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

### Technical Article

This example is described in technical literature [1] as example 9.5 and in [2] 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 M_{cr} is validated with an idealized 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

#### Design Loads

LC 1 Selfweight:

g_{d} = 1.42 kN/m

LC 2 Imposed load:

${\mathrm f}_{1,\mathrm d}\;=\;\frac{145.4\;\mathrm{kN}\;\cdot\;2}{4\;\mathrm m}\;=\;72.70\;\mathrm{kN}/\mathrm m$

${\mathrm f}_{2,\mathrm d}\;=\;\frac{198.5\;\mathrm{kN}\;\cdot\;2}{4\;\mathrm m}\;=\;99.25\;\mathrm{kN}/\mathrm m$

#### Design Internal Forces

Figure 03 - Distribution of Bending Moment My for Load Combination CO1 = LC1 + LC2

#### Stability Analysis without Considering Secondary Beams According to [3] 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 M_{cr} of 368 kNm is determined in RF-STEEL EC3 in line with the verification according to [3] 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 [3] 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 [2] and adjusted only to the notation of annex BB.2.2.

C_{θ,R,k} = 11,823 kNm (portion from flexural deformation of secondary beams)

C_{θ,D,k} = 359 kNm (portion from cross-section deformation of principal beam, connection to web is considered)

Conversion to continuous rotational restraint C_{θ} 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

K_{υ} = 0.35 for the elastic cross-section ratio

K_{θ} = 10 according to DIN EN 1993-1-1/NA, Table BB.1

A reduction of C_{θ,min} by (M_{Ed} / M_{el,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:

C_{θ,avail} = 134 kNm/m < C_{θ,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 [3] Clause 6.3.4

Determination of the available discrete rotational restraint:

The values are taken from [2] and adjusted only to the notation of annex BB.2.2.

C_{θ,R,k} = 11,823 kNm (portion from flexural deformation of secondary beams)

C_{θ,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.

Figure 04 - Rotational Spring in Window 1.7

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

Figure 05 - Factor αcr,op Determined by RF-STEEL EC3

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 M_{cr} 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 α_{cr,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 center, it is possible to ignore the slight exceeding and consider the design to be fulfilled.

Figure 06 - Design in RF-STEEL EC3

#### Determination of M_{cr} on FEM Model

With the "Generate Surfaces from Member" function and other available modeling 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 M_{y} 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.

Figure 07 - My in Beam (above) and Critical Buckling Load Factor in RF-STABILITY (below)

With this FEM model, we get a moment M_{cr} 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 modeling of the connections of the secondary beams.

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