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video. When a bar is loaded in uniaxial tension,
it will fail when the normal stress in the bar exceeds the yield or tensile strength
of the material. And if it's loaded in compression it will
fail by crushing when the compressive strength of the material is exceeded. But there's an additional way the bar can
fail when in compression, which is by buckling. Buckling is a loss of stability that occurs
when the applied compressive load reaches a certain critical value, causing a change
in the shape of the bar. An initially straight member will buckle suddenly,
producing large displacements. This doesn't always result in yielding or
fracture of the material, but buckling is still considered to be a failure mode since
the buckled structure can no longer support a load in the way it was designed to. The most simple example of a structure at
risk of buckling is a column. But individual members in trusses and frames
can also be loaded in compression, and so are at risk of buckling. There are other less obvious examples too. When railway tracks heat up on a hot day,
the steel the tracks are made of tends to expand. But expansion in the axial direction is prevented,
and so a compressive axial force builds up, which can lead to buckling. A very similar issue can occur in subsea pipelines
that carry hot product. The compression that builds up due to the
thermal expansion of the pipe steel can cause the pipeline to buckle on the seabed. Buckling can clearly lead to catastrophic
failure, so how do we take it into account in engineering design and analysis? To answer this question we need to travel
back to the mid 18th Century. In 1744, the mathematician Leonhard Euler
published a book in which he laid out a new method for analysing functions, called the
calculus of variations. To illustrate how this new method could be
applied, Euler included in an appendix the derivation of an equation for the axial load
that will cause a column to buckle. This is the Euler buckling formula, a simple
equation that engineers continue to use almost 300 years later to design columns and other
members that are loaded in pure compression. It's probably the oldest engineering design
equation that's still in regular use. The critical load at which a column will start
buckling depends on only three parameters, the Young's modulus of the column material,
the area moment of inertia of its cross-section, and its length. It doesn't depend on the strength of the material
at all. This form of the equation is valid for a column
that is pinned at both ends, meaning that the ends can rotate but can't translate horizontally. So a 2 meter tall steel column that has a
circular cross section with a radius of 40mm would be expected to be able to
support a load of around 1000 kN before buckling, not including any safety
factors. This assumes an idealised perfectly straight
column. At the critical buckling load, any small perturbation,
whether it's a lateral force or a small imperfection, will cause the column to bend. If the cross-section has a smaller area moment
of inertia about a particular axis, the column will buckle in that direction, and the smaller
value of I must be used to calculate the critical buckling load. If the end conditions change so that the column
is fixed at one end and free at the other, it will clearly only be able to support a
much smaller load before buckling, and the buckled shape is different. We can easily modify Euler's formula to account
for different end conditions by introducing the concept of an effective length. The effective length can be defined as the
distance between inflection points on the deflected shape. The column that's pinned at both ends has
an effective length equal to the column length. But for a column that's free at the top and
fully fixed at the bottom the distance between inflection points is twice the column length. Here are a few other common end conditions
and the associated effective lengths. We can replace the column length in Euler's
formula with the effective length to make the equation applicable for all of these end
conditions. Or we can keep the column length and add in
an effective length factor K. End conditions clearly make a huge difference
to the critical buckling load, and must be considered very carefully. In real life applications it isn't always
clear which effective lengths should be used, and the amount of restraint at the ends will
depend on the stiffness of the adjacent members. Design codes often provide guidance on conservative
assumptions that can be used for these scenarios. Even without knowing anything about Euler's
formula it's intuitively quite obvious that slender columns are at much greater risk of
buckling than stocky ones. It's why you would never design a truss structure
with long compressive members like this. This member is under compression, and since
it's so long and thin it's at risk of buckling. Members of a truss that are in compression
are sometimes designed to be thicker than those in tension to reduce the risk of buckling,
and long compressive members are prevented from buckling by the use of bracing members. Euler's formula confirms this intuition about
slender columns. The length term is squared, and so doubling
the length of a column means that it can only support a quarter of the weight before buckling. To better understand the effect of slenderness,
it's useful to introduce a non-dimensional parameter called the slenderness ratio. First, let's divide the critical load by the
cross-sectional area to obtain a critical stress. Then, if we define the radius of gyration
