 # Lateral Torsional Buckling of a Principal Beam with I-Section According to EN 1993-1-1

### Technical Article

001512

28 March 2018

This example is described in technical literature  as example 9.5 and in  as example 8.5. A lateral-torsional buckling analysis must be performed for a principal beam. This beam is a uniform structural member. Therefore, the stability analysis can be carried out according to clause 6.3.3 of DIN EN 1993-1-1. Due to the uniaxial bending, it would also be possible to perform the design by the general method according to clause 6.3.4. Additionally, the determination of the moment Mcr is validated with an idealised member model in line with the method mentioned above, using a FEM model.

#### System

Cross-sections:
Principal beams = IPE 550
Secondary beams = HE-B 240
Material:
Structural steel S235 according to DIN EN 1993-1-1, Table 3.1

LC 1 Selfweight:
gd = 1.42 kN/m
${\mathrm f}_{1,\mathrm d}\;=\;\frac{145.4\;\mathrm{kN}\;\cdot\;2}{4\;\mathrm m}\;=\;72.70\;\mathrm{kN}/\mathrm m$
${\mathrm f}_{2,\mathrm d}\;=\;\frac{198.5\;\mathrm{kN}\;\cdot\;2}{4\;\mathrm m}\;=\;99.25\;\mathrm{kN}/\mathrm m$

#### Stability Analysis without Considering Secondary Beams According to  Clause 6.3.2

Under the assumption of a lateral and torsional restraint available at the member's start and end, an ideal critical moment for lateral torsional buckling Mcr of 368 kNm is determined in RF-STEEL EC3 in line with the verification according to  clause 6.3.2. So, the design according to equation 6.54 results in 1.64. Hence, the ultimate limit state design cannot be fulfilled without the stabilizing effect of the secondary beams.

#### Stability Analysis Considering Secondary Beams According to  Annex BB.2.2

The rules of DIN EN 1993-1-1 annex BB.2.2 assume a continuous rotational restraint over the beam length. Therefore, the discrete rotational restraint available in the model is "smeared" to a continuous rotational restraint.

Determination of available continuous rotational restraint:
The values are taken from  and adjusted only to the notation of annex BB.2.2.
Cθ,R,k = 11,823 kNm (portion from flexural deformation of secondary beams)
Cθ,D,k = 359 kNm (portion from cross-section deformation of principal beam, connection to web is considered)

Conversion to continuous rotational restraint Cθ with average distance of secondary beams:
${\mathrm x}_\mathrm m\;=\;\frac{2.5\;\mathrm m\;+\;2.7\;\mathrm m}2\;=\;2.6\;\mathrm m$
${\mathrm C}_\mathrm\theta\;=\;\frac1{\left({\displaystyle\frac1{11,823}}\;+\;{\displaystyle\frac1{359}}\right)\;\cdot\;2.6}\;=\;134\;\mathrm{kNm}/\mathrm m$

Determination of required rotational restraint:
${\mathrm C}_{\mathrm\theta,\min}\;=\;\frac{{\mathrm M}_{\mathrm{pl},\mathrm k}^2}{{\mathrm{EI}}_\mathrm z}\;\cdot\;{\mathrm K}_\mathrm\theta\;\cdot\;{\mathrm K}_\mathrm\upsilon\;=\;\frac{65,330^2}{21,000\;\cdot\;2,670}\;\cdot\;10\;\cdot\;0.35\;=\;266.4\;\mathrm{kNm}/\mathrm m$
where
Kυ = 0.35 for the elastic cross-section ratio
Kθ = 10 according to DIN EN 1993-1-1/NA, Table BB.1

A reduction of Cθ,min by (MEd / Mel,Rd)² is possible:
${\mathrm C}_{\mathrm\theta,\min}\;=\;266.4\;\ast\;\left(\frac{452.7}{521.3}\right)^2\;=\;200.9\;\mathrm{kNm}/\mathrm m$

Verification:
Cθ,avail = 134 kNm/m < Cθ,min = 200.9 kNm/m

The design in the form of the verification of a sufficient restraint for the principal beam's lateral deformation according to annex BB.2.2 cannot be performed.

#### Stability Analysis Considering Secondary Beams According to  Clause 6.3.4

Determination of the available discrete rotational restraint:
The values are taken from  and adjusted only to the notation of annex BB.2.2.
Cθ,R,k = 11,823 kNm (portion from flexural deformation of secondary beams)
Cθ,D,k = 359 kNm (portion from cross-section deformation of principal beam, connection to web is considered)
${\mathrm C}_\mathrm\theta\;=\;\frac1{{\displaystyle\frac1{11,823}}\;+\;{\displaystyle\frac1{359}}}\;=\;348\;\mathrm{kNm}/\mathrm{rad}$

With this rotational spring it is possible to describe the structural model of the notionally singled out set of members for the design according to clause 6.3.4 in module window 1.7.

During the verification according to 6.3.4 a resolver for eigenvalues implemented in RF-STEEL EC3 determines the factor αcr,op, by which the smallest ideal critical buckling load can be reached with deformations from the structural system plane.

The critical buckling load factor is shown among the intermediate values (see result windows) and the corresponding mode shape can be displayed in a separate window. Thus, the result is a moment Mcr of 452.65 kNm ∙ 2.203 = 997.2 kNm.

Hence, the design according to equation 6.63 results for the model in 1.01. For the calculation of αcr,op the load application point was applied in accordance with the detail settings in a destabilizing way on the upper flange. Keeping in mind that the real point of load application is between the upper flange and the shear centre, it is possible to ignore the slight exceeding and consider the design to be fulfilled.

#### Determination of Mcr on FEM Model

With the "Generate Surfaces from Member" function and other available modelling tools, it is possible to easily create a surface model of the structure with a minimum amount of time. Using the "Result Beam" member type it is possible to determine and graphically display the moment My in the beam. The critical buckling load factor which is needed can be calculated on the entire model with the RF-STABILITY add-on module.

With this FEM model, we get a moment Mcr of 447.20 kNm ∙ 2.85 = 1,274.5 kNm. This is a bit higher than the result on the member model with corresponding discrete rotational springs. Consideration could be given to an even more accurate modelling of the connections of the secondary beams.

#### Reference

  Kuhlmann, U.: Stahlbau-Kalender 2013 - Eurocode 3 - Anwendungsnormen, Stahl im Industrie- und Anlagenbau. Berlin: Ernst & Sohn, 2013  Lindner, J.; Scheer, J.; Schmidt, H.: Stahlbauten - Erläuterungen zu DIN 18800 Teil 1 bis Teil 4. Berlin: Beuth, 1993  Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings; EN 1993-1-1:2010-12  National Annex - Nationally Determined Parameters - Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings; DIN EN 1993-1-1/NA:2015-08 