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Figure 1: Elastic buckling is a geometric instability. Long columns fail by sudden bowing, not by material yielding.
Entering the Euler Domain
In Column Design (Part 3), we established the "Decision Rule." If your actual Slenderness Ratio (KL/r) is greater than the Column Constant (Cc), your column is classified as Long.
For these slender members, failure occurs via Elastic Instability. We calculate the Critical Load (Pcr) using the famous formula derived by Swiss mathematician Leonhard Euler in the 18th century.
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The Euler Formula
The critical buckling load is defined as:
Pcr =
Ï€2 E A
(KL / r)2
We can also express this in terms of the Moment of Inertia (I) by substituting r2 = I/A. This is often the more convenient form for design:
Pcr =
Ï€2 E I
(KL)2
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Engineering Insight: Stiffness vs. Strength
Look closely at the variables in the formula above. The buckling load depends on:
- Length (L): Longer columns buckle easier.
- Cross-section (I or A): Thicker columns resist buckling.
- Material Stiffness (E): The Modulus of Elasticity.
Notice what is missing? The Yield Strength (Sy) is not in the equation.
Critical Design Note
For long columns, there is no benefit to using a high-strength alloy steel over a standard low-carbon steel. Both have roughly the same Modulus of Elasticity (E ≈ 207 GPa).
If you need to increase the buckling load of a long column, you must increase the Moment of Inertia (I) or change the End Fixity (K), not the material grade.
What about Short Columns?
If your column was classified as "Short" (where KL/r < Cc), the Euler formula is invalid because the material will yield before it buckles. For this, we need a different equation.
Continue to Part 5:
Column Design (Part 5): The J.B. Johnson Formula
References
- Robert L. Mott, Machine Elements in Mechanical Design
- Wikipedia: Buckling
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