Column Buckling — Euler’s Formula
Critical Load, Effective Length, End Conditions & Slenderness Ratio — Complete Guide
Last Updated: March 2026
📌 Key Takeaways
- Buckling is a sudden lateral failure of a slender column under compressive load — it fails by bending, not by crushing.
- Euler’s formula: Pcr = π²EI / (Leff)² — the critical load at which buckling begins.
- Effective length Leff = KL depends on end conditions: K = 1 (pinned-pinned), K = 0.5 (fixed-fixed), K = 0.7 (fixed-pinned), K = 2 (fixed-free).
- Slenderness ratio λ = Leff/rmin determines whether a column is short (fails by crushing) or long (fails by buckling).
- Euler’s formula is valid only for long, slender columns where buckling stress < yield stress.
- Buckling always occurs about the axis with the minimum moment of inertia (weakest axis).
1. What is Buckling?
Buckling is a mode of failure that occurs in slender compression members (columns, struts) when the applied compressive load reaches a critical value. Instead of simply crushing under the load (which is what short, stocky members do), a slender column suddenly deflects laterally — it bows sideways and loses its load-carrying capacity.
Buckling is dangerous because it is sudden — there is little visible warning before failure. A column can carry its design load with negligible deformation, and then catastrophically buckle with only a tiny load increase. This is fundamentally different from material failure (yielding/fracture), which shows gradual warning signs.
Buckling is a stability problem, not a strength problem. The column material may not be anywhere near its yield stress when buckling occurs. The failure is geometric — it depends on the column’s length, cross-section shape, and end conditions, not just on material strength.
2. Euler’s Formula
Euler’s Critical Buckling Load
Pcr = π²EI / (Leff)²
Where:
- Pcr = critical buckling load (N) — the maximum load before buckling
- E = Young’s modulus (Pa)
- I = minimum moment of inertia of the cross-section (m⁴)
- Leff = effective length = KL (m)
Euler’s Critical Stress
σcr = Pcr/A = π²E / (Leff/r)² = π²E / λ²
Where: r = √(I/A) = radius of gyration, λ = Leff/r = slenderness ratio
Key observations:
- Pcr ∝ EI — stiffer material and larger moment of inertia resist buckling better.
- Pcr ∝ 1/L² — doubling the length reduces the buckling load by a factor of 4.
- Buckling occurs about the axis with minimum I (the weakest axis).
- Euler’s formula does not depend on material strength — only on stiffness (E) and geometry.
3. End Conditions & Effective Length
The effective length Leff = KL accounts for how the column ends are supported. Different end fixities change the buckled shape and the critical load:
| End Condition | K (Effective Length Factor) | Leff | Pcr (relative) |
|---|---|---|---|
| Both ends pinned (hinged-hinged) | 1.0 | L | 1× (baseline) |
| Both ends fixed | 0.5 | 0.5L | 4× (strongest) |
| One fixed, one pinned | 0.7 (≈ 1/√2) | 0.7L | 2× (approximately) |
| One fixed, one free (flagpole) | 2.0 | 2L | 0.25× (weakest) |
Fixed-fixed is 4 times stronger than pinned-pinned, and 16 times stronger than fixed-free — the end conditions have an enormous effect on buckling capacity. This is why column connections in buildings are designed to provide as much fixity as possible.
4. Slenderness Ratio
Slenderness Ratio
λ = Leff / rmin
Where: rmin = √(Imin/A) = minimum radius of gyration
Dimensionless. Higher λ = more slender = more prone to buckling.
The slenderness ratio is the single most important parameter for column classification:
| Radius of Gyration | Cross-Section | Formula |
|---|---|---|
| Circle (D) | r = D/4 | I = πD⁴/64, A = πD²/4 |
| Rectangle (b × d, d > b) | rmin = b/√12 | About weaker axis |
| Hollow circle (Do, Di) | r = √(Do² + Di²)/4 | — |
5. Short vs Long Columns
| Feature | Short Column (low λ) | Long Column (high λ) |
|---|---|---|
| Failure mode | Crushing (material yields) | Buckling (lateral deflection) |
| Failure stress | σy (yield strength) | σcr = π²E/λ² (< σy) |
| Depends on | Material strength | Material stiffness (E) and geometry |
| Euler’s formula valid? | No — overestimates capacity | Yes |
The transition slenderness ratio (where Euler buckling stress equals yield stress) is:
λtransition = π√(E/σy)
For mild steel (E = 200 GPa, σy = 250 MPa): λtransition = π√(200,000/250) = π × 28.3 ≈ 89
Columns with λ > 89 → Euler applies. Columns with λ < 89 → use Rankine or empirical formulas.
