Most engineers and designers are clear about understanding failure from tension or bending loads. However, buckling is a form of structural instability—and ultimately failure—caused by compressive forces. A normal stress analysis won’t provide any sort of information about buckling as a mode of failure. This article will outline how buckling is calculated, when it should be considered as a failure mode and point to resources that will help you when performing a buckling analysis.

Here are a few types of buckling.

- Column buckling (Will be discussed further below)
- Snap through buckling (Will also be discussed further below)
- Flexural-torsional buckling
- Lateral-torsional buckling

While there are several different types of failure with buckling at its core, overall buckling is a failure characterized by sudden movement of the geometry of the structure in response to a compressive load. If a column is loaded axially downward, then the column’s rapid deformation to the left or right is due to buckling.

The mechanism of column buckling is caused by the fact that loading something perfectly in line with its geometric center is an impossibility. Nothing is ever perfectly straight, nothing is ever perfectly loaded and no atomic structure is ever arranged without flaws. Our world just doesn’t work that perfectly—but that is all fine provided this behavior is understood and checked for safety.

Here are some rules to determine whether buckling needs to be considered:

- Are compressive forces present? If not, forget about it—tensile forces don’t lead to buckling.
- Is the component long, or does it contain slender features? A column that has a small cross-section is an example of this.

For number 1, compressive forces can lead to failure through crushing as well. That isn’t the same as a buckling failure. In buckling, failure typically happens before reaching the yield stress, which makes it the governing failure mode in those cases.

### The Mathematics of Buckling

In the image below is the mathematical derivation, that will hopefully provide some insight into the mechanics of buckling. The mathematics start with this assumption that no matter how well-crafted, a structural member can never be manufactured completely straight.

As a result, a compressive load is never perfectly aligned with a straight axis, meaning that any axial load creates a moment in the structural member that causes it to start bending, which increases the misalignment of the load and the center of the beam leading to an increase in the moment of the beam. It’s basically a feedback loop.

When the mathematics of buckling were first being explored by Leonhard Euler, he didn’t understand how to calculate the stiffness of the beam, but he understood it was a constant, and so he used the variable C, in order to represent the stiffness of the beam’s cross-section. The image below shows the equation he came up with as well as the column to the right that he assumed represented any column.

Euler assumed that the beam had an infinitesimally small deformation at the middle, and that its deformation was the shape of a sine curve to make the math simpler. Below is an image of the breakdown of these mathematics.

Going through the equations above, equation
1 is from general beam theory. It says the moment at any point is related to
the 2^{nd} derivative of the displacement function multiplied by the
stiffness of the beam, C in this case, and acts in the opposite direction of
the displacement.

Equation 2, w(x), represents the displacement function. Euler assumed the shape of a half sine curve with a maximum amplitude of an infinitely small misalignment, ẟ. From there, he calculated the first derivative and the second derivative of the displacement function, equations 3 and 4 above.

The image on the right is the free-body diagram Euler used to plug in values into equation 1, along with the equation 4. By calculating a value for M and plugging in the second derivative of the displacement function, Euler was left with equation 5. The sine functions on both sides of the equation cancel out, as do the ẟ’s, leaving Euler with equation 6, also known as the critical buckling load calculation.

This is the pure compressive load that a member can hold before it buckles. It should be noted that *n* was an integer which was removed because checking values with higher order sine functions would result in a displacement shape with more “waves.” It is clear that *n*=1 creates the lowest critical buckling load and as a result would be the worst-case scenario, so *n *can be removed from the equation entirely.

It should also be noted that decades after Euler’s work was done, it was discovered that the constant *C* was calculated as *E*I*, where *E* is the modulus of elasticity of a material, while *I* is the moment of inertia of the cross-section. Euler didn’t discover this portion because his forte was mathematics, while those who experimented to learn this were scientists.

### Determining If Buckling is a Governing Failure Mode

With the mathematics of buckling explained, it’s important to understand when it governs as a failure mode. As mentioned, buckling only happens under compressive loads because tensile loads would “flatten” out the sine curve. Compressive loads instead exacerbate the issue. If high compressive forces exist a system can either fail via buckling, or it can fail due to crushing—so determining which failure mode governs is critical for engineers.

