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## Introduction

If a beam bent in a plane is free to move and twist between its two support points, then in addition to bending, there can be sudden perpendicular displacement and twisting: the beam will turn out of plane. This phenomenon is illustrated in Figure 1, which shows a beam of cross-section I with two supports bent about the strong axis: during bending in the vertical plane, when the moment reaches a critical value, the beam suddenly moves laterally and twists between the two 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 against flexural buckling of a compressed column. The analogy is illustrated in Table 1, where the corresponding parameters affecting the two buckling resistances are shown:

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 cross-section I:

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

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## Designing a lattice girder

The design of the bars of a truss (lattice girder) structure does not require any special theoretical knowledge: normally, the truss bars are designed as **compressed** and/or **tensioned** bars, neglecting bending moments and shear forces. The dimensioning of compression bars is nowadays carried out using a model-based computer procedure. For details, see the knowledge base material **Design of columns against buckling**. Here, only the determination of the deflection length of the compressed bars is presented.

The most important parameter for the dimensioning of a compressed bar is the slenderness:

$$\frac{}{\lambda}=\sqrt\frac{Af_y}{N_{cr}}$$

where

$$N_{cr}=\frac{\pi^2El}{(kL)^2}$$

where the buckling length factor *k* is recommended by EN1993-1-1 to facilitate manual calculations:

Type of the bar | Direction of buckling | k |
---|---|---|

chord | – in-plane – out-of-plane | 0.9 0.9 |

bracing | – in-plane – out-of-plane | 0.9 1.0 |

Software using model-based computational methods (e.g. Consteel software) determines the elastic critical force *N _{cr} *directly by

**finite element numerical methods,**taking into account the behaviour of the entire lattice girder, instead of the above conservative formula. The following example is intended to illustrate the relationship between the manual design procedure proposed by the standard and the results of the modern model-based numerical procedure.

- Let the structural model of the lattice girder under consideration be the Consteel model shown in Figure 1.
- Let the load shown correspond to the design load combination of the girder.
- Determine the deflection length of the most stressed compressed chord member using finite element numerical stability analysis.

## Relationship between procedures

The steps of the calculation are:

### Buckling stability analysis

The stability analysis of the elastic model shows the governing buckling mode of the lattice structure and the corresponding **elastic critical load factor α_{cr}** (Figure 2).

We can see that the upper chord of the perfectly elastic model is deflected laterally under load. The load that causes this elastic buckling is the **critical load**, whose value is given by the product of the design load and the critical load factor *α _{cr}*=5.99.

## 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.

gateIn Consteel, the calculation of cross sectional interaction resistance for Class 3 and 4 sections is executed with the modified Formula 6.2 of EN 1993-1-1 with the consideration of warping and altering signs of component resistances. Let’s see how…

## Application of EN 1993-1-1 formula 6.2

For calculation of the resistance of a cross section subjected to combination of internal forces and bending moments, EN 1993-1-1 allows the usage -as a conservative approximation- a linear summation of the utilization ratios for each stress resultant, specified in formula 6.2.

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}\leq 1$$

As Consteel uses the 7DOF finite element and so it is capable of calulcating bimoment, an extended form of the formula is used for interaction resistance calculation to consider the additional effect.

$$\frac{N_{Ed}}{N_{Rd}}+\frac{M_{y,Ed}}{M_{y,Rd}}+\frac{M_{z,Ed}}{M_{z,Rd}}+\frac{B_{Ed}}{B_{Rd}}\leq 1$$

Formula 6.2 ignores the fact that not every component results the highest stress at the same critical point of the cross-section.

In order to moderate this conservatism of the formula, Consteel applies a modified method for class 3 and 4 sections. Instead of calculating the maximal ratio for every force component using the minimal section moduli (W), Consteel finds the most critical point of the cross-section first (based on the sum of different normal stress components) and calculates the component ratios using the W values determined for this critical point. **Summation is done with considering the sign of the stresses caused by the components corresponding to the sign of the dominant stress in the critical point.**

(For class 1 and 2 sections, the complex plastic stress distribution cannot be determined by the software. The Formula 6.2 is used with the extension of bimoments to calculate interaction resistance, but no modification with altering signs is applied)

## Example

Let’s see an example for better explanation.

GATE**Did you know that you could use Consteel to determine the optimum number of shear connectors for composite beams?**

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**Did you know that you could use Consteel to** **determine automatically the second order moment effects for slender reinforced concrete columns?**

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**Did you know that you could use Consteel to** **perform local and distortional buckling checks for cold-formed members?**

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**Did you know that you could use Consteel to** **calculate effective cross-section properties for Class 4 sections?**

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