Understanding Buckling

Summary

This video explores the concept of buckling in engineering, explaining how structures like columns and trusses can fail due to loss of stability under compressive loads. It introduces Euler's buckling formula, a fundamental equation used to calculate the critical load that causes buckling, and discusses its limitations and real-world applications. The video also touches on the impact of imperfections and the importance of considering slenderness ratios and end conditions in column design.

Takeaways

• 🔴 The script discusses the concept of buckling in engineering, explaining it as a failure mode where a structure loses stability under compressive load, causing a change in shape without necessarily yielding or fracturing the material.
• 📚 It mentions the historical context of Euler's buckling formula, introduced by Leonhard Euler in 1744, which is still widely used in engineering to design columns and other members under pure compression.
• 📏 The critical load for buckling depends on three parameters: Young's modulus of the material, the area moment of inertia of the cross-section, and the length of the column, and is independent of the material's strength.
• 🔍 The script highlights the importance of considering end conditions in column design, as they significantly affect the critical buckling load, and introduces the concept of effective length to account for different end conditions.
• 📉 The slenderness ratio, defined as the ratio of the column's length to the radius of gyration, is a key factor in determining the likelihood of buckling, with slender columns being more susceptible.
• 📊 The critical buckling stress varies with the slenderness ratio, indicating that very slender columns have a lower critical buckling stress, while stockier columns can withstand higher stresses before buckling.
• 🚫 The script points out limitations of Euler's formula, such as the assumption of a perfectly straight column and the exact alignment of the applied load with the centroid of the column's cross-section.
• 🔧 It discusses the impact of imperfections and eccentric loading on the critical buckling load, which can reduce the load a column can support before buckling due to the introduction of additional bending moments.
• 🛠️ The importance of considering different types of buckling, such as flexural, torsional, and torsional-flexural buckling, is emphasized, especially for members with thin-walled open cross-sections.
• 🛡️ The script touches on the use of design codes and safety factors in engineering practice to account for the complexities and limitations of theoretical formulas like Euler's.
• 📚 It concludes with a mention of the need for detailed non-linear analysis, such as finite element method, for structures like plates and shells, which are more sensitive to imperfections and harder to predict in terms of buckling behavior.

Q & A

What is the primary cause of failure in a bar under uniaxial tension?

A bar under uniaxial tension fails when the normal stress in the bar exceeds the yield or tensile strength of the material.

How does a bar fail when loaded in compression?

A bar loaded in compression can fail by crushing when the compressive strength of the material is exceeded, or by buckling when the applied compressive load reaches a critical value causing a change in the shape of the bar.

What is buckling in the context of structural failure?

Buckling is a loss of stability that occurs when the applied compressive load reaches a certain critical value, causing an initially straight member to bend suddenly and produce large displacements without necessarily resulting in yielding or fracture of the material.

Why are columns a simple example of structures at risk of buckling?

Columns are at risk of buckling because they are typically long and slender, making them susceptible to a loss of stability under compressive loads, which can lead to failure even without material yielding or fracture.

What is the significance of the Euler buckling formula in engineering?

The Euler buckling formula is significant because it provides a simple equation that engineers use to design columns and other members loaded in pure compression, and it is one of the oldest engineering design equations still in regular use.

What parameters determine the critical load at which a column will start buckling according to Euler's formula?

The critical load at which a column will start buckling depends on the Young's modulus of the column material, the area moment of inertia of its cross-section, and its length.

How does the end condition of a column affect its critical buckling load?

The end conditions of a column significantly affect its critical buckling load. Different end conditions, such as pinned, fixed, or free ends, alter the effective length and the deflected shape of the column, thus changing the load it can support before buckling.

What is the slenderness ratio and how does it relate to the critical buckling stress of a column?

The slenderness ratio is a non-dimensional parameter defined as the ratio of the column's length to the radius of gyration (L/R). It is used to understand the effect of slenderness on the critical buckling stress; very slender columns with a large slenderness ratio have a very low critical buckling stress, while stocky columns with a low slenderness ratio have a high critical buckling stress.

