# Designing a bent beam to resist lateral torsional buckling

## 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:

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