Buckling stress explained for beginners: Euler's formula and empirical formula

Now, I think you should understand what buckling is in the explanation from last time.

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Buckling explained for beginners (buckling load, deformation mode) In the previous lesson, we finished explaining beam deflection, a highlight of the middle section of mechanics of materials. Up until now, we've introduced deformations such as tension, twisting, and bending. The deformation we'll begin explaining from this lesson onwards…

This time, we will explain buckling stress based on the buckling load discussed in the previous lesson.

Furthermore, the formula for buckling is not applicable to all column lengths, and in some cases, it may be necessary to use an empirical formula.

Let me also introduce that experimental formula.

I'd also like to introduce a useful coefficient for quickly considering buckling during the design process. This is very important.

Let's begin the explanation.

Buckling loads of various columns

Last time, we explained the buckling of a column with one side fixed and the other side free, which is a simple deformation.

Now, let's look at other patterns of column buckling.

Buckling load at both free ends

As usual, let's start by setting up an example problem.

Consider the case where a load P is applied from both ends of a length l area with a second moment of inertia I and elastic modulus E, and both ends are free, as shown in the following figure.

The basic deformation is as shown in the diagram, and looking at the shape of the deflection, the amount of deflection y is at its maximum in the middle, so the deflection angle θ is 0 in the middle.

If you look closely at the diagram, you'll see that at the halfway point in the length, it becomes exactly the same as the fixed-end, free-end configuration we introduced last time.

Therefore, the buckling load can be obtained by simply replacing the l in the buckling load $Per=\frac{π^2EI}{4l^2}$ that we calculated last time with $ \frac{l}{2}$, which is as follows.

$ Per=\frac{π^2EI}{l^2} $ (both free ends)

This represents the buckling load at both free ends.

Buckling load at both fixed ends

Next, let's consider the case where both ends are fixed. We fix both ends using the same settings as with the case where both ends are free.

If you look closely at the shape of the deflection deformation, you'll see that the amount of deflection y is at its maximum at the halfway point. Looking even more closely, the deflection angle θ becomes zero in the middle and reaches its maximum at l/4.

Looking at the diagram, you can see that there are four columns with one fixed end and one free end, arranged in a ratio of l/4.

Therefore, we can simply replace the l in the buckling load $Per=\frac{π^2EI}{4l^2}$ with $ \frac{l}{4}$, resulting in the following equation.

$ Per = \frac{4π^2EI}{l^2} $ (fixed ends on both sides)

This allows us to determine the buckling load at both fixed ends.

Generalization of buckling load

As many people have probably noticed by now, buckling load can be generalized using a coefficient.

If the coefficient is n, the buckling load can be expressed by the following equation.

$ Per=n\frac{π^2EI}{l^2} $

This is called Euler's formula.

As you can see from the formula, the pillarsThe shorter the length, the greater the buckling load. Therefore, caution is needed when lengthening it.

Depending on the difference in n, the modes of deformation are as follows:

In actual design, the value of n is determined by predicting deformations and conditions.

Also, if you're only slightly modifying an existing structure, you should have previous data, so you can just retrieve the value of n from there.

However, in the case of a completely new structure, the true value cannot be determined without actually deforming it, so I strongly recommend conducting experiments.

buckling stress

Let's consider the buckling stress from the buckling load we have calculated.

In ideal buckling, the column is considered to not deform at all unless the load exceeds the buckling load, so the compressive stress generated in the column just before buckling is expressed as follows.

$ σer=\frac{Per}{A} $(A is the area of ​​the cross-section)

This is called buckling stress.

Substituting Euler's formula into the buckling load Per, we get $ σer = nπ^2E\frac{I}{l^2A} $.

Now let $k^2=I/A$

$ σer=nπ^2E(\frac{k}{l})^2 $

Let's consider the meaning of 'k' here.

By rearranging the expression for k, we can obtain the following expression.

$ k^2A=I $

Does this formula look familiar? In fact, it's quite similar to the definition of the second moment of area, $ I = \int_{A}y^2dA $.

Although it's a bit of a stretch, if we consider y and k to be the same, we can think of it as the distance from the neutral axis or centroid.

Based on these observations, k is called the radius of gyration of the second moment of area, the radius of gyration, or the radius of gyration.

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First and Second Moment of Area Explained for Beginners As mentioned at the end of the previous discussion on beam stress, we will now explain the second moment of area. The second moment of area is a very important concept when designing structures in mechanical engineering. It is very…

Up to this point, we have considered buckling, buckling load, and buckling stress under the condition that the material is within its elastic limit.

However, if the column is short, the buckling load and buckling stress may exceed the elastic limit of the material before the buckling load is reached. In such cases, Euler's formula cannot be applied. Incidentally, if a load exceeding the yield stress is applied without buckling, the loaded surface may collapse.

Therefore, we can determine how many columns Euler's formula can be applied to, given that the yield stress of the material is σs, using the following equation.

$ σs=nπ^2E(\frac{k}{l})^2 $

$ \frac{l}{k}=π\sqrt{\frac{nE}{σs}} $

This should show you that the range in which Euler's formula can be applied is determined by the yield stress, elastic modulus, and deformation mode of the material.

Here, $ \frac{l}{k}$ is called the slenderness ratio of the column. Euler's formula holds when the slenderness ratio is $ \frac{l}{k}>=π\sqrt{\frac{nE}{σs}}$. This means that it will not hold unless l is reasonably long.

