Euler buckling load explained
So far a member failed by yielding or by breaking under stress. A slender column fails in a completely different way. Stand a ruler on end and press it from above. Under a light press it stays straight, and the moment the force crosses a threshold it suddenly bends sideways. The stress has not even come near yield. That is buckling. The material is not weak. The straight shape itself has become unstable, and that is what collapses. In 1744 Euler wrote the critical load as a formula. Pcr = π²EI(KL)². What is surprising is that the strength of the material does not enter this formula. It is fixed only by the stiffness EI, the length L, and how the two ends are held, namely K. This lesson looks at the limit of stability, the reason a long ladder, a slender column, and a thin can suddenly fold. The threshold is that stability itself.
First, simply press a column that stands straight. Move P and, until the load reaches the threshold, the column stays straight and only its length shortens by a very little. This is the axial compression seen in B1. It shortens by δ = PLAE and does not bend sideways. The stress on the cross-section is also uniform, σ = PA, and it has not come near yield. A short, stocky column would stay straight and simply crush when the stress reaches yield, and buckling would not happen. Compression itself is not buckling. Buckling is the event in which a straight column under compression, the moment it passes a certain load, gives up straightness and runs sideways. Where that threshold sits is the next question. While straightness holds, it is only a short compression.
Now push the load past the threshold. Move P past Pcr and see what happens. The column that was straight suddenly bends sideways like a bow. Raise P only a little more and the bend grows quickly. This is a bifurcation. Below Pcr the straight shape is the only stable answer, but once the load passes Pcr the straight shape becomes unstable and a bent shape appears, like a fork in the road. The reason it is sudden is this. If the column bends a little, the load P makes a bending moment through that sideways distance, that moment bends the column further, and further bending makes the moment larger again. The instant this feedback beats the material's restoring force is exactly Pcr. The column collapses because the stability of the shape collapses, not because the stress has reached a limit. The moment the load crosses the threshold, straightness stops being the answer.
That critical load is Euler's formula. For a column pinned at both ends, Pcr = π²EIL². Here EI is the bending stiffness. E is the elastic modulus of the material, I is the second moment of the cross-section, the same I seen in B3. EI tells how hard the column resists bending. The real point is that L² sits in the denominator. Stretch L to twice the length and Pcr drops to a quarter. Length cuts the buckling strength by its square. So the same cross-section is pitifully weak once the column is long. One more thing. Yield strength and ultimate strength, the strength of the material, do not enter this formula at all. Only stiffness and length do. The formula nails down that buckling is a problem of stiffness and shape, not a problem of strength.
Which column dies by yield, and which dies by buckling? The number that splits them is the slenderness λ = KLr. Here r is the radius of gyration, a measure of how efficiently the cross-section is spread out, and r = √(I/A). Written again as a critical stress, σcr = π²Eλ². Move the slenderness and the split shows itself. When the slenderness is large, that is when the column is long and thin, the critical stress is so low that the column buckles before the stress ever reaches yield. When the slenderness is small, that is when the column is short and stocky, the critical stress sits above the yield strength, so the column yields first, before buckling. The transitional slenderness where the two curves meet, λ₁ = π√(E/σy), is that boundary. In design, the first thing to read off a column is its slenderness. Whether the column is long or short decides which limit governs. Slenderness decides which limit is called.
Last, how the two ends are held changes the strength by a lot. The K of Euler's formula carries that. Change the end condition and the difference is plain. Pin and pin gives K=1, and that is the reference. Fix one end completely and leave the other free, like a flagpole, and K=2, so for the same length Pcr falls to 1/4. That is the weakest case. Fix both ends firmly and K=0.5, so Pcr jumps by a factor of four, and fixed and pin sits in between at K≈0.7. The meaning of K is an effective length. The real length L multiplied by K, the product KL, is the length reckoned as a pin and pin column. In Pcr = π²EI(KL)², K enters as a square, so merely holding the ends well changes the buckling strength several fold. Column design has to look at the joints at both ends together with the cross-section.