Why are slender columns more at risk of buckling than stocky ones?

Slender columns are more at risk of buckling than stocky ones because the length term in Euler's formula is squared, meaning that increasing the length of a column significantly reduces the load it can support before buckling occurs.

What is the impact of imperfections on the critical buckling load of a column?

Imperfections in a column, no matter how small, can reduce the critical buckling load. They introduce bending, which has a similar effect to eccentric loading, thus lowering the load at which the column will buckle.

How do design codes address the limitations of Euler's formula in practical engineering applications?

Design codes address the limitations of Euler's formula by providing design curves that engineers can use directly to design columns. These curves are calibrated using experimental data and are applied with suitable safety factors to ensure a safe design.

What are the different types of buckling that can occur in members with thin-walled open cross-sections?

Members with thin-walled open cross-sections can buckle by twisting, known as torsional buckling, or by a combination of twisting and bending, known as torsional-flexural buckling.

How does the presence of imperfections affect the buckling of plates and shells compared to columns?

The presence of imperfections has a more significant impact on the buckling of plates and shells than on columns. Imperfections make the effects more difficult to predict, and analytical equations for calculating critical buckling loads are often considered to provide an upper limit, with detailed non-linear analysis using finite element methods often required.

Outlines

00:00

📚 Introduction to Buckling in Structural Engineering

This paragraph introduces the concept of buckling as a failure mode in structural elements under compression, distinct from material yielding or fracture. It explains that buckling is a sudden loss of stability at a critical load, causing a change in the shape of the structure without necessarily leading to material failure. The paragraph provides examples of buckling in everyday scenarios, such as railway tracks and subsea pipelines, and emphasizes the importance of considering buckling in engineering design. The historical context is given by mentioning Leonhard Euler's contribution through the Euler buckling formula, which is still widely used today. The formula's dependence on material properties, cross-sectional geometry, and length is highlighted, along with the exclusion of material strength from the critical load calculation.

05:02

🔍 Deep Dive into Euler's Buckling Formula and its Implications

The second paragraph delves deeper into Euler's buckling formula, discussing the importance of end conditions on the critical buckling load and introducing the concept of effective length and the effective length factor K. It explains how different end conditions, such as pinned, fixed, or free ends, affect the load a column can support before buckling. The paragraph also touches on the intuitive understanding that slender columns are more prone to buckling and how design codes provide conservative assumptions for real-world applications. The concept of the slenderness ratio is introduced, along with its impact on the critical buckling stress, and the theoretical and practical differences between elastic and inelastic buckling are explored. Limitations of Euler's formula, such as the assumption of a perfectly straight column and the effects of load eccentricity, are also discussed.

10:06

🏗️ Practical Considerations and Advanced Topics in Buckling

The final paragraph addresses practical considerations in applying Euler's formula to real-world structures, including the effects of imperfections, the P-Delta effect, and the use of design codes and curves to ensure safe design practices. It acknowledges the limitations of the Euler formula in cases of large displacements and introduces additional buckling modes such as torsional and torsional-flexural buckling. The discussion extends to the buckling of plates and shells, which are more sensitive to imperfections and require detailed non-linear analysis. The paragraph concludes with a hypothetical scenario of a column buckling under its own weight due to gravity and mentions a companion video on Nebula that explores this topic further. It also includes a promotional note about the Nebula and CuriosityStream bundle deal, encouraging viewers to support the channel and explore a wide range of educational content.

Keywords

💡Uniaxial Tension

Uniaxial tension refers to the condition where a material is subjected to a tensile force along a single axis. In the video, it is mentioned as a loading condition that can cause a bar to fail when the normal stress exceeds the material's yield or tensile strength. This concept is crucial for understanding material failure under tension.

💡Compressive Strength

Compressive strength is the capacity of a material to withstand compression before it fractures or is crushed. The script discusses how a bar loaded in compression can fail when the compressive strength of the material is exceeded, highlighting a key factor in structural integrity under compressive loads.

