Designing a bent beam to resist lateral torsional buckling
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 My,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 kz is the coefficient of restraint about the weak axis of the cross-section, G is the shear modulus, and It and Iω are the pure (St. Venant) and warping torsional moments of inertia of the cross-section. The value of the factor C1 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, C1=1.0. The formula for the other design parameters, in particular the buckling reduction factor $\chi_{LT}$, depends on the design standard considered.
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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