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Column Buckling (Euler) Calculator

Q: What is Euler buckling?

Euler buckling is the sudden lateral deflection of a slender column under axial compression. Euler's formula gives the critical load at which buckling occurs for an ideal elastic column.

Q: What is the effective length factor K?

K accounts for end restraint: pinned-pinned K=1.0, fixed-fixed K=0.5, fixed-free K=2.0, fixed-pinned K=0.7. Smaller K means more restraint and higher buckling load.

Q: What is slenderness ratio?

Slenderness ratio λ = KL/r measures column slenderness. High λ means slender column prone to elastic buckling. Low λ means stocky column that yields before buckling.

Q: When does Euler buckling apply?

Euler buckling applies when λ ≥ λc (elastic buckling). For λ < λc, inelastic buckling occurs and the Johnson parabola or other methods should be used.

Q: What is a typical factor of safety for column buckling?

AISC 360 uses FS = 1.67 for ASD. Other codes use FS = 1.67–2.0 depending on slenderness and loading conditions.

Q: How does material affect buckling?

Higher E gives higher buckling load. Steel (E=200 GPa) is much stiffer than aluminum (E=69 GPa) or timber (E=10 GPa). However, steel's higher strength may not help if elastic buckling governs.

Q: What sections are most efficient for columns?

Hollow sections (pipes, HSS) and I-sections are efficient because they have high I relative to area, increasing the radius of gyration and reducing slenderness.

Q: What is the difference between elastic and inelastic buckling?

Elastic buckling occurs at stress below yield (σcr < σy). Inelastic buckling occurs when σcr ≥ σy — the column yields before Euler buckling, and the inelastic modulus governs.

Calculate Euler critical buckling load, critical stress, slenderness ratio, and allowable load for columns. Supports various end conditions and section shapes.

Structural Analysis Structural engineers, civil engineers Commercial Intent: VERY HIGH

Buckling Mode Shapes

Engineering Formulas

Euler's Critical Load

P_cr = π²EI / (KL)²

P_cr: Critical buckling load (N)

E: Young's modulus (Pa)

I: Moment of inertia (m⁴)

K: Effective length factor

L: Column length (m)

Slenderness Ratio

λ = KL / r
r = √(I/A)

λ: Slenderness ratio

r: Radius of gyration (m)

A: Cross-sectional area (m²)

Critical Stress

σ_cr = π²E / (KL/r)²

σ_cr: Critical buckling stress (Pa)

Transition Slenderness

λ_c = √(2π²E/σ_y)

λ_c: Transition slenderness

σ_y: Yield stress (Pa)

Worked Example

Steel Column with Pinned Ends

endCondition: pinned-pinnedL: 3material: steelE: 200yieldStress: 250sectionShape: rectangleb: 0.1d: 0.15D: 0.15bf: 0.15tf: 0.012tw: 0.008d_ibeam: 0.3b_sq: 0.15t_sq: 0.01D_hr: 0.15t_hr: 0.01

Pinned-pinned: K = 1.0

Section

A = 0.1 × 0.15 = 0.015 m², I = 0.1 × 0.15³ / 12 = 2.813 × 10⁻⁵ m⁴

Gyration

r = √(2.813×10⁻⁵ / 0.015) = 0.0433 m

Slenderness

λ = 1.0 × 3 / 0.0433 = 69.3

Buckling

Pcr = π² × 200×10⁹ × 2.813×10⁻⁵ / (3)² = 6,171 kN

Allowable

Pall = 6,171 / 1.67 = 3,695 kN (elastic buckling)

Result: Pcr = 6,171 kN, Pall = 3,695 kN, λ = 69.3 (Elastic buckling mode)

Engineering Notes

For λ < λc, use Johnson's parabolic formula instead of Euler.

AISC 360 specifies φc = 0.90 for LRFD and Ωc = 1.67 for ASD.

Consider weak-axis buckling — the minimum I governs.

For real columns, initial crookedness and residual stresses reduce capacity.

Bracing at intermediate points reduces effective length and increases capacity.

Assumptions

• Perfectly straight column (no initial imperfection)

• Centric axial load (no eccentricity)

• Linear elastic material (Hooke's law applies)

• No lateral bracing along column length

• Small deflections (linearized theory)

Common Mistakes

✕ Using Euler formula for stocky columns where inelastic buckling governs

✕ Forgetting to use consistent units (E in Pa, L in m, I in m⁴)

✕ Confusing effective length KL with actual length L

✕ Not checking critical stress against yield stress

✕ Using wrong K factor for the actual end restraint conditions

Frequently Asked Questions

What is Euler buckling?

Euler buckling is the sudden lateral deflection of a slender column under axial compression. Euler's formula gives the critical load at which buckling occurs for an ideal elastic column.

What is the effective length factor K?

K accounts for end restraint: pinned-pinned K=1.0, fixed-fixed K=0.5, fixed-free K=2.0, fixed-pinned K=0.7. Smaller K means more restraint and higher buckling load.

What is slenderness ratio?

Slenderness ratio λ = KL/r measures column slenderness. High λ means slender column prone to elastic buckling. Low λ means stocky column that yields before buckling.

When does Euler buckling apply?

Euler buckling applies when λ ≥ λc (elastic buckling). For λ < λc, inelastic buckling occurs and the Johnson parabola or other methods should be used.

What is a typical factor of safety for column buckling?

AISC 360 uses FS = 1.67 for ASD. Other codes use FS = 1.67–2.0 depending on slenderness and loading conditions.

How does material affect buckling?

Higher E gives higher buckling load. Steel (E=200 GPa) is much stiffer than aluminum (E=69 GPa) or timber (E=10 GPa). However, steel's higher strength may not help if elastic buckling governs.

What sections are most efficient for columns?

Hollow sections (pipes, HSS) and I-sections are efficient because they have high I relative to area, increasing the radius of gyration and reducing slenderness.

What is the difference between elastic and inelastic buckling?

Elastic buckling occurs at stress below yield (σcr < σy). Inelastic buckling occurs when σcr ≥ σy — the column yields before Euler buckling, and the inelastic modulus governs.