This article presents an overview on the design of structural steel beams for lateral torsional buckling to EN 1993-1-1-2005, and the UK National Annex to EN 1993-1-1-2005. For convenience EN 1993-1-1:2005 is routinely referred to as the standard, while the UK National annex to the standard is sometimes referred to as the UK NA.

Lateral torsional buckling occurs when a beam is subjected to bending and its compression flange is unrestrained. The unrestrained length of the compression flange tends to buckle sideways; the tension flange resists this lateral movement which causes the compression flange to twist about the axis of the beam. This phenomenon is called lateral torsional buckling.

Exceptions for Lateral Torsional Buckling

Lateral torsional buckling is an open-section (UB, UC, EFA etc.) phenomenon. Beams with closed cross-sections such as square or circular hollow sections, circular tubes or square box sections are not susceptible to lateral torsional buckling.

Also, open sections with sufficient restraint to their compression flange are also not susceptible to lateral torsional buckling. Steel beams in building supporting concrete slab in a composite construction have full lateral restraint to their compression flange and do not experience lateral torsional buckling. Other source of restraints to a compression flange could also come from members such as purlins, secondary beams, bracings etc. It should however be noted that if the restraints are at intervals, and not fully along the compression flange, such as in the case of purlins and secondary beams, the designer must check the length of the beam between intermediate restraints for resistance to lateral torsional buckling.

Lateral Buckling Resistance

The lateral buckling resistance of unrestrained members subjected to major axis bending is given in expression 6.54 of the standard. The expression is reproduced below

$
\,\,\frac{M_{Ed}}{M_{bRd}}\,\,\leqslant \,\,1
$

where M_b,Rd  is the buckling resistance. This is given in clause 6.3.2.1(3) of the standard as:

$$
M_{b,Rd}\,\,=\,\,\chi _{LT}\,\,\frac{W_y\,\,.\,\,f_y}{\varUpsilon _m}
$$

W_y is the appropriate section modulus for each class. If we substitute W_y for  in each of the classes, we can render the expression as below:

$
M_{b,Rd}\,\,=\,\,M_{pl,Rd}\,\,=\,\,\chi _{LT}\,\,\frac{W_{ply}\,\,.\,\,f_y}{\varUpsilon _m}
$ (for Class 1 and 2 sections)

$
M_{b,Rd}\,\,=\,\,M_{pl,Rd}\,\,=\,\,\chi _{LT}\,\,\frac{W_{elmin}\,\,.\,\,f_y}{\varUpsilon _m}
$ (for Class 3 section)

$
M_{b,Rd}\,\,=\,\,M_{pl,Rd}\,\,=\,\,\chi _{LT}\,\,\frac{W_{effmin}\,\,.\,\,f_y}{\varUpsilon _m}
$ (for Class 4 section)

Χ_LT is the reduction factor for lateral torsional buckling

 

The Reduction factor for Lateral Torsional Buckling (Χ_LT)

The standard gives two methods for determining the reduction factor

1. A general case
2. Hot-rolled sections or equivalent welded sections

 

The General Case

The general case applies to all kinds of sections such as plate girders, castellated beams, rolled sections, and all other types of sections. The value of the reduction factor can be determined either by using expression (6.56) of EN 1993-1-1 or by using the buckling curves in Figure 6.4 of the standard.

Expression (6.56) of the standard is given below;

$
\,\,\chi _{LT}\,\,=\,\,\frac{1}{\phi _{LT}\,\,+\,\,\sqrt{\phi _{LT}^2\,\,+\,\,\lambda _{LT}^2}}
$

Φ_LT = 0.5 (1 + ∝_LT( λ_LT – 0.2) + λ_LT ²)

∝_LT is an imperfection factor for lateral torsional buckling and it is given in Table 6.3 of the standard. (It should be noted that table 6.3 can only be consulted after determining the appropriate buckling curve using table 6.4)

λ_LT is the non-dimension slenderness for lateral torsional buckling.

$
\,\lambda _{LT}\,\,=\,\,\sqrt{\frac{W_y\,\,.\,\,f_y}{M_{cr}}}
$

M_cr is the elastic critical moment for buckling

W_y is the appropriate section modulus for the section classification

Determining the Reduction factor (Χ_LT ) for general case using Buckling Curves

As stated earlier, the reduction factor can as well be read off from the buckling curve given in figure 6.4 of the standard having determined the slenderness due to lateral torsional buckling (λ_LT) rather than using expression 6.56 of the standard. Table 6.4 of the standard enables the designer to determine the appropriate buckling curve for lateral torsional buckling

 

Method for Hot-Rolled Sections or Equivalent Welded Sections.