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Elasticity and stress-strain diagrams (plasticity, material properties) that even beginners can understand Continuing from last time, we'll explain the very important concepts of elasticity and the basic properties of materials. This explanation is also extremely important in mechanical design, so please try your best to follow along. Even beginners can understand...

In mechanical designIn design, if the cross-section of a column or rib (height L) is close to a circular shape (diameter D), then $ \frac{l}{k}$ is expressed as L/D and simply called L-by-D (a rather forced interpretation).

Also, although it's a bit of a stretch, if the cross-section is not circular, we sometimes use the cross-sectional area as A and L/D as L/A.

Within the scope of Euler's formula,As the L/D ratio increases, the columns become longer, the buckling stress decreases, and the structure becomes weaker.

By comparing this L/D ratio, the strength of structures can be easily compared (although the absolute value is unknown, the comparison is important).

ThereforeBy comparing the L/D ratio of the design target with structures that have been used in the past and have a proven track record, it is easy to compare the strength of the structures.

Please remember this.

empirical formula for buckling stress

For mechanical design, using Euler's formula is usually sufficient (I don't know about architecture or civil engineering). This is because, depending on the size of the structure, the L/D ratio rarely becomes extremely large.

However, in some cases, short columns are used where Euler's formula cannot be applied, so I will introduce an empirical formula that can be used within that range. Incidentally, since it is an empirical formula, there is no need to think about or memorize the meaning of the formula at all. It is enough to know that it exists as a tool and how to use it.

Here, we will consider the buckling stress in cases where Euler's formula cannot be applied, using long-grade (heat-treated) carbon steel S30C and S35C, which are standard materials for iron, as examples.

There are mainly three empirical formulas: Johnson's formula, Tetmaier's formula, and Rankine's formula.

Before going into a detailed explanation, let me first show a graph where the horizontal axis is $λ0=\frac{l}{k}$ (slenderness ratio, L/D) and the vertical axis is buckling stress.

Rankine's formula

Of the three, Rankine's formula is the most important (the green line on the graph).

This is used when a large load is applied to a short column.

As you can see from the graphRankine's formula calculates the smallest buckling stress, so if you use this formula, there shouldn't be any problems regarding strength alone.

The formula itself is simple and is as follows. In Rankine's formula, σex is the buckling stress, σs is the yield stress, and σer is the buckling stress in Euler's formula. λ0 is the slenderness ratio.

$ \frac{1}{σex}=\frac{1}{σs}+\frac{1}{σer} $

$ σex=\frac{σs}{1+σs\frac{λ0^2}{π^2E}} $

However, this often does not match the actual buckling stress, so Rankine proposed the following equation.

$ σex=\frac{σ0}{1+a0(λ0)^2} $

Here, σ0 is the compressive strength (experimental value) and a0 is an experimental constant, which makes for a rather ambiguous equation.

For reference, when the transformation mode n=1 (both ends are free)

Simply put, the harder it is, the smaller the L/D ratio (length-to-width ratio) it can only be used within.

In any case, these are experimental values, so if it's an existing material or structure, any reputable company should have the data, and if it's a new material or structure, the only option is to actually test it (and there are people who dislike new materials and structures because it's troublesome).

As will be explained later in the section on metal materials, the materials departments of companies like automobile manufacturers are responsible for generating these kinds of figures. Having access to this kind of fundamental data is also an important parameter that indicates the strength of a company or organization.

Johnson's formula

Next, I'll introduce Johnson's formula, which I don't recall using very often myself. However, knowing it will expand your knowledge base, so I encourage you to take a look.

Now, let's look at the formula. Here, the upper limit of the buckling strength is defined as the yield point σs.

$ σex = σs + C(λ0)^2 $ (where C is a constant)

Since the maximum buckling stress σex is less than or equal to the yield point σs, if we set the above equation to be tangent to Euler's buckling stress σer,

$ C=-\frac{σs^2}{4π^2E} $

Therefore, σex becomes as follows.

$ σex=σs-\frac{σs^2(λ0)^2}{4π^2E} $

The applicable range is $ \frac{σs}{2} $ < σex < σs.

This seems to be fairly accurate.

Tetmaier's formula

Tetmaier's equation is well-suited for mild steel (such as S30C). In other words, it is well-suited for materials with a gentle elastic region in their stress-strain diagram.

Let me introduce the formula.

$ σex = σ0(1-a0λ0) $ (both σ0 and a0 are experimental values)

It's very simple, and in deformation mode, for n=1, σ0 is about 304 MPa, a0 is about 0.00368, and λ0 is less than 105, making it usable.

It generally works well with soft, flexible materials (like brass).

This concludes the introduction of the experimental formula.

Summary

Now let's summarize buckling.

Becomes

We have now covered the basics of stress, strain, and deformation in structures.

Next time, we'll delve into the topic of failure in earnest, but before that, let me briefly explain stress concentration.

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Stress Concentration Explained for Beginners: Shape Factor, Stress Concentration Factor, and Practical Techniques So far, we've covered most of the basic deformation of objects. Up until now, we've discussed stress and deformation in uniform (same-shaped) cross-sections, but this time we'll look at the stress generation when the cross-sectional shape changes abruptly...

Once the destruction section is complete, the fundamental parts of materials mechanics will be finished, so please keep going a little longer.

To those who found this article helpful in understanding design:

While there is basically no textbook covering this content, and it is my own original work, I will introduce the textbook that I have been using since I was a student.

Basic Mechanics of Materials (Revised Edition)

created by Rinker

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