If P/A > σ then crushing will happen before buckling. In this case:

- P = Compressive Load Applied (the critical buckling load can be substituted here if the actual applied loads are unknown. This would tell you which would govern, if the critical buckling load/cross sectional area is above the yield stress, material failure is going to govern the design.)
- A = Cross-sectional area of the column
- σ = Ultimate compressive stress of the material or yield stress depending on whether it is a brittle material or ductile.

In general, buckling can be prevented by using a larger cross-section or stiffer material. Whatever can be done to increase the stiffness of the cross-section, E*I will help. Additionally, it can be seen in the critical load calculation that the buckling load is inversely proportional to the length of the structural member squared, so if required, reducing the length of the structural member or bracing the member can be used to increase the critical buckling load.

### Tools for Determining Buckling Loads

There are many tools available for calculating the critical buckling load, including spreadsheets, tables and FEA software. Each has its own merits and benefits:

**Spreadsheets are easy; just plug in data and go.** They can be cumbersome to build, or can be purchased for a nominal cost. However, they usually aren’t customizable, so if your project is slightly different, you’re out of luck.

**Tables are cheap and easy to use.** The most common ones provide for determining different effective lengths that modify the calculation of the critical load. One such source is the *Manual of Steel Construction* that many civil engineers use, as well as mechanical engineers that work on parts of larger scale. Discussion of effective lengths will be provided later in this article.

Finally, **software such as SOLIDWORKS Simulation professional** can be used to run a buckling study to calculate the critical load. This can be particularly beneficial when the geometry involved is that of a system acting more as a larger system in a composite action, or if it has irregular cut outs that make the stiffness change along the length of the beam. That is something that Euler’s calculation didn’t account for specifically, as it assumes the moment of inertia is constant along the length of the beam, without the cross-section changing. If the cross-section does change, it requires a numerical approach to solve efficiently and accurately.

Spreadsheets can be created by a user.
Tables can be found from sources like the *Manual of Steel Construction, *and
as mentioned Finite Element Analysis software like *SOLIDWORKS Simulation*
*Professional* can be used.

### Learning How to Use Simulation for Buckling Analysis

For *SOLIDWORKS Simulation Professional *users,
there is a tutorial built right-in. Open SOLIDWORKS, make sure the simulation
tool add-in is turned on, then go to *Help > Tutorials > Simulation
Tutorials > Simulation Professionals* then look for the buckling tutorial
under the Frequency and Buckling tutorials section.

This tutorial will walk you through the basics of any buckling analysis:

**1. Parts definition – Materials and mesh type**. The goal of this step is to tell the software the properties and how to derive the stiffness.

**2. Loads**. This determines what the structure is attempting to withstand.

**3. Fixtures**. This tells the software how the structural system is restrained from moving.

**4. Meshing**. This describes the shape and stiffness of the structure. This step relates to the accuracy of the analysis. If the mesh is too coarse at this step, the model results will over-predict the size of the critical buckling load—a negative trait in this case.

**5. Run**. Takes all the inputs and solves them.

**6. Processing results, also known as post-processing.** Every software is a little bit different, but in SOLIDWORKS Simulation, for example, the result provided is called a buckling load factor. It essentially tells you how much you would have to scale the loads by in order to cause buckling.

When looking at the results, buckling load factors are the factor of safety on the critical load. A buckling load factor of 3 means the applied load would have to be increased by a factor of 3 for buckling to happen. It is also possible to have negative buckling factors of safety. These mean that the system is in tension, so the load direction would have to be reversed and multiplied by that amount in order to see buckling occur.

When the buckling load factor is anywhere between 0 and 1, the design doesn’t work. In those cases, that means the software is predicting a buckling load failure.

### Effective Lengths and Boundary Conditions

Like any analysis, results are dependent on your boundary conditions. The table below shows how important the boundary conditions of your simulation are. The values from these tables don’t need to be input into your simulation; the behavior is already included in the analysis.

