Within SCIA Engineer by default the method according to ECCS 119 is applied. This can be freely modified by the user for any project. For more information reference is made to the Steel Code Check Theoretical Background Ref.[8].

## References

[1] Eurocode 3, Design of steel structures, Part 1 - 1 : General rules and rules for buildings, EN 1993-1-1:2005.

[2] SN003a-EN-EU, NCCI: Elastic critical moment for lateral torsional buckling, Access Steel, 2006.

[3] Rules for Member Stability in EN 1993-1-1, Background documentation and design guidelines, ECCS - N° 119, 2006.

[4] Eurocode 3, Design of steel structures, Part 1 - 1/ A1 : General rules and rules for buildings, ENV 1993-1-1:1992/A1, 1994.

[5] Staalconstructies TGB 1990, Stabiliteit, NEN 6771 - 1991.

[6] Déversement élastique d’une poutre à section bi-symétrique soumise à des moments d’extrémité et une charge répartie ou concentrée., Y. Galéa, CTICM, Construction Métallique, n° 2-2002.

[7] Lateral-Torsional buckling of steel beams: A general expression for the moment gradient factor., A. López, D. J. Yong, M. A. Serna, Stability and Ductility of Steel Structures, 2006.

[8] Steel Code Check – Theoretical Background, SCIA, 2012.

Peter Van Tendeloo

09/02/2013