💡Buckling

Buckling is defined as a sudden change in the shape of a structural element under compressive load, leading to a loss of stability. The video emphasizes buckling as a critical failure mode, especially in columns and other compression-loaded members, where it can occur without material yielding or fracture.

💡Euler Buckling Formula

The Euler buckling formula is a fundamental equation derived by Leonhard Euler that calculates the critical axial load at which a column will buckle. The video explains its significance in engineering design, as it is used to determine the stability of columns under compression, with parameters such as Young's modulus, area moment of inertia, and length.

💡Effective Length

Effective length is a concept introduced in the script to modify Euler's formula for different end conditions of a column. It represents the distance between inflection points on the deflected shape and is crucial for calculating the critical buckling load in various column configurations.

💡Slenderness Ratio

The slenderness ratio is a non-dimensional parameter that describes the relative slenderness of a column, defined as the ratio of the column's length to its radius of gyration. The video uses this concept to explain the relationship between a column's slenderness and its susceptibility to buckling.

💡Inelastic Buckling

Inelastic buckling is a complex failure mode where buckling occurs in combination with material yielding. The script contrasts this with elastic buckling, explaining that inelastic buckling does not align perfectly with theoretical models like Euler's and requires more complex analysis.

💡Secant Formula

The Secant formula is used in the video to calculate the maximum compressive stress in a column when the load is eccentrically applied. This formula accounts for the additional bending moment caused by the load offset, which is a limitation of the Euler buckling formula.

💡P-Delta Effect

The P-Delta effect refers to the significant change in the moment acting on a column as deformation progresses, especially in cases of large displacements. The video mentions this as a limitation of Euler's formula, which assumes small displacements and thus does not account for this geometric non-linearity.

💡Torsional-Flexural Buckling

Torsional-flexural buckling is a type of buckling that involves both twisting and bending, typically affecting columns with thin-walled open cross-sections. The script discusses this as one of the various buckling modes that engineers need to consider during the design process.

💡Finite Element Method

The finite element method is a numerical technique mentioned in the script for detailed non-linear analysis of structures, such as plates and shells, which are susceptible to buckling. This method is particularly useful for complex structures where analytical equations may not provide accurate predictions.

Highlights

CuriosityStream sponsors the video discussing structural failure modes.

Bars fail in uniaxial tension when normal stress exceeds material's yield or tensile strength.

Bars loaded in compression can fail by crushing when compressive strength is exceeded.

Buckling is an additional failure mode in compression, characterized by a loss of stability and shape change.

Buckling can occur without material yielding or fracture, but still signifies failure.

Columns are a simple example of structures at risk of buckling.

Truss members and subsea pipelines are also susceptible to buckling due to thermal expansion.

Buckling is a critical consideration in engineering design and analysis.

Leonhard Euler's calculus of variations and the Euler buckling formula are foundational in column design.

The critical buckling load depends on Young's modulus, area moment of inertia, and column length.

Euler's formula is applicable for columns pinned at both ends.

Effective length and end conditions significantly affect the critical buckling load.

Slender columns are at a higher risk of buckling compared to stocky ones.

The slenderness ratio is a non-dimensional parameter that indicates the risk of buckling.

Euler's formula has limitations, such as assuming perfect alignment of the applied load.

Imperfections in real columns can reduce the critical buckling load.

Design codes provide conservative assumptions for uncertain end conditions.

Inelastic buckling is a complex combination of plastic failure and elastic buckling.

Eccentric loading and imperfections introduce moments that affect buckling.

The P-Delta effect accounts for geometric non-linearity in structural analysis.

Design codes offer design curves calibrated with experimental data for safe column design.

Columns with thin-walled open cross-sections are prone to torsional and torsional-flexural buckling.

Plates and shells are susceptible to buckling, requiring detailed non-linear analysis.

Euler's struggle with self-buckling due to gravity is covered in a companion video on Nebula.

CuriosityStream and Nebula offer a bundle deal for ad-free educational content.

Transcripts

Thanks to CuriosityStream for sponsoring this 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.

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And that's it for this introduction to buckling.

Thanks for watching!