R of the column as the square root of I divided by A, we can write the equation for critical
stress in a new form. The term L over r is the slenderness ratio. Let's take a look at how the Euler critical
buckling stress varies with the slenderness ratio. Very slender columns have a large slenderness
ratio and a very low critical buckling stress. For stocky columns with low slenderness ratios
the critical buckling stress will be very large. If we draw the compressive yield strength
of the column material on this graph, we can see that for these slenderness ratios the
strength of the material will be exceeded before the buckling limit is reached. This means we can define two distinct regions,
where beams fail by crushing because the stress in the column exceeds the material yield strength,
and where they fail due to buckling. The limiting slenderness ratio depends on
the material Young's modulus and yield strength. For steel columns the limiting slenderness
ratio is around 90. But this curve only represents the theoretical
behaviour of columns. If we plot buckling stresses determined experimentally
for real columns we can see it doesn't exactly match the theoretical behaviour. In particular the transition between plastic
failure and elastic buckling failure is much more gradual. This is because for columns in this transition
range, buckling is actually a complex combination of these two failure modes. This is called inelastic buckling, and the
theoretical behaviour can be modelled using methods like Engesser's theory or Shanley's
theory. There's a much better correlation between
the test data and Euler's formula for very slender columns. But even for these columns there are some
limitations to Euler's formula that the engineer needs to be aware of. One of these limitations is that the formula
assumes that the applied load acts exactly through the centroid of the column cross-section. But the applied load will always be slightly
offset from the centroid, even if it's by only a very small amount. This eccentricity introduces a moment that
acts in addition to the axial load, which reduces the critical buckling stress and significantly
changes how buckling occurs. If the load is applied at the centroid of
the cross-section the force-displacement curve looks like this. There is no displacement until the critical
buckling load is reached, at which point the displacement suddenly becomes very large. But in the case of an eccentric load the additional
moment causes the column to bend as soon as the load is applied. Because of the bending, the stress in the
column isn't uniform. The maximum compressive stress occurs on the
inner surface halfway up the column, and can be calculated using the Secant formula. Another limitation of Euler's formula is that
it assumes that the column is perfectly straight before the load is applied. But real columns contain imperfections and
however small they may be, these can reduce the critical buckling load. Since imperfections introduce bending, they
have a similar effect to eccentric loading and so can be modelled in the same way. Euler's formula and the Secant formula also
assume that displacements are small. If displacements are large the moment acting
on the column will change significantly throughout the deformation, introducing significant geometric
non-linearity. In structural analysis this is called the
P-Delta effect. Most design codes deal with these limitations
of Euler's formula and the complexities of inelastic buckling by providing design curves
that engineers can use directly to design columns. These curves are calibrated using experimental
data and are applied in combination with suitable safety factors to ensure a safe design. Euler's formula is used to calculate the critical
buckling load for a column where the displacement occurs by bending. This is called flexural buckling. But members can buckle in other ways too,
and these also need to be checked during design. Columns with thin-walled open cross-sections
tend to have low torsional stiffness. Under certain conditions these columns can
buckle by twisting, called torsional buckling, or by a combination of twisting and bending,
called torsional-flexural buckling. So far we've considered buckling of columns
and other straight members, like those you might find in a truss or a frame. But thin plates and shells like those you
would find in a storage tank are also susceptible to buckling. Buckling in these types of structures is even
more sensitive to the presence of imperfections than it is in columns, and the effects are
more difficult to predict. And so, although analytical equations do exist
to calculate critical buckling loads for plates and shells, they're usually considered to
provide an upper limit on the buckling load. Detailed non-linear analysis using the finite
element method is often required for this type of structure. We've seen that slender columns can buckle
when compressive loads act on them. But what about the effect of gravity? Is it possible that a column could be built
tall enough that it would buckle because of nothing other than its own weight? This is a problem that even Euler struggled
with, but since it didn't really fit into this introduction to buckling I've covered
it in a short companion video that you can watch right now over on Nebula. Nebula is a streaming platform that I've been building
with a group of independent educational creators. It's a place where you can watch our videos
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channel. And that's it for this introduction to buckling. Thanks for watching!
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