Lateral-torsional buckling; flange curtailment

Lateral-torsional buckling

When a beam bends about its major axis, the compression flange behaves like a column, but it is restrained by the tension flange and the web. If the compression flange is not held laterally, at a certain load the beam suddenly deflects sideways and twists at the same time. This failure by combined lateral bending and twisting is lateral-torsional buckling (LTB). It reduces the bending strength below the plastic moment of the section.

• The strength is governed by the elastic critical moment $M_{cr}$, which depends on the unbraced length $L_{LT}$, $EI_y$, the torsional rigidity $GI_t$ and the warping rigidity $EI_w$ (IS 800:2007 cl. 8.2.2, Annex E).
• Slenderness $\lambda_{LT} = \sqrt{\beta_bZ_pf_y/M_{cr}}$; then $f_{bd} = \chi_{LT}f_y/\gamma_{m0}$ and $M_d = \beta_bZ_pf_{bd}$.
• It is avoided or reduced by closer lateral bracing, a slab connected to the compression flange, wider flanges (larger $I_y$), or closed sections (box sections, which have a high $I_t$).
• Short, stocky or fully restrained beams ($\lambda_{LT} \le 0.4$) are not affected.

Curtailment of flanges in a plate girder

The bending moment in a simply supported girder is largest at mid-span and zero at the supports. A flange sized for the maximum moment is more than is needed near the ends, so the flange thickness or width is curtailed (reduced, or extra cover plates are stopped short) where the bending moment is smaller. This saves steel.

   actual    theoretical                 theoretical    actual
    end        cut-off                      cut-off       end
     |<-ext.->|                               |<-ext.->|
     +================================================+   <- curtailed outer plate
 +--------------------------------------------------------+  <- full-length flange plate
 |                          web                           |
 ^                                                        ^
support                                                support

• The theoretical cut-off point is where the moment of resistance of the reduced section, $M_{d,red}$, equals the applied moment. For a UDL $w$ on span $L$: $w\,x(L-x)/2 = M_{d,red}$, giving $x = \dfrac{L}{2} - \sqrt{\dfrac{L^2}{4} - \dfrac{2M_{d,red}}{w}}$ from the support.
• The cut plate must be extended beyond the theoretical point (by enough length, with enough bolts or welds, to develop the force in the plate) so that it can take its share of the force, and its end is welded all round to avoid fatigue cracks. The welds or bolts between the flange and the plate over the extension must carry the plate force.
• Curtailment is done in stages to keep the changes in section small, and the web-to-flange connection is checked at each change because the shear flow changes with the section.