Push on the ends of a thin ruler, and you’re almost certain to bow it out to the side, similar to the (1) in the figure below. You haven’t crushed or yielded the material that makes up the ruler, but it has still exceeded its safe capacity to take more load. This sort of instability failure is known as ‘buckling‘, and in columns, the name for the most famous mathematical model for it is Euler Column Buckling.
Video – What is (Euler) Column Buckling?
If you prefer videos, check out my video on column buckling:
What is Buckling?
Buckling is a form of premature failure brought on by instability in either the loading or the structure.
Loads are very rarely applied perfectly concentrically with a structural element, so all real compressive loads impart at least some small amount of bending into the member. Real structural members don’t come perfectly straight either, and the small amounts of bow or warp will act to turn even a perfectly-concentric axial load into a load/moment pair which will put the column into bending.
Even if you could somehow produce a perfectly-straight column, and load it perfectly concentrically, there would still be instability introduced by variability in the materials the column is constructed from.
For a column that’s short and stout enough, the actual material strength will govern before buckling starts to become a concern, but the longer and more slender a column becomes, the more likely buckling will govern.
Column buckling can be mitigated by providing lateral bracing to shorten the unbraced length, restraining more degrees of freedom on the column ends to cut down on the effective length factor, or choosing a section with a higher radius of gyration.
History
According to Stephen Timoshenko’s “History of Strength of Materials”, Leonard Euler first derived his column buckling equation circa 1757, and marketed it with a pretty high level of certainty. In fact, Euler went so far as to claim:
“…unless the load P to be borne be greater than Cπ²/4𝓁², there will be absolutely no fear of bending; on the other hand, if the weight P be greater, the column will be unable to resist bending.”
Leonard Euler, as translated by stephen timoshenko in “history of strength of materials”
Since Euler’s day, many refinements and adjustments have been made, but the basic form of the “Euler Critical Load” remains in many building codes around the world. For certain very slender members, there are now other equations that are a better fit than Euler’s, but they’re sort of special-case spin-offs.
Modern Buckling Equation
The form appearing in the “AISC Steel Construction Manual, 15th Edition” as Eq E3-4, is:
Where:
- E is the Young’s Modulus of the material in question
- K is the effective length factor (see next paragraph)
- L is the laterally unbraced length of the member
- r is the radius of gyration (as described at-length in my article “What is the Radius of Gyration“)
Effective Length Factors
Euler’s original equation assumes no rotational restraint at either end of the column, and needs that adjustment factor “K” to help extend it to other end conditions.
The theoretical K values and “Recommended design values” vary in the Steel Construction Manual, below is my adaptation of their recommendations for our figure.
| Case Below | 1 | 2 | 3 | 4 | 5 |
| Theoretical K Value | 1.0 | 0.5 | 1.0 | 0.7 | 2.0 |
| Recommended design K value | 1.0 | 0.65 | 1.2 | 0.8 | 2.1 |
Case 1 is the default case, with pins at both the top and bottom. Case 2 features rotation restraints at both ends of the column. Case 3 has rotational restraints, but also allows the top of the column to “sway”, this case is similar to a moment frame. Case 4 is rotationally-restrained at the bottom, but free to rotate at the top. Case 5 is rotationally restrained at the bottom, and free to both displace laterally and rotate at the top.
Buckling Safety Factors
Buckling can be a sudden, catastrophic failure, often occurring with little or no warning. As a rule, that’s our least-favorite kind of failure, so it shouldn’t be a big surprise that we load up on safety factors for buckling.
For steel, we end up with stacking safety factors. The ASD safety factor at the beginning of Chapter E – Design of Members for Compression is only 1.67, but we don’t get to use our calculated F.e directly. The F.cr calculated from our F.e actually takes only 87.7% of our theoretical Euler Buckling Stress, which brings our true effective Safety Factor up to 1.90, fairly high for a structural member in Steel Design.
Depending on the end conditions, the Effective Length Factor, K, has an additional built-in safety factor, based on uncertainty in those conditions.
Other Kinds of Buckling
Euler Column Buckling is just one of the most-accessible sorts of buckling, and we’ll have more articles soon on the others.
Euler Buckling is a type of “global buckling”, where the full length of the member is involved, and is joined in that category by Lateral-Torsional Beam Buckling, one of the major design constraints on steel beams.
Local Buckling, the other major category, is when a portion of the section buckles, dropping the capacity of the overall member. A good example of this is web buckling on a channel shape, where most of the strength & stiffness of the member comes from having two flanges spaced far apart. If the web crinkles and lets the flanges come closer together, the whole member loses capacity and quickly hinges at that location.
Watch for more articles and some videos soon to come on these other topics, and as always, please shoot me an email if you have any questions or topics you’d like to see covered!
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