This method is peculiar to hot-rolled sections or equivalent welded sections (i.e. welded sections with similar dimensions as rolled sections). The value of the reduction factor can only be determined by using expression (6.57) of EN 1993-1-1. The standard does not give buckling curve for hot rolled and equivalent sections similar to that of the general case. However, buckling curve for hot rolled sections can be found as Non-contradictory Complementary Information (NCCI) in “Steel Building Design: Concise Eurocode (SCI P362)”

Expression (6.57) of the standard is given below

$
\,\,\chi _{LT}\,\,=\,\,\frac{1}{\phi _{LT}\,\,+\,\,\sqrt{\phi _{LT}^2\,\,+\,\,\ βλ _{LT}^2}}
$

Φ_LT = 0.5 (1 + ∝_LT( λ_LT – λ_LT,0) + βλ_LT ²)

Although the standard treats hot-rolled sections and equivalent welded section as the same, the UK NA has a few different recommendations that is peculiar to each type of section. EN 1993-1-1 recommends 0.4 as the maximum value of λ_LT,0 (The UK national annex recommends 0.2 for welded sections), and also 0.75 as the minimum value of  β (The UK national annex recommends 1.0 for welded sections).

∝_LT is an imperfection factor for lateral torsional buckling and it is given in Table 6.3 of the standard. (It should be noted that table 6.3 can only be consulted after determining the appropriate buckling curve (this buckling curve for hot-rolled and equivalent welded section is not explicitly plotted in the standard) using table 6.5 (replaced with Table NA.1 in the UK NA)).

$
\,\lambda _{LT}\,\,=\,\,\sqrt{\frac{W_y\,\,.\,\,f_y}{M_{cr}}}
$

W_y  is the appropriate section modulus for the section classification

M_cr  is the elastic critical moment for buckling

 

Elastic Critical Moment (M_cr ) for Lateral Torsional Buckling

Elastic critical moment is an important parameter to determine the non-dimension slenderness. Conversely, the reduction factor for lateral torsional buckling is also highly dependent on the value of the non-dimension slenderness. In spite of its importance, the standard does not give an expression to determine the elastic critical moment (M_cr), it only gives information that the critical buckling moment should be based on gross cross-sectional properties and that it also depends on the loading, real moment distribution, and lateral restraints.

To make up for this omission in the standard, Access Steel document SN003 provides an expression for determining elastic critical moment for doubly symmetric sections such as UB and UC that are not supporting destabilizing loads. The main expression can be simplified for linear moment between restraint as provided below:

$M_{C r}=C_1 \frac{\pi^2 E I_z}{L^2} \sqrt{\frac{I_w}{I_z}+\frac{L^2 G I_t}{\pi^2 E I_z}}$

where;

E is the young modulus (210000N/mm²)

G is the shear modulus (80770 N/mm²)

L is the length of the beam between point of restraint

I_z  is the second moment area about weak axis

I_w is the warping constant

I_t is the torsion constant

C1 is a parameter that depends on the shape of the bending moment diagram (it can be conservatively taken as zero). It values for different shape of bending moment diagram for members with end moment loading and members with transverse loadings respectively are given below.

 

Click here to study a worked example on the design of an un-restrained I-beam

Alternative Methods for evaluating the Non-dimension Slenderness

Rather than using the expression in the standard for non-dimension slenderness which is dependent on determining the elastic critical moment, other conservative approaches can be used which precludes determining the value of M_cr . These alternative approaches are contained in Handbook for Structural Steelwork

Alternative Approach 1

The non-dimension slenderness (λ_LT) where load is not destabilizing can be determined using the expression below:

$
\,\lambda _{LT}\,\,=\,\,\frac{1}{\sqrt{C_1}}UV\lambda _z\sqrt{B_w}
$

U is section properties given in properties table. It can be conservatively taken as 0.9 for rolled section and 1.0 for welded section

V is a property that is related to slenderness. For symmetric section where the load is not destabilizing, it can be conservatively taken as 1.

λ_z = KL/Iw (k can be conservatively taken as 1.0)

$
\,\sqrt{\beta _w}\,\,=\,\,\sqrt{\frac{W_y}{W_{ply}}}
$

1/√C1 is a parameter that is dependent on the shape of the bending moment diagram. It can be conservatively taken as 1 for non-destabilizing loads.

 

 

Alternative Approach 2

Another simplified but highly conservative alternative to determine the non-dimension slenderness () is using Table 3.3 of Handbook of Structural Steelwork which is reproduced below. The table is only valid for doubly symmetric, hot-rolled I and H sections with lateral restraint at both ends of segment being considered for lateral torsional buckling and also support non-destabilizing load.

 

References

EN 1993-1-1:2005: Eurocode 3, Design of Steel structures – Part 1:1: General rules for buildings.

NCCI: Elastic Critical Moment for Lateral Torsional Buckling (Access Steel document SN003)

Handbook of Structural Steelwork

Steel Building Design: Concise Eurocode (SCI P362)