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
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