The reason I’m including the table here is to illustrate that if you mismanage your boundary conditions, setting up your analysis with a fully fixed condition when in fact the true behavior is rotationally fixed-pinned, it’s possible to be off by a factor of 4! That’s not safe, so be certain your boundary conditions are correct in the simulation, or else use the worst-case scenario shown below.

The table below comes from the *Manual of Steel Construction *and is used to modify the hand calculations for different end mounting conditions, modifying the length in the critical buckling load calculation. The reasoning is that when Euler was doing his work, he was assuming both ends are pinned, so his deformation function representing half of a sine wave was correct.

However, other end conditions change this behavior. Both ends fixed “shortens” the sine function as seen in (a) in the table below, so the effective length of the column becomes half its normal length in the critical buckling load equation.

For other end conditions like (e) and (f) below, they only represent a quarter of a sine curve, as such they double the effective length in the critical buckling load calculation. The table shows you the numbers for other conditions as well.

### Snap-Through Buckling

At this point, we’ve talked about column buckling and linear simulation methods that can be used to determine that behavior. However, there is also a more complex form of buckling called snap-through buckling. The most common example of this would be a Snapple cap. Taking that cap and pressing the “button” on it, it goes from bulging outward towards you to bulging away from you. That system has a “switch” and once it’s pushed to a neutral or flat position, just another small increase in force, and it “snaps through” and becomes resistant to the applied load in tension.

This is a non-linear phenomenon because it involves a significant change in the geometry as it happens. Linear analysis tools can’t predict this behavior; it must be solved iteratively, which is the realm of non-linear solvers.

Luckily, SOLIDWORKS also has a tutorial
built into it for this purpose; the catch is that you need *SOLIDWORKS Simulation
Premium* to have access to the non-linear analysis tools needed to analyze
this. You can find this tutorial in SOLIDWORKS at *Help > Tutorials >
Simulation Tutorials Tab > Simulation Premium > “Snap through/Snap back
Analysis of a cylindrical sheet”.*

This is the model used in the tutorial. It is a thin sheet with a slight curvature to it. The tutorial utilizes symmetry to run a ¼ model, saving computational time. It will also walk you through the setup and running of this type of analysis.

Snap-through behaviors come from geometries that are thin sheets with curvature to begin with. Depending on the material used and the geometry, snap-through behavior may not damage a structure, as can be illustrated with the Snapple cap. It can be pressed repeatedly, however, if its purpose is to support a load and it snaps through, then this could be considered a failure.

I know far fewer rules of thumb for this sort of analysis, so it is generally only calculated using tables or FEA tools such as *SOLIDWORKS Simulation Premium.*

A couple questions to ask after running this sort of analysis:

**Is there a material failure prior to this behavior?**That can depend on your definition of failure. However, it could be that the yield is reached, or that ultimate failure is reached. If the yield stress is reached, the analysis needs to be re-run using a more complex material curve, like a bilinear model or a stress strain curve if linear behavior was assumed. If those were used and failure stress was reached prior to the snap-through behavior, then snap-through isn’t a governing failure mode.**What is the deformation that happens after the snap-through and is it acceptable to operation?**If a component is attached to this, or if this behavior is represented as something like a mechanical switch, will it contact the piece it’s supposed to?

Buckling is one possible failure mode in a system that has compressive loading present. It needs to be understood and checked to make sure that designs are safe. By following the guidelines outlined above you can better analyze your designs and be more confident that they will operate as they are supposed to. While this article can’t possibly hope to cover every possible consideration for buckling, it has introduced some of the basic concepts and hopefully given you an understanding as a foundation.

*Disclaimer: When designing any structural component, make sure that you are qualified to do so and have proper credentials and do proper safety checks. There are numerous factors that this article cannot possibly account for.*

Learn more with the whitepaper Design Through Analysis: Simulation-Driven Design Speeds System Level Design and Transition to Manufacturing.

**About the Author**

Brandon Donnelly is a Technical Account Specialist with GSC. He produces a podcast for engineers called, “Stories for Engineers” that can be found on Spotify, Apple Podcasts, Stitcher and iHeartRadio. He is always available for questions or conversations about SOLIDWORKS, Simulation, 3D Printing and general engineering talk.