Buckling
Buckling resistance of uniform
members in bending
Richard Henderson of the Steel Construction Institute discusses the phenomenon of
lateral-torsional buckling.
Introduction
A grid of beams is usually divided into primary and secondary beams
and where there is no floor slab to provide continuous support to the
compression flanges, the secondary beams provide discrete restraints to
the primary. An end plate connection to the primary beam web detailed in
accordance with the Green Book rules may be considered to provide a fork
end restraint. The secondary beams also apply point loads to the primary
and, for this type of connection, the loads are not destabilizing. The system
of point loads results in a shear force diagram for the primary beam with
constant values between the point loads and a bending moment diagram
made up of straight lines (ignoring the effect of the primary beam selfweight).
In determining the resistance of the beam to bending, especially in
hand calculations, it is common to consider the primary beam in segments
defined by the incoming secondary beams where the segments have
defined end restraints and end moments taken from the bending moment
diagram of the full beam. This approach corresponds to the conditions
set out in clause 6.3.3 of Eurocode 3 which deals with uniform members
in bending and axial compression and the effect of these two actions in
combination. Note 1 to clause 6.3.3(2) states: “The interaction formulae are
based on the modelling of simply supported single span members with
end fork conditions and with or without continuous lateral restraints, which
are subjected to compression force, end moments and/or transverse loads”.
Taking the segments one by one is usually on the safe side as the study
described in the following sections shows. The purpose of the study is to
determine what effect continuity of the beam beyond the segment being
considered has on the beam’s calculated bending resistance.
Beams studied
A series of loading arrangements on a 610 × 229 UB 140 was examined. All
the arrangements were chosen to result in a 3 m segment of beam subject
to a uniform moment of 1200 kNm. The point loads were always applied
at restraint positions and beams of length 9 m and 15 m were considered.
The loads and restraint positions were chosen such that the lengths of the
segments were not always the same so that the half-wave lengths of the
buckled shape were uneven. The arrangements are set out in Table 1.
Beam Length
(m)
No of
point
loads /
restraints
Segment length (m)
1 2 3 4 5 6 7
1 9 2 3 3 3
2 9 2 3.5 3 2.5
3 15 4 3 3 3 3 3
4 15 4 3.5 2.5 3 3.5 2.5
5 15 6 2 2 2 3 2 2 2
6 15 4 3.5 2.5 3 2.5 3
As an illustration, the bending moment diagrams for beams 2 and 6
(neglecting the beam self weight) are shown in Figure 1.
16 NSC
Technical Digest 2018
3.5 3.0 2.5
3.5 2.5 3.0 2.5 3.5
Beams 1 and 3 have equally spaced loads and restraints, forming
segments 3 m long. The buckled shape of the beam calculated by LTBeamN
in determining Mcr is shown in plan in Figure 2. The top compression flange
buckles into a series of half-waves. In each case, the central segment has a
uniform bending moment and the adjacent segments have either triangular
or trapezoidal-shaped bending moment diagrams. The amplitude of
the half-waves can be seen to reduce where the bending moment is not
uniform.
Where the bending moment is uniform over the whole beam, the half-waves
of the buckled shape can be seen to have the same amplitude as shown in
Figure 3.
Beam Resistances
The resistances of beam segments and beams identified in Table 1 have
been calculated for comparison. The segments examined all have a
maximum bending moment of 1200 kNm with a bending moment diagram
Table 1: Arrangement of beams and beam segments
Figure 1: Bending moment diagrams, beams 2 and 6
1200 kNm
1200 kNm
Figure 2: Buckled shape: 3-segment and 5-segment beams
Figure 3: Buckled shape: 5-segment beam, uniform moment