When beams get loaded, they deflect in the direction of loading. This is very
intuitive and logical. However, as the load increases, the beam begins to also deflect
laterally. This lateral deflection is not as intuitive and requires a deeper investigation.
This phenomenon of sideways deflection and twisting is known as Lateral Torsional Buckling
and can be extremely destructive to beams. Lateral Torsional Buckling was the cause for the Station
Square Collapse in 1988 but we will save that investigation for another time. For now,
let’s break down why this type of buckling occurs and why is it so “fatal” for beams.
For the geeky part of the audience that enjoys going deeper into technicalities,
near the end of the video we will dive right to the bottom of the physics of this phenomenon to
explain the root cause of lateral-torsional buckling and the governing equation.
For now, let’s qualitatively show how lateral-torsional buckling occurs. When a simple
beam is loaded, the top flange gets compressed while the bottom flange gets stretched. As we
displayed in this previous video, when members are compressed, they tend to deflect laterally. For a
beam, the situation is slightly more complicated since half of the beam is in compression and half
is in tension. For that reason, the top half of the beam wants to buckle due to compression while
the bottom half wants to straighten out due to the pulling/tensile forces. This tendency of the
flanges to move in opposite directions twists the beam, causing it to deflect laterally.
Beams that are relatively strong at resisting torsion compared to their bending resistance do
not suffer from torsional buckling because their bending strength usually gets exhausted before
lateral-torsional buckling can be initiated. Such examples are closed sections like pipes
or square tubes. Furthermore, these sections have the same stiffness in-plane and out-of-plane and
therefore “show no tendency to buckle laterally” [2] (CISC steel handbook, 2-39, paragraph 4).
In contrast, wide flange sections, as we have seen before, are poor at resisting torsional
loads. To make things worse, W-sections are also generally weaker in bending about their
weak axis (y axis). This weak axis makes them even more susceptible to lateral-torsional buckling.
Taking a closer look at the deflected shape of a buckled beam, we can identify 3 types of
deformations of the section. First, we have the vertical deflection of the beam that is
intuitively due to the load application. Secondly, we have the lateral deflection,
and lastly, we have the twisting of the section. Each one of these deformations
induces stresses in the material. This is where the destructiveness of torsional
buckling lies. The load that was initially applied in the direction where the beam is strongest gets
propagated to other axes where the beam has very low stiffness. Since the displacement is inversely
proportional to the stiffness [3], it takes a relatively small load to cause large deformations.
This is exactly why lateral-torsional buckling occurs at relatively small
loads and is followed by large deformations. Based on that it follows that lateral-torsional
buckling is most pronounced for sections that are significantly weaker in torsion and
bending about the y-axis. Such examples are deep W-sections or very slender rectangular beams.
On the other hand, for stockier beams, vertical deflection also plays an important role. Stocky
beams need special attention because they would most likely experience premature yielding of the
material and would fail in combined stability and material failure mode [4]. The yielding,
in this case, occurs due to the fact that when the beam is laterally buckled
it undergoes cross-bending, meaning that it’s simultaneously bent about the X and Y-axis.
This bi-axial bending causes significant damage to the flange tips that considerably weakens
the section and accelerates buckling. To illustrate the strength reduction due to
lateral-torsional buckling we will investigate a wide flange beam and compare its capacity.
We will compare the same wide flange beam for 3 different scenarios:
Simple beam with a point load at mid-span applied at the centroid of the section
Simple beam with a point load at mid-span applied at the centroid of the top flange, and lastly,
The same simple beam with a point load at mid-span but also with regular lateral bracings.
The buckling analysis was performed in Abaqus for an 8 meter long beam. The following failure
loads were obtained for each scenario. The clear winner from the comparison is
the braced beam which could not deflect laterally and therefore achieved its full
bending capacity.
In essence achieving material failure rather than becoming prematurely unstable.
An experiment was also conducted comparing the tendency for lateral-torsional buckling
between open and closed sections. To ensure the same amount of material, both sections were
constructed out of two-channel sections. We know that this introduces other issues like shear flow
but with adequate fastener spacing, these effects were successfully minimized.
When the box section was loaded, nearly all of the deformations were in the direction of
loading as expected. As the load increased, yielding and local buckling of the top
flange could be noticed due to the channels acting separately between fastener locations,
but still, no lateral deflections were present. Finally, the beam collapsed at
around 72 kg or 160 lb in what seemed to be a combination of yielding and local buckling.
In contrast, the I section, displayed a very different behavior and failure mode. At around
36 kg (80 lb) a slight lateral deflection started to appear. As the load was increased
to 40 kg and later 42 kg the lateral deflection was apparent. The beam failed soon after
when the load was increased to 45 kg or around 100 lb. From the footage, it is clear that this
was a different type of failure and at a much lower load. Even though both beams used the same
amount of material the box beam achieved a 60% higher capacity compared to the I-section.
The good thing about lateral-torsional buckling is that it’s relatively easy to deal with.
The most common solution is to laterally brace the compression flange of the beam.
This way, the beam is held in place, and it can reach its full bending capacity without
stability issues. In some situations, for single beams or warehouse cranes,
lateral bracing is not easily achievable, in that case, engineers are required to carefully select
the type of section used and ensure the beam will not get unstable before reaching its capacity.
But what is the actual cause of this instability? The root cause of lateral-torsional buckling is
the imperfect manufacturing and construction process. This is nothing new and it is true for
any material. Even though not visible, when beams get installed, they are already slightly bent.
When a load is applied to them, this imperfection gets amplified. At this stage, the beam is in
a position where it is experiencing bending moments or torque around all three of its axes.
As mentioned before, if the beam is particularly strong in resisting torsion and bending about the
y-axis or it is laterally braced, then the process of lateral buckling will halt here. However, if
none of these conditions are true, then the beam will continue to deflect laterally. More lateral
deflection will induce even more bending which will quickly degenerate into an unstable beam
or in essence a failure. Here we can also see why having the load applied at the top of the beam
makes things worse. The additional horizontal eccentricity between the centroid of the beam
and the load application further increases the bending about the z-axis or torsional load and
accelerates the buckling process. Since lateral-torsional buckling,
among other things, is load-specific, design codes around the world have had trouble fully
capturing its effects with one equation. Many codes usually rely on the equation derived for
a beam subjected to end moments. This equation is comprised of two main terms that contribute
to the buckling resistance of the beam that is the torsional and warping stiffness of the beam.
To extend its utility to multiple cases, design codes add a modification factor that accounts for
different bending moment distributions. But still, this equation has many limitations.
Besides the torsional capacity, bending resistance about the weak axis, and the point of load
application there are also other factors that influence lateral-torsional buckling. For example,
less symmetric sections have stress amplification factors that cause localized stress concentrations
and accelerate buckling. If the end connections do not fully restrain the twisting of the beam,
the critical load is further lowered. A varying bending moment diagram, the slenderness of the
beam and the load type could also impact the buckling susceptibility. As engineers,
we constantly evaluate many possible failure scenarios and account for different factors so
that we can design safe structures. This type of thinking takes a while to develop on its own.
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