## Introduction

When a beam, bent in a plane, is allowed to move and twist freely between its two support points, in addition to bending, sudden perpendicular displacement and twisting may occur: causing the beam to deviate out of its original plane. This phenomenon is illustrated in Figure 1, showing a single supported beam with I-section bent around the strong axis. As the bending moment in the vertical plane increases, reaching a critical value, the beam undergoes abrupt lateral movement and twisting between the supports. This phenomenon is called **lateral torsional buckling (LTB)**, which is a loss of stability mode that can apply to both perfect beams and real beams.

The design of the beam against LTB is fully analogous to the design of a compressed column against flexural buckling. The analogy is illustrated in Table 1, where the corresponding parameters are shown that affect the two buckling resistances:

Flexural (column) buckling | Lateral torsional buckling |
---|---|

design force ($N_{Ed}$) | design moment ($M_{Ed}$) |

critical force ($N_{cr}$) | critical moment ($M_{cr}$) |

column slenderness ($\frac{}{\lambda}$) | beam slenderness ($\frac{}{\lambda}_{LT}$) |

buckling reduction factor ($\chi$) | buckling reduction factor ($\chi_{LT}$) |

buckling resistance ($N_{b,Rd}$) | buckling resistance ($M_{b,Rd}$) |

The critical moment of the perfect beam is determined at the location of the maximum value of the *M _{y,Ed }*design bending moment diagram. For a doubly symmetrical I cross-section:

$$M_{cr}=C_1\frac{\pi^2EI_z}{(k_z⋅L)^2}\left[\frac{I_\omega }{I_z}+ \frac{(k_zL)^2GI_t}{\pi^2EI_z}\right] ^{0.5} $$

where *k _{z}* is the coefficient of restraint about the weak axis of the cross-section,

*G*is the shear modulus, and

*I*and

_{t}*I*are the pure (St. Venant) and warping torsional moments of inertia of the cross-section. The value of the factor

_{ω}*C*depends on the shape of the bending moment diagram and its value can be found in appropriate tables and manuals. For a constant moment diagram,

_{1}*C*=1.0. The formula for the other design parameters, in particular the buckling reduction factor $\chi_{LT}$, depends on the design standard considered.

_{1}## Lateral torsional buckling resistance by EN1993-1-1

The design of the beam against LTB (load capacity check) according to EC3-1-1 shall be carried out in the following steps:

gate## The evolution of compressed bar (column) design

One of the characteristic features of steel structures made of bars (e.g. lattice girders) is the compressed bar. We speak of a compressed bar when the structural element, which usually has a straight axis, is loaded by a compressive force * P* applied centrally (Figure 1).

Figure 2 illustrates the evolution of compressed bar (column) design. In the beginning (in the old days), master builders determined the load-bearing capacity of compressed columns of different materials and sizes on the basis of the **experience accumulated over the centuries**, passed down from master to apprentice. A significant change was brought about by the application of classical mathematical differential analysis to engineering. The Swiss mathematician and physicist Euler (1707-1783) solved the problem of the deflection of a compressed elastic line, which could be applied to the solution of the elastic compressed bar (**Euler’s force**). In the following centuries, engineers recognised that Euler’s force only gave an acceptable approximation to the real load capacity of a compressed bar in certain cases (mainly for large slender bars). Many solutions for the bearing capacity of a compressed bar were developed that were more advanced than the Euler formula, but it was not until the huge structural engineering boom following World War II that significant changes were made. **Compression bar experiments** were carried out in every major structural laboratory in the world, and a database of over two thousand experiments was compiled from the results. The load capacity of the pressure bar was given by a formula based on the database, using the method of mathematical statistics.

This methodology is still dominant today:

‘the dimensioning of the compressed bar has become a political issue for the steel construction profession…’. Understanding the principle of compressed bar design is therefore essential for the structural engineer.

The right side of the Figure 2 also contains a hint for the future. At the level of scientific research, it is already present that the load capacity of a real compressed column can be determined by mathematical-mechanical simulation. Indeed, in the near future, databases that go beyond anything we know today can be created using supercomputers. On the basis of such a gigantic database, **artificial intelligence** could, at least in principle, supersede existing engineering knowledge and methodology. But the reality is that structural engineering is not one of the pull sectors (such as the defense or automotive industries), so this new shift in design theory is certainly a long way off.

In the following, the **Euler force** and the **experimentally based** standard design formula, which are of major importance to structural steel engineering today, are discussed in detail.

## Buckling strength of the ideal columns: the Euler force

Assume that the hinged compressed column shown in the Figure 3 has the following properties:

- perfectly
**straight**, - its material is perfectly linearly
**elastic**, **centrally**compressed.

Under the above conditions, perform the compressed column experiment using Consteel software: run the Linear Buckling Analysis (LBA) calculation. The result is illustrated in Figure 3.

gate**Did you know that you could use Consteel to Consider the shear stiffness of a steel deck as stabilization for steel members?**

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**Introduction**

This verification example studies a simple fork supported beam member with welded section (flanges: 200-12 and 100-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment distribution from concentrated transverse force is examined for both orientations of the I-section. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.