6. Rankine’s Formula — Intermediate Columns
Euler’s formula overestimates the capacity of short and intermediate columns. Rankine’s formula provides a smooth transition between crushing failure and buckling:
Rankine’s Formula
1/PR = 1/Pc + 1/PE
Where: Pc = σyA (crushing load), PE = π²EI/(Leff)² (Euler load)
This can be written as:
PR = σyA / [1 + a(Leff/r)²]
Where a = σy/(π²E) = Rankine constant. For mild steel: a ≈ 1/7500.
Rankine’s formula works for all column lengths — for very short columns it gives PR ≈ Pc, for very long columns it gives PR ≈ PE, and for intermediate columns it gives a value between the two.
7. Worked Numerical Examples
Example 1: Euler Buckling Load — Pinned-Pinned
Problem: A steel column (E = 200 GPa) has a circular cross-section of diameter 50 mm and length 2 m. Both ends are pinned. Find the Euler buckling load.
Solution
I = πD⁴/64 = π × 50⁴/64 = π × 6,250,000/64 = 306,796 mm⁴
Leff = 1.0 × 2000 = 2000 mm
Pcr = π²EI/(Leff)² = π² × 200,000 × 306,796 / 2000²
= 1,974 × 306,796 / 4,000,000 = 605,658,504 / 4,000,000
Pcr = 151,415 N ≈ 151.4 kN
Example 2: Effect of End Conditions
Problem: If the same column from Example 1 has both ends fixed instead of pinned, find the new buckling load.
Solution
Leff = 0.5 × 2000 = 1000 mm
Pcr = π² × 200,000 × 306,796 / 1000² = 605,658,504 / 1,000,000
Pcr = 605,659 N ≈ 605.7 kN
Fixing both ends increases the buckling load by 4 times (from 151.4 to 605.7 kN). ✓
Example 3: Check if Euler Applies
Problem: A mild steel column (E = 200 GPa, σy = 250 MPa) has Leff = 1.5 m and cross-section 40 mm × 40 mm. Does Euler’s formula apply?
Solution
rmin = b/√12 = 40/√12 = 40/3.464 = 11.55 mm
λ = Leff/r = 1500/11.55 = 129.9
λtransition = π√(E/σy) = π√(200,000/250) = π × 28.28 = 88.9
Since λ = 129.9 > 88.9 → Yes, Euler’s formula applies — the column is long enough to buckle before yielding.
σcr = π²E/λ² = π² × 200,000/129.9² = 1,974,000/16,874 = 117 MPa (< σy = 250 MPa) ✓
8. Common Mistakes Students Make
- Using the wrong effective length factor: K = 1 is only for pinned-pinned. Fixed-fixed is K = 0.5. Getting K wrong changes the answer by a factor of 4 or more.
- Using maximum I instead of minimum I: Buckling occurs about the weakest axis. For a rectangular section (100 × 50 mm), use I about the 50 mm direction (the minimum), not the 100 mm direction.
- Applying Euler to short columns: Check if σcr < σy before using Euler. If σcr > σy, the column is short and fails by yielding, not buckling. Euler overestimates the capacity of short columns.
- Confusing radius of gyration with actual radius: r = √(I/A) is NOT the physical radius. For a circle, r = D/4 (not D/2). For a rectangle, r = dimension/√12.
- Forgetting the π² in Euler’s formula: Pcr = π²EI/L² — the π² is essential. Omitting it gives an answer off by a factor of about 10.
9. Frequently Asked Questions
What is Euler’s buckling formula?
Pcr = π²EI/(Leff)² gives the critical compressive load at which a slender column buckles laterally. It depends on material stiffness (E), cross-section geometry (I), and effective length (Leff = KL). The formula is valid only for long, slender columns where the buckling stress is below the yield stress.
What is the slenderness ratio?
The slenderness ratio λ = Leff/rmin is a dimensionless measure of how prone a column is to buckling. Higher λ means more slender and more likely to buckle. Columns with λ above the transition value (≈89 for mild steel) fail by Euler buckling; those below fail by material yielding/crushing.