Euler Column Buckling
Slender columns fail by buckling long before the material reaches its yield strength. Euler's formula gives the critical load — exceed it and the column collapses sideways.
Euler's Critical Load
P_cr = π²EI / (L_e)²
E = Young's modulus (Pa)
I = minimum second moment of area (m⁴)
L_e = effective length = K × L
Effective length factors K:
Both ends pinned: K = 1.0
One end fixed, one pin: K = 0.7
Both ends fixed: K = 0.5
One end free (flagpole): K = 2.0Worked Example
Steel column: L=4m, both-end pinned, 100×100×6mm SHS
I = 1.67×10⁻⁶ m⁴, E = 200 GPa
L_e = 1.0 × 4 = 4m
P_cr = π² × 200e9 × 1.67e-6 / 4² = 206 kNSlenderness Ratio
λ = L_e / r r = √(I/A) (radius of gyration)
λ > 120: slender — Euler buckling governs
λ < 60: stocky — yielding governs
60-120: intermediate — use interaction formulasCalculate buckling load: Free Column Buckling Calculator
Euler's Buckling Formula
Critical buckling load: P_cr = π²EI / (KL)², where E = Young's modulus, I = minimum second moment of area, L = column length, K = effective length factor. K values: both ends pinned: K = 1.0; one end fixed, one pinned: K = 0.7; both ends fixed: K = 0.5; one end fixed, one free (flagpole): K = 2.0. The effective slenderness ratio = KL/r, where r = √(I/A) is the radius of gyration. Columns fail by yielding if slenderness is low; by buckling (elastic instability) if slenderness is high.
Practical Implications
- Critical slenderness ratio: Below ~100 (steel), yield governs. Above ~100, elastic buckling governs.
- Section choice: Hollow circular or square sections have high I in all directions — efficient for compression. Wide-flange (H/I) sections have different I about each axis; weaker axis may govern.
- Bracing: Lateral bracing of columns reduces effective length KL and dramatically increases buckling capacity. A column braced at midpoint has L effectively halved, quadrupling P_cr.
- Imperfections: Real columns buckle at lower loads than the ideal Euler formula due to initial curvature, load eccentricity, and residual stresses. Design codes apply reduction factors.
Frequently Asked Questions
What is the difference between buckling and crushing?
Crushing (yielding) is a material failure — stress reaches yield strength and section deforms plastically. Buckling is an instability failure — the column deflects laterally and collapses before material yield in most cases (elastic buckling). Euler buckling is sudden and without warning; yielding gives visible deformation first. Slender columns (high KL/r) buckle; stocky columns (low KL/r) crush.
How are real structural columns designed?
Design codes (Eurocode 3, AISC 360) use design curves that transition between yield and buckling, accounting for real column imperfections. A "column curve" (Perry-Robertson or similar) reduces capacity below the theoretical Euler value at intermediate slenderness. The capacity is expressed as a utilisation ratio: applied load / design resistance ≤ 1.0.
Why do triangulated trusses not buckle like plain columns?
In a truss, compression members are braced at each node (panel point) — reducing their effective length to the panel spacing rather than the full truss length. This makes individual truss members far stiffer against buckling than a single column spanning the full height would be, allowing efficient long-span structures with relatively light members.