**Geometry**

**Normal orientation – wide flange in compression**

Constant bending moment distribution

Triangular bending moment distribution – load on upper flange

Triangular bending moment distribution – load on bottom flange

**Reverse orientation – narrow flange in compression**

Constant bending moment distribution

Triangular bending moment distribution – load on upper flange

Triangular bending moment distribution – load on bottom flange

**Calculation by hand**

Factors to be used for analitical approximation formulae of elastic critical moment are taken from *G. Sedlacek, J. Naumes: Excerpt from the Background Document to EN 1993-1-1 Flexural buckling and lateral buckling on a common basis: Stability assessments according to Eurocode 3 CEN / TC250 / SC3 / N1639E – rev2*

**Normal orientation – wide flange in compression**

Constant bending moment distribution

**Reverse orientation – narrow flange in compression**

**Computation by Consteel**

Version nr: Consteel 15 build 1722

**Normal orientation – wide flange in compression**

Constant bending moment distribution

**7 DOF beam element**

First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=25). The eigenshape shows lateral torsional buckling.

**Superbeam**

First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).

**Introduction**

This verification example studies a simple fork supported beam member with welded section (flanges: 200-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment dsitribution from concentrated transverse force is examined. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.

**Geometry**

Constant bending moment distribution

Triangular bending moment distribution – load on upper flange

Triangular bending moment distribution – load on bottom flange

**Calculation by hand**

Constant bending moment distribution

Triangular bending moment distribution

**Computation by Consteel**

Version nr: Consteel 15 build 1722

Constant bending moment distribution

**7 DOF beam element**

First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=16). The eigenshape shows lateral torsional buckling.

**Superbeam**

First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).

Triangular bending moment distribution – load on upper flange

**7 DOF beam element**

First buckling eigenvalue of the member which was computed by the Consteel software using the 7 DOF beam finite element model (n=16).

**Superbeam**

First buckling eigenvalue of the member which was computed by the Consteel software using the Superbeam function (δ=25).

Triangular bending moment distribution – load on bottom flange

(more…)Perfect the understanding of your structure with advanced buckling sensitivity results illustrated on proper mode shape and colored internal force diagrams.

gateConsteel 14 is a powerful analysis and design software for structural engineers. Watch our video how to get started with Consteel.

**Contents**

- Set analysis parameters
- Perform first and second order analysis
- Perform buckling analysis
- Analysis results in graphics and in tables
- Results: deformation, internal forces, reactions

## Part 2 – Imperfection factors

The Eurocode EN 1993-1-1 offers basically two methods for the buckling verification of members:

(1) based on buckling reduction factors (buckling curves) and

(2) based on equivalent geometrical imperfections.

In the first part of this article, we reviewed the utilization difference and showed the relationship between the two methods. It was concluded that the method of chapters 6.3.1 (reduction factor) and 5.3.2 (11) (buckling mode based equivalent imperfection) are consistent at the load level equal to the buckling resistance of the member, so when the member utilization is 100%. The basic result of the procedure in 5.3.2 (11) is the amplitude (largest deflection value) of the equivalent geometrical imperfection. However, the Eurocode gives another simpler alternative for the calculation of this amplitude for compressed members in section 5.3.2(3) b) in Table 5.1, where the **amplitude of an initial bow** is defined as a portion of the member length for each buckling curves (Fig. 1.). We use the first column (“elastic analysis”) including smaller amplitude values.

*It is an obvious expectation that these two standard procedures should yield at least similar results for the same problem. However, this is by far not the case in general.*

In order to show the significance of the imperfection amplitudes this part is dealing with these two calculation methods, the variation of their values and the effect on the buckling utilization.

Let’s see again the simple example of Part 1: a simply supported, compressed column with a Class 2 cross-section (plastic resistance calculation allowed). The column is 6 meters high and has an IPE300 cross-section made of S235 steel. The two methods are implemented into Consteel and on Figure 2. it can be seen, that the two values for the amplitude of the geometrical imperfection is very different – *e _{0} = 24 mm* by the 5.3.2(3) b) Table 5.1 (L/250) and

*e*by the 5.3.2 (11) (same as in Part 1).

_{0}= 13,4 mmClick the button bellow to download and read the full article at page 187-195.

In this paper a numerical study is presented which examines a steel frame with two different finite element programs. Stability failure is more frequent in a lot of cases than strength failure hence it is important to focus on these failure modes: global, in-plane-, out-of-plane -, lateral-torsional- and local buckling. Three models were used with different elements such as shell elements and 7 DOF beam elements. 7 DOF beam elements were used in the first model, shell elements were used in the other two. The first of the shell models gave too much local buckling shapes therefore it was improved with local constraints and that is the third model where global buckling shapes can be examined. There are three different procedures to calculate the resistance: (i) the general method, (ii) the method of the reduction factors, and (iii) the simulation. The analysis results of the different programs and design methods were compared to each other and to the manual calculation based on the Eurocode 3 standards.

gateThe new versions of the EN 1993-1-1 (EC3-1-1) and the EN 1993-1-5 (EC3-1-5) standards have introduced the general method designing beam-column structures; see [1] and [2]. The design method requires 3D geometric model and finite element analysis. In a series of papers we present this general design approach. The parts of the series are the following:

- Part 1: 3D model based analysis using general beam-column FEM

Click the button bellow to download and read the full article.

gate